Navier-Stokes Equations: Theory and Numerical Analysis

NAVIER-STOKES EQUATIONS THEORY AND NUMERICAL ANALYSIS

ROGER TEMAM Unioersite de Paris-Sud, Orsay, France Ecole Polytechnique, Palaiseau, France

N'H

(l\>l9I5=

--

1977 NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM' NEW YORK' OXFORD

© North-Holland Publishing Company 1977 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN: 07204 2840 8

Published by: North-Holland Publishing Company Amsterdam' New York· Oxford Sole distributors for the U.S.A. and Canada: Elsevier North-Holland, Inc. 52 Vanderbilt Avenue New York, NY 10017

Library of Congress Cataloging in Publication Data Temam, Roger. Navier-Stokes equations. Bibliography: 15 pp. 1. Navier-Stokes equations. I. Title. QA372.T38 515'.352 76-51536 ISBN 0-7204-2840-8

Printed in England

FOREWORD In the present work we derive a number of results concerned with the theory and numerical analysis of the Navier-Stokes equations for viscous incompressible fluids. We shall deal with the following problems: on the one hand, a description of the known results on the existence, the uniqueness and in a few cases the regularity of solutions in the linear and non-linear cases, the steady and time-dependent cases; on the other hand, the approximation of these problems by discretisation: finite difference and finite element methods for the space variables, finite differences and fractional steps for the time variable. The questions of stability and convergence of the numerical procedures are treated as fully as possible. We shall not restrict ourselves to these theoretical aspects: in particular, in the Appendix we give details of how to program one of the methods. All the methods we study have in fact been applied, but it has not been possible to present details of the effective implementation of all the methods. The theoretical results that we present (existence, uniqueness,...) are only very basic results and none of them is new; however we have tried as far as possible to give a simple and self-contained treatment. Energy and compactness methods lie at the very heart of the two types of problems we have gone into, and they form the natural link between them. Let us give a more detailed description of the contents of this work: we consider first the linearized stationary case (Chapter 1), then the non-linear stationary case (Chapter 2), and finally the full non-linear time-dependent case (Chapter 3). At each stage we introduce new mathematical tools, useful both in themselves and in readiness for subsequent steps. In Chapter I, after a brief presentation of results on existence and uniqueness, we describe the approximation of the Stokes problem by various finite-difference and finite-element methods. This gives us an opportunity to introduce various methods of approximation of the divergence-free vector functions which are also vital for the numerical aspects of the problems studied in Chapters 2 and 3. In Chapter 2 we introduce results on compactness in both the continuous and the discrete cases. We then extend the results obtained for the linear case in the preceding chapter to the non-linear case. The chapter ends with a proof of the non-uniqueness of solutions of the stationary Navier-Stokes equations, obtained by bifurcation and topological methods. The presentation is essentially self-contained. v

vi

Foreword

Chapter 3 deals with the full non-linear time-dependent case. We first present a few results typical of the present state of the mathematical theory of the Navier-Stokes equations (existence and uniqueness theorems). We then present a brief introduction to the numerical aspects of the problem, combining the discretization of the space variables discussed in Chapter I with the usual methods of discretization for the time variable. The stability and convergence problems are treated by energy methods. We also consider the fractional step method and the method of artificial compressibility. This brief description of the contents will suffice to show that this book is in no sense a systematic study of the subject. Many aspects of the Navier-Stokes equations are not touched on here. Several interesting approaches to the existence and uniqueness problems, such as semi-groups, singular integral operators and Riemannian manifolds methods, are omitted. As for the numerical aspects of the problem, we have not considered the particle approach nor the related methods developed by the Los Alamos Laboratory. We have, moreover, restricted ourselves severely to the Navier-Stokes equations; a whole range of problems which can be treated by the same methods are not covered here. Nor are the difficult problems of turbulence and high Reynolds number flows. The material covered by this book was taught at the University of Maryland in the first semester of 1972-3 as part of a special year on the Navier-Stokes equations and non-linear partial differential equations. The corresponding lecture notes published by the University of Maryland constitute the first version of this book. I am extremely grateful to my colleagues in the Department of Mathematics and in the Institute of Fluid Dynamics and Applied Mathematics at the University of Maryland for the interest they showed in the elaboration of the notes. Direct contributions to the preparation of the manuscript were made by Arlett Williamson, and by Professors J. Osborne, J. Sather and P. Wolfe. I should like to thank them for correcting some of my mistakes in English and for their interesting comments and suggestions, all of which helped to improve the manuscript. Useful points were also made by Mrs Pelissier and by Messrs Fortin and Thomasset. Finally, I should like to express my thanks to the secretaries of the Mathematics Departments at Maryland and Orsay for all their assistance in the preparation of the manuscript.

CHAPTER I THE STEADY-STATE STOKES EQUATIONS Introduction In this chapter we study the stationary Stokes equations; that is, the stationary linearized form of the Navier-Stokes equations. The study of the Stokes equations is useful in itselfj it also gives us an opportunity to introduce several tools necessary for a treatment of the full Navier-Stokes equations. In Section 1 we consider some function spaces (spaces of divergence› free vector functions with L 2 -co m po ne nt s ) . In Section 2 we give the variational formulation of the Stokes equations and prove existence and uniqueness of solutions by the projection theorem. In Sections 3 and 4 we recall a few definitions and results on the approximation of a normed space and of a variational linear equations (Section 3). We then propose several types of approximation of a certain fundamental space V of diver› gence-free vector functions; this includes an approximation by tae finite› difference method (Section 3), and by conforming and non-conforming finite-element methods (Section 4). In Section 5 we discuss certain approximation algorithms for the Stokes equations and the corresponding discretized equations. The purpose of these algorithms is to overcome the difficulty caused by the condition div u = 0. As it will be shown, this difficulty, sometimes, is not merely solved by discretization. Finally in Section 6 we study the linearized equations of slightly com› pressible fluids and their asymptotic convergence to the linear equations of incompressible fluids (i.e., Stokes’ equations).

§ 1. Some function spaces In this section we introduce and study certain fundamental function spaces. The results are important for what follows, but the methods used in this section will not reappear so that the reader can skim through the proofs and retain only the general notation described in Section 1.1 and the results summarized in Remark 1.6.

1.1. Notation In Euclidean space tRn we write el = (1, 0, ... , 0), e2 = (0, 1, 0, ... , 0), ... , en = (0, ... , 0, 1), the canonical basis, and x = (xl’ ... , Xn),

en. 1, § 1

The steady-state Stokes equations

2

y = (Yl’ ... , Yn ) , Z = (Zl’ ... , Zn)’ ... , will denote points of the space.

The differential operator

will be written Dj and if i = (i!, ... differentiation operator DJ =Dh 1 ’ " Din n =

in) is a multi-index,

J)i will be the

aU) .

axIl ... ax~n .

(1.1)

where (1.2)

Ifi; = 0 for some t, IY;i is the identity operator; in particular if [j] = 0,

o! is the identity.

The set n Let n be an open set of tR n with boundary r. In general we shall need some kind of smoothness property for n. Sometimes we shall assume that n is smooth in the following sense: The boundary r is a (n-l j-dimensional manifold of class lfr (r ~ I which must be specified) and n is locally located on one side of r.

(1.3)

We will say that a set n satisfying (1.3) is of class’s". However this hypothesis is too strong for practical situations (such as a flow in a square) and all the main results will be proved under a weaker condition: The boundary of n is locally Lipschitz.

(1.4)

This means that in a neighbourhood of any point x E r, r admits a representation as a hypersurface Yn = O(Yl’ ... , Yn-l) where 0 is a Lipschitz function, and (Yl’ ... , Y n ) are rectangular coordinates in tRn in a basis that may be different from the canonical basis elo ... , en’ Of course if n is of class lfl, then n is locally Lipschitz.

Ch. I, § 1

Some function spaces

3

It is useful for the sequel of this section to note that a set .Q satisfying

(1.4) is "locally star-shaped". This means that each point Xj E r, has an .Q n is star-shaped with respect open neighbourhood O’ such that to one of its points. According to (1.4) we may, moreover, suppose that the boundary of is Lipschitz. If r is bounded, it can be covered by a finite family of such sets 0, ; E J; if r is not bounded, the family (OJ)j E J can be chosen to be locally finite. It will be assumed that .Q will always satisfy (1.4), unless we explicitly state that .Qis any open set in din or that some other smoothness property is required.

0’=

0

0

IY and Sobolev Spaces Let Iz be any open set in din. We denote by LP(.Q), I

(L

lu(x)IPdx ) l/p

(1.5)

or lIull£"o(.o)= ess. sup lu(x) I). .0

For p

= 2, L 2 (.Q) is a Hilbert space (u. v)

=

L

u(x)v(x)dx.

with the scalar product (1.6)

The Sobolev space Wm,P(.Q) is the space of functions in LP(.Q) with

Ch. I, § 1

The steady-state Stokes equations

4

derivatives of order less than or equal to m in LP (0) (m an integer, I ~ p ~ + 00). This is a Banach space with the norm (1.7)

When p = 2, Wm, 2(0) «u, V»Hm

(n

=H"’ (0) is a Hilbert space

)=

If)

with the scalar product

2:<. m (Diu, Div)

(1.8)

Let ~(o) (or ~(o» be the space of
IF (0)

= {LP(12nn,

Hm (0)

= {H Tn (o)}n,

Wm,p (0) ~

= {Wm,p(12nn

(12) = {.9))(12)}n,

and we suppose that these product spaces are equipped with the usual product norm or an equivalent norrn (except .9))(0) and .9))(0), which are not normed spaces). The following spaces will appear very frequently

The scalar product and the norm are denoted by (’,.) and 1’1 on L 2 (12 ) orL 2(O)(or [.,.] and [.] on Hb(12) orHb(12». . We recall that if 12 is bounded in some direction (1) then the Poincare inequality holds: (1.9)

n lies within a slab whose boundary is two hyperplanes which are orthogonal to this direction. The minimal distance of such pair of hyperplanes is called the thickness of n in the corresponding direction.

(1) i.e.,

Ch I, § 1

Some function spaces

5

where D is the derivative in that direction and c(n) is a constant de› pending only on .0 which is bounded by 21, 1 = the diameter of .0 or the thickness of 0 in any direction. In this case the norm [.] on HA(O) (or HA(O» is equivalent to the norm:

lIuli =

(

~ IDIUI’)1/2

(I.lO)

The space HA (0) (or HA (0» is also a Hilbert space with the associated scalar product 11

«u, v»

= i=l L

(1.11)

(Diu, Div).

This scalar product and this norm are denoted by HA (.0) and HA(0) (0 bounded in some direction). Let fbe the space (without topology) "Y= {u E ~(O):

div u =

« .,. » and

OJ.

11•11 on

(1.12)

The closures of f in L2 (0) and in HA (.0) are two basic spaces in the study of the Navier-Stokes equations; we denote them by Hand V. The results of this section will allow us to give a characterization of Hand V

1.2. A density theorem Let £(0) be the following auxiliary space:

This is a Hilbert space when equipped with the scalar product «u, v»E(n)

= (u,

v)

+ (div

u, div v).

( 1.13)

It is clear that (1.13) is a scalar product on £(0); it is easy to see that

£(0) is complete for the associated norm

(1)

lIuIIE(n) = {«u, U»E(n) F/2 (l) For if urn is a Cauchy sequence in E(n), then "m is also a Cauchy sequence in

L’(n); u m converges to some limit u in L’(n) and div "m converges to some Iimitg in L’(n), necessarily g = div u, and so U E E(n) and u m converges to U in E(n).

The steady-state Stokes equations

6

Ch. I, § 1

Our goal is to prove a trace theorem: for u E E(n) one can define the value on r of the normal component u ’v, v = the unit vector normal to the boundary. The method we use is the classical Lions-Magenes [1] one. We begin by proving Theorem 1.1. Let n be a Lipschitz open set in /RI!, Then the set of vector functions belonging to~(n) is dense in E(n). Proof. Let u be some element of E(n). We have to prove that u is a limit in E(n) of vector functions of~(n). (i) We first approximate u by functions of E(n) with compact support in n. Let

0 let

* is the convolution

operator

if*~~)=fnf~-YM~)~ ~

lffEL1(~n),

gE LP(R n), 1 <. P

< oo, thenf.g makes

sense and belongs to LP(~n).

Some function spaces

Ch. I, § I

7

(iii) For the general case, ,Q =1= tRn , we use the remark after (1.14):

,Q ~locally star-shaped. The sets fz, (mi)iEJ’ form an open covering of,Q. Let us consider a partition of unity subordinated to this covering.

(1.15)

We may write U

= qm + iEJ 2:

¢•u, J

the sum };,i is actually finite since the support of u is compact. Since the function ¢u has compact support in,Q it can be shown as in (ii) that ¢u is the limit in E (,Q) of functions belonging to!!} (,Q) (the function ¢u extended by 0 outside fz belongs to E(tRn ) and for e sufficiently small, p *(¢u) has compact support in ,Q). Let us consider now one of the functions ui = ¢iu not identically equal to zero. The set m = mi n ,Qis star-shaped with respect to one of its points; after a translation in tRn we can suppose this point is O. Let ax, A =1= 0, be the linear (homothetic) transformation x -+ AX. It is clear, since is Lipschitz, and star-shaped with respect to 0, that:

i

m;

for A> 1,

Let 010.. V denote the function x 1-+ v(ox (x)); because of Lemma 1.1 of the function ax ui’ A > 1, converges to below, the restriction to IIj in E(mj) (or E(n)) as A \-+ 1. But if l/Jj E@ (a~(m;n and l/Jj = Ion mj the function l/Jiax II) clearly belongs to E(dP). Hence we must only approximate in place of the function ui’ a function vi E E(,Q) which is the restriction to ,Q of a function wi E E(tRn ) with compact support (take wi = l/J/a"A u)). The result follows then from point (ii). 0 It remains only to prove Lemma 1.1 giving some results that we used in the above proof and other results which will be needed later.

m;

0

Lemma 1.1. Let

mbe an open

set which is star-shaped with respect to O.

Ch. I, § 1

The steady-state Stokes equations

8

(i) If p E ~’ «(!)) is a distribu tion in (!), then a dis tribu tion ax p can be defined in~’ (ax (!)) by 0

(ax p, r/>) 0

1

= An

(p, aI/X r/>), V r/> E 0

~(ox

(!))

(A> 0).

(1.16)

The derivatives of aX p are related to the derivatives of p by the formula Dj(oXop) = AO X (Djp),

l:s;;;,i:s;;;,n.

0

0.17)

If A > 1, A ~ 1, the restriction to (!) of aX p converges in the distribu› tion sense to p. 0

(ii) IfpELOI.«(!)), 1 :s;;;,a<+oo, then 0xopELOI.(ox(!))’ ForA> 1, A ~ 1, the restriction to (!) ofo xo p converges to pin LOI.«(!)).

Proof. It is clear that the mapping r/> f-+ O/A") tp, aI/X r/>) is linear and con› tinuous on ~(ox (!)) and hence defines a distribution which is denoted by xo p. The formula 0.17) is easy, 0

(Dj(ox

0

p) = -

-

-

0

1 (p, An

-

(Djr/»)

0l/X o

1

- - (p D.(Ul/"'o¢)

= =

«ax p), Djr/>}

AI1 -

1

+ AI1 -

=A

l

l

’I

"

(Djp,

l/X r/>)

(ax -Djp, r/>)

When A> 1, A ~ 1, the functions aI/X or/> have compact support in for A-I small enough, and converge to r/> in ~«(!)) as A ~ 1. (ii) It is clear that

J

I(ox p) (y)I Ct

dy

o~

= An

J ~

o

lIox pIILOI.(ox~)

= IIpIlLOI.(~)•

Ip(x)ICt dx,

(!)

§1

Some function spaces

9

It is then sufficient to prove that a7\ p restricted to (9 converges to p, for the p’s belonging to some dense subspace of LOt. «(9). But f!) «(9) is dense in LOt.( (9), and the result is obvious if P E@( (9). 0

1.3. A trace theorem We suppose here that n is an open bounded set of class~2. It is known that there exists a linear continuous operator ’Yo E £ (HI (n), L2 (f» (the trace operator), such that ’You = the restriction of u to I’ for every function u E HI (n) which is twice continuously differentiable in n. The space (n) is equal to the kernel of ’Yo’ The image space ’Yo (HI (n) is a dense subspace of L 2 ( r) denotedHl/2(r); the space Hl/2(r) can be equipped with the norm carried from HI (I’) by ’Yo’ There exists moreover a linear continuous operator Qn E £(H 1 / 2 ( r) , HI (n» (which is called a lifting operator), such that ’Yo•~n = the identity operator in HI /2(I’). All these results are given in Lions [I] , Lions & Magenes [I]. We want to prove a similar result for the vector functions in E(n). Let H-1/2 (r) be the dual space of H 1/2 (f); since H 1/2 (I’) C L 2 (f) with a stronger topology, L2 (f) is contained in H-l/2 (I’) with a stronger topology. We have the following trace theorem (which means that we can define uovl r when u E E):

H5

Theorem 1.2. Let n be an open bounded set of class (C 2 Then there exists a linear continuous operator ’Yv E £(E(n), H- 1/2(r» such that ’Yv u

= the restriction

of u-v to f. for every u E~(n).

(1.18)

The following generalized Stokes formula is true for all u E E(n) and

w EH1 (n)

(u, grad w) + (div u, w) = <’Yv u, ’Yo w)

(1.19)

Proof. Let ifJ E H1/2(f) and let w E HI (n) with ’Yo w = ifJ. For u E E(n), let us set X.

(~)

i

[div u(x) ",(x)

+ u(x) grad

",(x) I dx

n

= (div u,

w)

+ (u, grad w).

Lemma 1.2. Xu (ifJ) is independent of the choice of w, as long as w EH1(n) and ’Yo w = ifJ.

10

The steady-state Stokes equations

Proof. Let

wI

’Yo wI

and

Ch. I, § 1

belong to HI (n), with

w2

= ’Yo w2 = ¢

and let W = wI - w2’ We must prove that (div u, WI) + (u, grad wI) = (div u, W2) + (u, grad w2), that is to say (div u, w) + (u, grad w) = O.

(1.20)

But since W E HI (n) and ’Yo W = 0, W belongs to H6 (n) and is the limit in HI (n) of smooth functions with compact support: W = lim w m , wm E ~(n). It is obvious that (div U, w m ) + (u, grad w m ) = 0, V wm E

and (1.20) follows as m -+ Let us take now equality

W

00.

~(n)

0

= Qn ¢ (see

above). Then by the Schwarz in›

IXu(¢)1 ~ lIuIIE(n ) IIwlI H l (n)’

and since Qn E £(H1 / 2 (r) , HI (n» IXu(¢)1 ~

coliuIlE(n)II¢lI u lI l

(r)’

(1.21)

where Co = the norm of the linear operator Qn . Therefore the mapping ¢ t-+ Xu (¢) is a linear continuous mapping from H1/ 2 (I’) into (fl. Thus there exists g = g (u) E H- 1/2 (I’) such that X u (¢ )

= (g, ¢>.

(1.22)

It is clear that the mapping u

IlKllu -

1I1

(r) ~

f+

g (u) is linear and, by (1.21),

Co lIuII E(.o.) ;

(1.23)

this proves that the mapping ul-+ g(u) = ’You is continuous from E(n) into H- 1/ 2 ( r) . The last point is to prove (1.18) since (1.19) follows directly from the definition of ’YlJu.

Lemma 1.3. Ifu E ~(n),

then

’YlJu = the restriction of u-u on I’,

Ch. I, § 1

Some function spaces

Proof. For such a smooth u and for any w twice continuously differentiable in .Q),

x;.how) -

11 E~(.Q)

(or if u and ware

L

div (u w)dx

= IrW(UOV)df = Ir(UOV)(-roW)df

(by the Stokes formula)

= tu-v, ’Yo w). Since for these functions w, the traces ’Yow form a dense subset of

HI/2 (f), the formula

is also true by continuity for every E HI/2 (I’), By comparison with (l.22), we get ’Y"u = U ovl r . Remark 1.1. Theorem 1.1 is not explicitly used in the proof of Theorem 1.2, but the density theorem combined with Lemma 1.3 shows that the operator ’Y" is unique since its value on a dense subset is known. Remark 1.2. The operator ’Y" actually maps E(.Q) onto H-I/2 (f). Let be given in H- I /2 (I’), such that (, 1) = O. Then the Neumann problem .1p

ap

-

av

= 0 in.Q =onf

has a weak solution p = p(
(1.24)

The steady-state Stokes equations

12

Ch. I, §1

additive constant (See lions & Magenes [1]). For one of these solutions p let U

= grad p

It is clear that U E E(O) and ’YIIU = (j). In additionit is clear that there exists a vector function Uo with components in ~1 (0) such that ’Ylluo = 1. Then for any 1/1 in H1 /2 (r), writing

1/1 = (j) + <1/1, 1>, mes r

(j)=I/I_,

mes

(1.25)

r

one can define a U = U(1/1) such that ’YIIU = 1/1 by setting U

<1/1,1>

= grad p(l/I) + mes

r uo•

(1.26)

Moreover the mapping 1/11-+ U(1/1) is a linear continuous mapping from H-1 /2(r) into E(O) (Le., a lifting operator as.e n)• 0 Let Eo (0) be the closure of

~(O).

We have

Theorem 1.3. The kernel 01’Y1I is equal to Eo (0). Proof. If U E Eo (0), then by the definition of this space, there exists a sequence of functions um E ~(n) which converges to U in E(O) as m -+- 00. Theorem 1.2 implies that ’Yllum = and hence ’YII"U = limm->oo ’Y1I"Um = 0. Conversely let us prove that if U E E (0) and ’YII"U = 0, then U is the limit in E (0) of vector functions in ~(o)n. Letlbe any function in£» (tRn), and (j) the restriction of«Jl to O. Since ’YlloU = 0, we have <’YlloU, ’Yo(j)} = which means

L uo~ [div

+ uograd

~l

dx

= O.

Hence

and so

r [~"(j)+u"grad(j)]dx=O, i, divu=d~

V(j)E~(tRn) (1.27)

Ch. I, § 1

Some function spaces

13

where v denotes the function equal to v in f2. and to 0 in en. This implies that Ii E E(GiIl). Following exactly the same steps as in proving Theorem 1.1 (in particular points (i) and (ii)) we may reduce the general case to the case where the function u has its support included in one of the sets (Pj n O. For such a function u we remark that Ii E E(Gin) and that Ox’ Ii has a compact support in (Pi, for 0 < A < 1 «(Pi is supposed star› shaped with respect to 0). According to Lemma 1.1, Ox’ U res tricted to (P; (or n)converges to u in E( (Pi) (or E(O)) as A-+> 1. The problem is then reduced to approximating a function u with compact support in o in terms of elements of ~(Gin); this is obvious by regularization (as in point (ii) in the proof of Theorem 1.1).

Remark 1.3. If the set 0 is unbounded or if its boundary is not smooth, some partial results remain true: for example, if u E E (0), we can defme ’Yvu on each bounded part roof I’ of class ~2, and ’Yv u E HI (I’0)’ If 0 is smooth but unbounded or if its boundary is the union of a finite number of bounded (n-l )-dimensional manifolds of class ~2 , then ’Yv u is defined, in this way, on all I’. Nevertheless the generalized Stokes formula (1.19) does not hold here. The results will be more precise if we know more about the trace of functions in HI (0). Let us assume that the following results hold: - there exists ’Yo E £ (HI (f2.), L 2 ( r) ) such that ’You = u\r for every u E ~(O). We denote ’Yo (HI CP)) by ’JC 1/2(r). - there exists a lifting operator Qn E £(;’J(’1/2(r), HI (nn, such that ’Yo•Qn = the identity; ;](’1/2(r) is equipped with the norm carried by ’Yo• - 0 is a lipschitz set. Then all the preceding results can be extended to this case. Theorems 1.1 and 1.3 are true. The proof of Theorem 1.2 leads to a definition of ’Yv ’ u as an element of ;](’- 1/2 (I’) = the dual space of ;](’1/2 (I’). The gen› eralized Stokes formula (1.19) is valid. 1.4. Characterization of the spaces H and V We recall the notation at the end of Section 1.1: "Y = {u E !?j(0), div u = O}, H = the closure of "Y in L 2 (0), V

= the closure

of "Y in

H6 (0).

14

The steady-state Stokes equations

Ch. I, § 1

Characterization of the gradient ofa distribution.

Let 0 be an open set in 6{n and let p be a distribution on 0, p It is easy to check that for any u E "f/we have n

(gradp, v)

= L (DiP, i=I

E ~’ (0).

II

v;)

=-

L tp, D;V;) = tp, div v) = O. ;=1

(1.28)

The converse of this property is true. This important property can be proved by using a profound result of De Rham’U: Proposition 1.1. Let

Ii

E

n be an open

set of ~n and I = {It, ... ,In},

i = I, ... ,n. A necessary and sufficient condition that

~’(n),

I = grad p,

(1.29)

for some p in ~’(n),

is that

(f,v}=O

VvEi":

(1.30)

This result is essential for the interpretation, later, of the variational formulation of Navier-Stokes and related equations. Let us now state a number of connected results. Proposition 1.2. Let n be a bounded Lipschitz open set in 6{1l. (i) If a distribution p has all its first-order derivatives DiP, 1’:< i ~ n, in L2 ( n ), then p EL 2(0)and (1.31 ) (l) We sketch an outline of this proof which lies outside the scope of this book. We consider

the currents

II

f=

and the-e

oo

L fidxi,[i

;=1

E 5f(n),

forms of degree (n-1) with compact support: n

ii =

L

i=I

(-dVidxl

A

AdXi_I

A

dx i+ 1

A .

A

dx ll ,

Vi

EJ)(Sl).

Then dii = 0 if and only if v = {VI’ ... , VII} E V (i.e. div v = 0). Theorem 17’ on p.114 of De Rham [1) states that for f Ef)’(n),f = dg, g E~’(n) (i.e, f;=Djg, 1<;;<;n), if and only if if. v> = 0 for every j; of the previous type. This is equivalent to the statement in Proposition 1.1. (J.L. Lions, private communication). See also in Remark 1.9 a simple proof of Proposition 1.1 based on elementary arguments and Proposition 1.2.

Ch. I, § 1

Some function spaces

15

(ii) If a distribution p has all its first derivatives DiP, 1 ~ i H-l(O), then p E L2(O) and

IIp’’L2 (n)/6/. ~c(O)"

~

n, in

(1.32)

grad p "8-1 (n)’

In both cases, if 0 is any open set in tRl!, p E Lt~c

(0).

Proof. Point (i) and (1.49) are proved in Deny & Lions [1] for a bounded star-shaped open set O. In our case, because of this result, p is L2 on every sphere contained in 0 with its closure, and on all the sets (1); defined following (1.4). Since a finite number of these balls and sets (1); cover 0 the result follows. Point (ii) is proven in J. Necas [2] . For a set without any regularity property, we apply the foregoing results on each ball contained in 0 with its closure, and we obtain merely that p E L~c(O). Remark 1.4. (i) Combining the results of Propositions 1.1 and 1.2, we see that if fE H-l (0) (orLToc(O» and (f, v) = 0, then f = grad p with p E Lfoc(O). If moreover, 0 is a Lipschitz open bounded set, then p E L2 (0) (or HI (0». (ii) Point (ii) in Proposition 1.2 implies that the gradient operator is an isomorphism from L 2 (O)/tR into H-l (0); hence the range of this linear operator is closed. We recall (0 bounded) that L2 (O)/tR is isomorphic to the subspace of L2 (0) orthogonal to the constants L 2 (O)/tR

={ p

E L

2

(0 )j;L

p(x)dx

=

o}.

See also in (6.12), a different version of (1.32). Characterization of the space H We can now give the following characterization of Hand H 1 (the orthogonal complement of H in L 2 (0».

Theorem 1.4. Let 0 be a Lipschitz open bounded set in tRl!. Then: Hl= {uEL 2(0): u =gradp,pEH1(0)},

H

= {u E

L 2(0): div u

= 0, ’Yvu = O}.

(1.33) (1.34)

Ch. I, § 1

The steady-state Stokes equations

16

Proof of (1.33). Let U belong to the space in the right-hand side of (1.33). Then for all v e«, (u, v)

= (grad p, v) = -(p,

div v)

=

.

(1.35)

Hence, U E Hi. Conversely, Hi is contained in the space in the right-hand side of (1.33). Indeed ifu E Hi, then,

VvEf

(u,v)=O

and according to Proposition 1.1, u Proposition 1.2, p E H1 en).

= grad p E ~’(n).

Then, by

Proof of (1.34). Let H• be the space in the right-hand term of (1.34) and let us prove thatHCH•.Ifu EH, thenu = limm~ ... u m ’ um Ef, this convergence in L 2 (n) implies convergence in the distribution sense; since differentiation is a continuous operator in distribution space and div u m = 0, we see that div u = 0. Since div um = div u = 0, u m and u belong to E(n) and (1.36)

so that um converges to u in E (n) and ’Yllu = limm~’" ’Yllum = (’Yllum = u m Vir = OVum En. Hence u E HO, Let us suppose that H is not the whole space H’ and let H• be the orthogonal complement of H in H•, By 0.33) every u E H" is the grad› ient of some p E HI (n); moreover p satisfies

.

iJ.p - div

U

= 0, u•vlr = apl av r = 0,

(1.37)

and this implies that p is a constant and u = 0; therefore H••= {O} and

H=Ho.

Remark 1.5, If n is any open set in /R n , the proof of (1.33) with very slight modifications shows that Hi = {u E L 2 (n): u

= grad p,

p E LIoc(n)}.

(1.38)

If n is unbounded but satisfies condition (1.4) then Hi = {u E

Theorem 1.5. Let

L2 (n): u = grad p, p E L1oc(nn. n

be an open bounded set of class rrl 2 . Then

(1.39)

Ch. I, § 1

Some function spaces

L2 (n)

17

= H (f) HI (f) H2 ,

(1.40)

where H, HI’ H2 are mutually orthogonal spaces HI = {Il E L2 (n): u = grad p, p E HI (n), Ap = O},

(1.41)

H2 = {IlEL 2 (0 ) : ll=gradp,pEHA(O)}.

(1.42)

Proof. It is clear that HI and H2 are included in H1 , and that the intersection of any two of the spaces H, HI’ H2 , is effectively reduced to {OJ. The spaces HI and H2 are orthogonal; if u = grad p E HI’ v = grad q E H2 , then u E E(O) and by using the generalized Stokes formula (1.19), (u, v) = (Il, grad q) = (’Yu"’ ’Yoq) - (div u, q)

and this vanishes since ’Yo q = 0 and div u = Ap = O. It remains for us to prove that any element u of L2 (0) can be written as the sum of elements " O, "l’ " 2 of H, HI’ H2 . For such a u let p be the unique solution of the Dirichlet problem Ap = div u E H-1 (0),

P E HA(O).

We take " 2

(1.43)

= grad p.

Let then q be the solution of the Neumann problem Aq = 0,

I

aq = ’}; (u - grad p). av r J/

(1.44)

We notice that div(ll- grad p) = 0, so that Il- grad p E E(O), and ’YJ/(Il - grad p) is defined as an element of H-1/2 (f), and by the Stokes formula (1.19)

(~,(u

- grad p), I) =

L

div (u - grad p) dx

= O.

According to a result of Lions & Magenes [1 ] , the Neumann problem (1.44) possesses a solution which is unique to within an additive constant. We take "l

= grad q,

(1.45) (1.46)

18

The steady-state Stokes equations

Ch. I, § 1

We have now to show that Uo E H. But div Uo = div (u - ul - u2) = div u -.1 p = 0, and

’Y.Po = ’Yv(u -ul - u2) = ’Yv(u - grad p) - 3q;3v = O. Remark 1.6. Let us write PH for the orthogonal projector in L 2 (0.) onto H; obviously PH is continuous into L2 (0.). In fact PH also maps HI (0.) into itself and is continuous for the norm of HI (0.). In the proof of Theorem 1.5, let us assume that u E HI (0.); then div u E Hb(n.) n 2 H (0.); u - grad p belongs to HI (0.) and ’Yv(u - grad p) belongs to HI/2 (I’). Finally we infer from (1.44) that q E H 2 (0.), and PHu = u - grad (p+q) E HI (0.).

It is clear also (see for instance Lions & Magenes [I]) that the mappings u ....p .... u - grad p .... q, are continuous in the appropriate spaces and we (0.) into HI (0.): conclude that PH is continuous from

nb

IIPHUIlHI(o)
(1.47)

If 0. is of class cer+l, r Integer ,> I, a similar argument shows that if u E H’(n.) then PHu E Hr(n.) and PH is linear and continuous for the norm of H’(n.): IIPHUIlH,(o) ~ c(r, 0.) lIuIlHrCo)’Vu enr(n.).

(l.48)

Characterization of the space V. Theorem 1.6. Let 0. be an open Lipschitz set Then V={uEHb(n.):

divu=O}.

(1.49)

Proof. Let V be the space in the right-hand side of (1.49). It is clear that V C V: for if u E V, u = lim urn’ urn E’1’; this convergence in Hb (0.) implies that div urn converges to div u as m -+- 00, and since div urn = 0, div u = O. To prove that V= V’, we will show that any continuous linear form Lon V’ which vanishes on V is identically equal to O. We first observe that L admits a (non-unique) representation of the type n

L (v)

I:

L (Q;, v;>,

;=1

Q; E

u:’ (0.).

(1.50)

Ch. I, § 1

19

Some function spaces

Indeed V is a closed subspace of HA(n) = [HA(n»)n; any linear con› tinuous form on V can be extended as a linear continuous form on Hb (0.) and such form is of the same type as the form in the right-hand side of (1.50). Now, the vector distribution Q = (Ql’ , Qn ) belongs to H-l (0.), and (Q, v) = 0, V v E "Y. Propositions 1.1 and 1.2 are applicable and show that

Q=gradp,

pEL 2(n);

thus For v E V•, n

L(v)

= .2: 1 (Q:, v.) = 1=

I

I

(P, div v)

= 0,

and L vanishes on all of V•. Remark 1.7. We can give a more direct and much simpler proof of (1.49) for a set 0. which is globally star-shaped with respect to one of its points. Let us suppose that 0. is star-shaped with respect to 0 and let ax denote as before the linear transformation x ~ >.x. Let" E V•. Then the function a"l\ u belongs to HA (a~ 0.) and div (J~" = O. The function aj, equal to in (J" 0. and to 0 in 0. - (J" 0. (0 < X < 1), is in HA(n) and div"" equals Xa" (div a) in a" 0. and 0 in 0. - (J~ 0.; hence div u~ = 0, u" E V• and has a compact support in n. In this case it is easy to check by regularization that "" E V, and since a; converges to u in HA (0.) as X ~ 1, u E V and V = V•. 0

(J"O"

Remark 1.8. The results which will be extensively used below are Propositions 1.1 and 1.2, Theorem 1.6 and less frequently, Theorem 1.5. Remark 1.9. A result weaker than Proposition 1.1 but sufficient for what follows can be proved by a more elementary method.

Assume that 0. is a Lipschitz bounded open set in tRn and that/EH-l(n), satisfies (f, v)= 0 V v E’f’(or V). Then f » grad p, p E L2(n).

(1.51)

Thestationary Stokes equations

20

Ch. I, §2

For an arbitrary open set n eRn, the same result holds, with only P E Ltoc(n).

(1.52)

Let us sketch the proof of this result due to L. Tartar, and based on Proposition 1.2 and Remark 1.4 (ii). It is clear that there exists an increasing sequence of open sets n m (n m c n m + 1)’ which are Lipschitz and whose union is n. Let A (or Am) be the gradient operator E £(L 2 (n), H-1 (n)) (or E £(L 2 (n m ), H-l(nm ))) , and let A * E £(Hb(n), L 2 (n )) (or A~ E £(HbCn m ) , L 2 ( n m)) ) be its adjoint. By Remark 1.4 (ii), the range of A (written R (A)) is a closed subspace of H-l (n)). Now it is known from linear operator theory that the orthogonal of Ker A’" is the closure of R (A) and this is therefore equal to R (A); Ker A’" is the Kernel of A’" Ker A’"

= {u EHA(n),

div u

= O}(l)

Similar remarks hold for Am’ Now let/satisfy the condition in (1.51) and let u EKer Ifii’is the function u extended by 0 outside n m , then Ii E (n) and div Ii = div u = O. Since Ii has a compact support in n, it is clear by regulariza› tion that ’i1 is the limit in Ha (n) of elements of "Y, hence ’i1 E V, and (f, U> = O. Therefore the restriction of / to n m is orthogonal to Ker A ~, and thus belongs to R (Am) : / = grad Pm on n m , Pm E L 2 (n m ). Since the n m are increasing sets, Pm + 1 - Pm = const on nm , and we can choose Pm + 1 so that this constant is zero. Hence t = grad p, P E Ll~c (n). This is sufficient for (1.52). For obtaining (1.51) (p E L (n)), we use the fact that n is locally star shaped (see Section 1.1). Let (1)j)l<.j<.1 be a finite covering of I’, such that 0/ = OJ n n is star-shaped, ’Vj. If u EHA(oj) and div u = 0, then for 0 = 0, and then / = grad qj in OJ, qj E L 2 (Oj~; qj =P on OJ, and P E L 2(0;) v], so that P E L 2 ( n ) .

HJ

A;.

§ 2. Existence and uniqueness for the Stokes equations

The Stokes equations are the linearized stationary form of the full Navier-Stokes equations. We give here the variational formulation of Stokes problem, an existence and uniqueness result using the projection (1) It is proved in Theorem 1.6 that this space is V, but this result is not yet known when we

prove Proposition 1.1.

21

Existence and uniqueness for the Stokes equations

§2

theorem, and a few other remarks concerning the case of an unbounded domain and the regularity of solutions. 2.1. Variational formulation of the problem

Let

n

be an open bounded set in dill with boundary I’, and let

fE L 2 ( n ) be a given vector function in n. We seek a vector function U = (u 1 ..., u ll ) representing the velocity of the fluid, and a scalar

function p representing the pressure, which are defined in n and satisfy the following equations and boundary conditions (v is the coefficient of kinematic viscosi ty, a constant): -vf1u+gradp=f

div U U

=0

=0

inn

(v>O)

in n,

(2.1) (2.2)

on I’,

(2.3)

Iff, u, and p are smooth functions satisfying (2.1) - (2.3) then, taking

the scalar product of (2.1) with a function v E f we obtain, (-v f1u

+ grad p, v ) = if,

v)

and, integrating by parts, the term (-ilu, v) gives(l) n

~ (grad Uj, grad v j) = ((u, r)

1=1

(2.4)

and the term (grad p, v) gives -(p, div v)

= 0,

and there results v ((u, v»=(f, v),

VvEf.

(2.5)

Since each side of (2.5) depends linearly and continuously on v for the nA(n) topology, the equality (2.5) is still valid by continuity for each vE V (1) See the notation at the end of Section 1.1.

The steady-state Stokes equations

22

01. I, §2

the closure of f in nA (0). If the set 0 is of class C(J2 then due to (2.3) the (smooth) function u belongs to nAco), and because of (2.2) and Theorem 1.6, u E V. We arrive then at the following conclusion: u belongs to V and satisfies v«u. v»

.J

= if, v), V i’E

V.

(2.6)

Conversely, let us suppose that u satisfies (2.6) and let us then show that u satisfies (2.1) - (2.3) in some sense. Since u belongs only to nA (0), we have less regularity than before and can only expect u to satisfy (2.1) - (2.3) in a sense weaker than the classical sense. Actually, u E nA(O) implies that the traces ’Yo ui of its components are zero in H 1 /2 ( r ) ; u E V implies (using Theorem 1.6) that div u = 0 in the dis› tribution sense; and using (2.6) we have (-v6.u-f,V)=O, V vEf.

Then by virtue of Propositions 1.1 and 1.2, there exists some distribution pEL 2 (0) such that -v6.u - f= - grad p

in the distribution sense in O. We have thus proved Lemma 2.1. Let 0 be an open bounded set of class The following conditions are equivalent

1’(j’2.

u E V satisfies (2.6) u belongs to nA (0) and satisfies (2.1) - (2.3) in the following weak sense: (i) (ii)

there exists p E L 2 (0) such that -v!:iu + grad p = f in the distribution sense in 0 ;

div u

=

’YoU = 0.

in the distribution sense in 0;

(2.7)

(2.8) (2.9)

Definition 2.1. The problem "find u E V satisfying (2.6)" is called the variational formulation of problem (2.1) - (2.3).

Ch. I, §2

Existence and uniqueness for the Stokes equations

23

Remark 2.1. Before studying existence and uniqueness problems for (2.6), let us make a few observations. (i) The variational formulation of problem (2.1) - (2.3) was intro› duced by J. Leray [1] [2] [3]. It reduces the classical problem (2.1)› (2.3) to the problem of finding only u; the existence of p is then a consequence of Proposition 1.1.

(ii) When the set n is not smooth, we have two spaces which we called Vand VO in the proof of Theorem 1.6 and which may be different:

V = the closure of f in Hij(n), V’ = {u E Hij(n), div u

= O},

VC VO.

We may then pose two (possibly different) variational formulations: either (2.6) exactly, or the same problem with V replaced by V Sets n such that V =1= V are unknown and, a fortiori, the relation between these two problems is not known. For technical reasons, particularly important in the non-linear case, we will always work with the space V and consider only the variational problem (2.6). Let us remark as a complement to Lemma 2.1 that for any set n, if u satisfies (2.6) (or (2.6) with V replaced by YO), then it satisfies (2.7) with the restriction that p E LYoc(n) only; it satisfies (2.8) without any modification; and it satisfies (2.9) in the sense that u E Hb (n), a more precise meaning depends on the trace theorems available for n. 0.

2.2. The projection theorem

Let

n be any

open set of tRn such that

n is bounded in some

direction.

(2.10)

According to (1.10), Hij(n) is a Hilbert space for the scalar product (2.4); fis defined in (1.12) and Vis the closure off in Hfi(n). Theorem 2.1. For any open set n c eRn which is bounded in some direction, and for every IE L2(n), the problem (2.6) has a unique solution u. (The result is also valid if f is given in H- 1 (n).)

24

The steady-state Stokes equations

Ch. I, §2

Moreover, there exists a function p E Lfoc (n) such that (2.7) › (2.8) are satisfied. If n is an open bounded set of class rc 2, then pEL 2 (n) and (2.7) › (2.9) are satisfied by u and p. This theorem is a simple consequence of the preceding lemma and the following classical projection theorem. Theorem 2.2. Let Wbe a separable real Hilbert space (norm 11 11 w) and let a(u, v~ be a bilinear continuous form on WXW, which is coercive, i.e., there exists a> 0 such that a(u,u);;;;’allull~,

(2.11)

VuEW.

Then for each Q in W’, the dual space of W, there exists one and only one u E Wsuch that a (u, v)

= (Q, v), V v E

(2.12)

W.

Of course to apply this theorem to (2.6), we take W = the space V equipped with the norm associated with (2.4), a(u, v) = v«u, v)), and for v ~ (Q, v) the form v ~ (f, v) which is obviously linear and con› tinuous on V. The space V is separable as a closed subspace of the separable space (n).

HA

Proof of theorem 2.2. Uniqueness. Let ul and u2 be two solutions of (2.12) and let u = u 1 - u2’ We have a (ul’ v)

=a

(u 2 , v)

a(u I-u 2,v)=0,

= (Q, v),

Vv E W,

VvEW.

Taking v = u in this equality we see from (2.11) that allulI~

~ a(u, u)

= 0,

and hence u = O. Existence. Since W is separable, there exists a sequence of elements wI"" Wm , ... , of W which is free and total in W. Let Wm be the space spanned by WI , ... , wm ’ For each fixed integer m we define an appro x-

25

Existence and uniqueness for the Stokes equations

Ch. I, §2

imate solution of (2.12) in Wm ; that is, a vector 11m E Wm m

= ~~~.

I,m W., I

a(u m , v)

= (Q, v),

U

m

i=1

~. I,m Edt

(2.13)

V v E Wm .

(2.14)

satisfying Let us show that there exists one and only one um such that (2.14) holds. Equation (2.14) is equivalent to the set of m equations (2.15) and (2.15) is a linear system of m equations for the m components

ofu m :

m

L ~i,m

a(w i, Wj)

i=1

= (Q, Wj)’

j

= 1, ... , m.

s; (2.16)

The existence and uniqueness of um will be proved once we show that the linear system (2.16) is regular. To show this it is sufficient to prove that the homogeneous linear system associated with (2.16), i.e., m

L ~I

1=1

a(w i , Wj) =

0,

j = 1, ...

.m,

(2.17)

has only one solution ~1 = ... = ~m = O. But if ~1’ ... , ~m’ satisfy (2.17), then by multiplying each equation (2 ..17) by the corresponding ~j and adding these equations, we obtain m

L

Ij = 1

~i~j

a(wi’ Wj)

=0

or, because of the bilinearity of a, a(

~ ~iwi’ ~ ~jWJ)

1=1

j=1

= 0;

using (2.1l) we find m

L

i=1

and finally

~iwi=O. ~1

= ... = ~m = 0 since WI , ... , w m are linearly independent.

26

The steady-state Stokes equations

Ch. I, §2

Passage to the limit: When we put v = urn in (2.14), we obtain a(u rn, Urn)

= (Q, urn)’

(2.18)

from which, by (2.11), it follows that

cxllu rn IItv E;;; a (urn ’ u",) = (Q, urn) E;;; IIQIIw,lIu", II,

lIurn II w

E;;;

a1 IIQllw"

(2.19)

which proves that the sequence urn is bounded independently of min W. Since the closed balls of Hilbert space are weakly compact, there exists an element U of Wand a sequence urn’ m’ ~ 00, extracted from urn ’ such that Urn’~U

in the weak topology of W, as m’ ~

00.

(2.20)

Let v be a fixed element of Wj for some j. As soon as m’ ~ t, v E Wm’ and according to (2.14) we have

a(u rn" v)

= (Q, v).

By using the following lemma, we can take the limit in this equation as m’ ~ 00, and obtain:

a(u, v) = (Q, v).

(2.21)

Equality (2.12) holds for each v E Uj= 1 Wj’ and as this set is dense in W, equality (2.21) still holds by continuity for v in W. This proves that U is a solution of (2.12).

Lemma 2.2. Let a(u, v) be a bilinear continuous form on a Hilbert space W. Let ¢rn (or ’/Irn) be a sequence of elements of Wwhich converges to ¢ (or 1/1) in the weak (or strong) topology ofW. Then (2.22) (2.23)

Ch.I,§2

Existence and uniqueness for the Stokes equations

27

Proof. We write

As the form a is continuous and the sequence cP m is bounded,

and this term converges to 0 as m 1-* 00. We notice next that the linear operator v ~a(l/I, v) is continuous on Wand hence there exists some element of W’, depending on 1/1 and denoted by A( 1/1), a(l/I, v)

= (4(1/1), v),

V v E W.

(2.24)

We can now write

and this converges to 0 as m ~ 00, as a consequence of the weak con› vergence of cP m . This proves (2.22). For (2.23) we have only to apply (2.22) to the bilinear form a*(u, v)

= a(v,

u).

Remark 2.2. (i) Theorem 2.1 is also true iff is given ;K’ -1 (n).

It can be proved that the sequence {um } constructed in the proof of Theorem 2.2, as a whole, converges to the solution u of (2.12) in the strong topology of W. We do not prove this result here; it will appear as a consequence of Theorem 3.1. (iii) Using the form A (1/1) introduced in the proof of Lemma 2.2 (cf. (2.24)), one can write equation (2.12) in the form (ii)

(4(u), v)

= (Q, v)

which is equivalent to A(u) = Q in W’.

An alternative classical proof of the projection theorem is to show

28

The steady-state Stokes equations

Ch. I, §2

that the operator u 1-+ A (u) is an isomorphism from W onto W’. A variational property. Proposition 2.1. The solution u of(2.6) is also the unique element of V such that E(u) :E;;; E(v),

Vv E V,

(2.26)

2(f, v).

(2.27)

where E(v)

= vllvll 2 -

Proof. Let u be the solution of (2.6). Then as lIu-vll 2 ~ 0, Vv E V,

we have vllull 2

+ vllvll 2

-

2v«u, v» ~ 0.

(2.28)

Because of (2.6) we have -vllull 2 = vllull 2 - 2(f, u) = E(u), -2v«u, v»

= -2(f,

v)

and thus (2.28) gives exactly (2.26). Conversely, if u E V satisfies (2.26), then for any v E V and A E lR, one has E(u) :E;;; E(U+AV).

This may be reduced to VA 211v1I 2+ 2Av«U,V»

- 2A(f, v) ~ 0, VA E lR.

This inequality can hold for each A E lR only if v«u, v»

= V, v),

and thus u is indeed a solution of (2.6). Remark 2.3. If the spaces V and V’ of Remark 2.1 are different,

(2.29)

29

Existence and uniqueness for the Stokes equations

Ch. I, §2

Theorem 2.2 also gives the existence and uniqueness of a ii E V• such that

v«ii,

v» = if. v),

(2.30)

Vv E V•.

Proposition 2.1 is also valid with u and V replaced by U and V•.

2.3. The unbounded case We consider here the case where n is unbounded and does not satisfy (2.10). If n is of class rc 2 , problem (2.1) - (2.3) is not equiv› alent to (2.6) as in Lemma 2.1. The reason is that if u is a classical solution of (2.1) - (2.3), it is not clear, without further information about the behaviour of u at infinity, that u E HI (n); hence perhaps u $ V and equation (2.5) cannot be extended by continuity to the closure V off. Further, there is a difficulty in trying to solve problem (2.6) directly using Theorem 2.2; hypothesis (2.11) is not satisfied, V is a Hilbert space for the norm [un

= Vlul 2 + lIul1 2 ,

which is not equivalent to the norm lIuli since we lose the Poincare inequality. In order to pose and solve a variational problem in the general case, let us introduce the space Y

= the completion

of f under the norm 11•11.

(2.31 )

It is clear, since II u II ~ [u], that Y is a larger space than V

(2.32)

VC Y.

Lemma 2.3. Y C {u E La (n): DiU E L2 (n), 1 ~ l ~ n}

with ex

= 2n/(n-2) if n ~

3, and

YC {uELToc(n):

for n

= 2. The injections

(2.33)

DiuEL2(n),i= 1,2},

Va~

1,

(2.34)

are continuous.

Proof. Let us prove (2.33). This is a consequence of the Sobolev in› equality (see Sobolev [1] , Lions [1 ] , and also Chapter II);

Ch. I, §2

The steady-state Stokes equations

30

114>II L a (o ) ~ c(a, n) llgrad 4>II L 2(o where a

)’

V 4> E

p)

(0),

(2.35)

= 2n/(n-2).

If u E Y there exists a sequence of elements um E fconverging to u in Y; by (2.35)

(2.36) where Z stands for the space on the right-hand side of inclusion (2.33) and 1I’lIz is the natural norm of Z:

As m and p tend to infinity, the right-hand side of (2.36) converges to O. Thus um is a Cauchy sequence in Z; its limit u belongs to Z. It is clear also that

lIuliz ~ e" lIuli with the same e" as in (2.36) (e" = c"(n)). The proof is similar for (2.34), by considering Cauchy sequences in La (0’) where 0’ is a compact subset of O. Theorem 2.3. Let 0 be any open set in lRn , and let! be given in y’, the dual of the space Yin (2.31). Then, there exists a unique u E Y such that JI«u, v)

= (f, v),

V v E Y.

(2.37)

There exists p E Ltoc(O), such that (2.7) is satisfied and (2.8) is true. Proof. We apply Theorem 2.2 with the space Y, a(u, v) = v «u, v)), and Q replaced by!; we get a unique u satisfying (2.37). Then, Proposition 1.3 shows the existence of some p E Ltoc(O) such that (2.7) is satisfied; (2.8) is of course easily verified. Finally, (2.9) is satisfied in some sense depending on the trace theorem

Existence and uniqueness for the Stokes equations

Ch. I, § 2

31

available for Y (or for the space Z on the right-hand side of inclusions (2.33) and (2.34)). If n is locally Lipschitz, Proposition 1.2 shows that p E L1oc(O). Remark 2.4. By Lemma 2.3, for fE D:I’(n) (l/a+ l/a’ = 1), V 1-+

f.n t-v dx

(2.38)

is a linear continuous form on Y. Thus one can take in Theorem 2.3, any fE La’(n) (a = 2n/(n-2), n ~ 3).

2.4. The non-homogeneous Stokes Problem We consider a non-homogeneous Stokes problem. Theorem 2.4. Let n be an open bounded set of class rt2 in lR n . Let there be given fE H- l (n), g E L2 (r), ¢ E H1/2(I’), such that

L/dx = fr

¢•vdf.

(2.39)

Then there exists

u E HI (n), p E L2 (n), which are solutions of the non homogeneous Stokes problem -

v~

+ grad p

div a =g in

’Yo"

=¢

=fin n

n

(2.40) (2.41 )

i.e. u = ¢ on I’,

(2.42)

u is unique and p is unique up to the addition of a constant. Proof. Since H1/2 (I’) = ’YoHI (n), there exists "0 E HI (n), such that

’Yo"o = ¢. Then, from (2.39) and Stokes’ formula,

f

n (g - div "0) dx

= O.

Using Lemma 2.4 below we see that there exists a"1 E HJ (n) such that div"1 = g - div "0 .

The steady-state Stokes equations

32

Ch. I, §2

Setting v =u - UO- Ul’ (2.40) - (2.42) reduces to a homogeneous Stokes problem for v: - v~v

+ grad p

div v

v

=[ -

v~(uo

+ ul)

E H-l (51),

= 0 in 51,

= 0 on an.

The existence and uniqueness of v and p (and therefore u and p) then follows. Lemma 2.4. Let 51 be a Lipschitz open bounded set in lHn . Then the divergence operator maps HJ (51) onto the space L 2(n )/ lH, i.e.,

(2.43) Proof. From Proposition 1.2, and Remark 1.4 (ii), A = grad E .£ (L2(n), H-l(n)) is an isomorphism from (2.43) onto its range, R (A). By transposition, its adjoint A * = - div E .£(H~ (51), L2 (n))1s an iso› morphism from the orthogonal of R (A) onto the space (2.43); in particular A * maps H~ (51) onto L2 (n)/lH. Remark 2.5. Theorem 2.4 is easily extended to the case where 51 is only a Lipschitz open bounded set in lHn , provided cP is given as the trace of a function cPo in HI (51) and (2.39) and (2.42) are understood as follows

fn

gdx

=

In div cPo

u - cPo E Hij (51) Wejust take Uo

dx,

(2.39’) (2.42’)

=cPo.

2.5. Regularity results A classical result is that the solution u E Hij (51) of the Dirichlet problem -tw +u =[belongs to H’" +2 (51)whenever [E H’" (51) (and 51 is sufficiently smooth). One naturally wonders whether similar regularity results exist for the Stokes problem.

33

Existence and uniqueness for the Stokes equations

Ch. I, §2

This result and the similar one in LP-spaces is given by the following Proposition.

Proposition 2.2. Let n be an open bounded set of class’s", r (m+2, 2), m integer> O. Let us suppose that

= max

1
(2.44)

are solutions of the generalized Stokes problem (2.40) - (2.42): divu =g in ’You

in n,

+ grad p = f

-v~u

= rJ>;

(2.40)

n,

i.e.,

U

(2.41 )

= rJ>

on r.

(2.42)

IffE Wm ,OI (n ), g E Wm + 1,00(n) and u E

rJ> E

Wm +2-1/0I,0I(r),

Wm +2,01 (n), p E Wm + I ,01 (n)

(1)

then (2.45)

d there exists a constant Co (a, v, m, n) such that lI u llw m +2,0I(n)

+ IIpllwm+I,OI(n)/iR

{llfIlw m ,OI(O) + Ilgllwm+I,OI(n) + 1IrJ>lI w m+2- 1/0I,0I (r) + dOl lIuIILOI(O) } ~ Co

dOl

= 0 for a ~

2, dOl = 1 for 1 < a

(2.46)

< 2.

Proof. This proposition results from the paper of Agmon-Douglis› Nirenberg [2], hereafter referred to as A.D.N., giving a priori estimates of solutions of general elliptic systems. Let u n+ 1 = 1Iv p, U = (u l , ... , u,,+1 ), f = (fl lv, ... , f n lv, g). Then equations (2.42) and (2.43) become n+l

j~ (1)

Q;j(D)uj=!J, 1

~i~n

+ 1,

wm + 2 - 1/0I,a ( r ) = 1’0 wn+ 2 ,0I( il) and is equipped with the image 1Iv"l wm+2 -1/01, OI(r) =

In! lIullwm+2, Ot(n)’ ’You=1/I

(2.47) norm

The steady-state Stokes equations

34

where 2il(~)’

~

2il

= (~l

(n = 1~12

’ ... , ~n

)

Ch. I, §2

E lR n , is the matrix

5 il, 1 E;;;; i, j E;;;; n, 1 E;;;;jE;;;;n

2n+lJ(~)=-2/,n+l(~)=~/’

2n + 1,n+ l (~) = 0, 1~12

= ~i + ... + ~~.

(2.48)

We take (see p. 38, of A.D.N.), si = 0, t j = 2, 1 E;;;; i E;;;; n, sn+l = -1, = 1. As requested, degree 2;/(~) E;;;; si + tl , 1 E;;;; i, j E;;;; n + 1, and we We easily compute L(~) = det 2;/(n = 1~12n, so that have ~/~) = 2;/(~)’ L(~) =F for real ~ =F 0, and this ensures the ellipticity of the system (Condition (1.5), p. 39). It is clear that (1. 7) on p. 39 holds with m = n. The Supplementary Condition on L is satisfied: L(~ + T~’) = has exactly n roots with positive imaginary part and these roots are all equal to

tn+ 1

r+(~,

f) = -s-r + iv’I~121~’i2-I~•~’12.

Concerning the boundary conditions (see p. 42), there are n boundary conditions and B hj = 5hl (the Kronecker symbol) for 1 E;;;; hE;;;; n, 1 E;;;; j E;;;; n + 1.

We take ’h = -2 for h = 1, ... , n. Then, as requested, degree B hl E;;;;’h + ~ and we have B hl =B hl. It remains to check the Complementing Boundary Condition. It is easy to verify that M+(~)

= (T -

r+(~)r

where r+(~) = r+(~, v). The matrix with elements !.f’!.lB;’/(~)I1k(~) 1 E;;;; h. k E;;;; n, -2 h n+l (~), simply the matrix with elements 2hk(~)’ 1 <: hE;;;; n. A combination Ej =1 Ch !.f,:l BhjLik is then equal to

is

n

(Gl(~+TV)2,, , Cn(~+TV)2,

2:

i= 1

Ci(~i+TVi»

and this is zero modulo M+ only if C1 =... = Cn = 0, and the Complement› ing Condition holds. We then apply Theorem 2.5, page 78 of A.D.N. in order to get (2.45) and (2.46) with da = 1 for all 0:. According to the remark after Theorem

35

Existence and uniqueness for the Stokes equations

Ch. I, §2

2.5, one can take dOl. = 0 for a ~ 2 since the solutions u and p of (2.41 ) - (2.44) are necessarily unique (p is unique up to an additive constant): if (u., P.), (u.. , P ) are two solutions, then u = u.-u .. , p = P.-P are solutions of (2.7) - (2.9) with f = 0 and hence u = 0 and p = constant

o

Remark 2.6. For a = 2 and mE cR, m ~ -1, one has results similar to those in P;oposition 2.2 by using the interpolation techniques of Lions-Magenes [ 1 J. 0 Proposition 2.2 does not assert the existence of u, p satisfying (2.42) - (2.46) (for givenf, g, tP) but gives only a result on the regularity of an eventual solution. The existence is ensured by Theorems 2.1 and 2.4 if a = 2. The following proposition gives a general existence result for n = 2 or 3. Proposition 2.3. Let 0 be an open set of class’s", r = max (m + 2, 2), m integer ~ -1, and let f E Wm,Q(O),g E Wm+I,Q(O), tP E Wm+ 2-Q,0I.(r) be given satisfying the compatibility condition

f

n gdx

=

f

r tP vd[’.

(2.49)

Then there exist unique functions u and p (p is unique up to a constant) which are solutions of (2.42) - (2.44) and satisfy (2.45) and (2.46) with dOl. = 0 for any a, 1 < ex < 00. Proof. This is precisely the result proved in Cattabriga [1] when n = 3 (and even for m = -1). For n = 2 one can reduce the problem to a classical biharmonic problem. There exists v E Wm+ 2,0I.(O), such that div v =

s,

(2.50) (2.51)

Such v may be defined by

v = grad 8 + where

(J E

{auaX2 ,_ ax!au }

Wm+ 3,0I.(O) is a solution of the Neumann problem

(2.52)

The steady-state Stokes equations

36

Ch. I, §2

= ginn,

(2.53)

ao =
(2.54)

AO

av

and a E Wm+3,Cl£(n) will be chosen later. The Neumann problem (2.53) - (2.54) has a solution 0 because of (2.49), and 0 E Wm+3,Cl£(n) by the usual regularity results for the Neumann problem. The conditions on a are only boundary conditions on r and these are

aa

aT

=the tangential

aa = the normal av

derivative of a

=0,

derivative of a = (/JoT _

ao . aT

Since <poT - aO/aT E Wm+Z-1/Cl£,Cl£(r), there exists a a E Wm+3,Cl£(r) with ’Yo a = 0, ’Yl a =
-vAw + grad p div w

=

r.r =f+ vAv,

(2.55)

= 0,

(2.56) (2.57)

If n is simply connected then, because of (2.56), there exists a function p such that

w = (Dzp, -D1 p). Condition (2.57) amounts to saying that p gives after differentiation

(2.58)

= ap/av = 0 on rand (2.55)

37

Existence and uniqueness for the Stokes equations

Ch. I, §2

Thus we obtain V~2p

= D1f’,.-D2fl E

Wm-I,Cll(n)

= 0, ~~ = 0 on r.

p

(2.59) (2.60)

The biharmonic problem (2.59) (2.60) has a unique solution p E Wm + 3, Cll (n), and the function w defined by (2.58) is a solution of (2.55) › (2.57) and belongs to Wm+ 2,Cll(n). If n is not simply connected, the condition (2.56) allows us to obtain (2.58) locally, i.e. p might not be a single valued function. A further

Fig. 2

argument using (2.57) and proved in the next Lemma shows that p is necessarily a single-valued function, such that p

= 0 on r o, p = constant on r i, i = 1, 2, ...,

ap av

= 0 on

r

’

(2.61) (2.62)

where r = ro u q u r2 , is the decomposition of an into its connex components, r o is one of these components, (the outside boundary of n if n is bounded for example), and r l , r 2 , ... , are the other compon› ents. Then the problem (2.55)-(2.57) is reduced for p to the equation (2.59) associated with the boundary condition (2.61), (2.62). As previously, the biharmonic problem (2.59), (2.61), (2.62) has a unique solution p E Wm+ 3, Cll (n ) and the function w defined by (2.58) is a solution of (2.55) - (2.57) and belongs to wm+2,Cll (.Q).

38

The steady-state Stokes equations

Ch. I, §2

Lemma 2.5. Assume that a vector function w in WA,l(n), n open set of ~2 , is divergence free. Then there exists a unique single-valued function p E W2 , l (n) which satisfies (2.61), (2.62) and

w = (D2 p,-D l pl.

(2.63)

Proof. Let us first show this result for smooth functions w. The result can be extended for the more general vector functions w by a density argument. Let r 0’ r I’ r 2"’" denote the connex components of r = an ~nd let us make some smooth cuts in n such that n = 1.: U n, where n is a simply connected open set. Then for the given smooth function w, there exists a function p such that (2.63) holds in It is clear that p is smooth in the closure of n and since w = 0 on an, grad p vanishes on an so that p is constant on each r j , and aplav vanishes on an. We make p unique by choosing the value of p on r 0 : p = 0 on r o: Now let P– denote the values of p on both sides of 1.:, 1.:+ and 1.:_, For an arbitrary w they could be different; but if w vanishes on an and p is constant on each r p they must be the same: if M–, P– are two points of 1.:–’ M– E an, we have

n.

p(P+) - p(M+)

=

J

w dx

=

I:(M+~P+)

= p(P_)

- p(M_),

2.6. Eigenfunctions of the Stokes Problem

Let n be any open bounded set in tRn The mapping A : f ~ 1lv U defined by Theorem 2.1 is clearly linear and continuous from L2 (n) into V and hence into HJ (n). Since n is bounded the natural injection of HJ (n) into L2 (n) is compact’P and therefore A considered as a linear operator in L 2 (n) is compact. This operator is also self-adjoint as

=V «u1’ U2)) = (fl’ Ah) when Af;=Uj’ i = 1,2. (Af1’ h)

(1) The Compactness Theorems in Sobolev Spaces are recalled in Chapter II, § 1.

Ch. I, §3

Discretization of the Stokes equations

39

Hence, this operator A possesses an orthonormal sequence of eigen› functions W( AWj = "A;Wj’ i ~ 1, Aj> 0, "A; -+- + 00, i -+- + 00, : (2.64)

As usual (Wj’ wk)

= fJ jk,

((Wj’ wk))

=:

Vi, k.

AjfJ jk,

(2.65)

Using again Theorem 2.1 we can interpret (2.64) as follows: each [, there exists PjE Lfoc(n) such that

- v Ii Wj+ grad p.J

= 0 in n, ’Yo Wj = 0 on r.

for

= A’w, in n ’ 1 J

div Wj

If n is a Lipschitz set then Pj EL 2 (n ), due to Proposition 1.2. If n is of class s?" (m an integer ~ 2), a reiterated application of Proposition 2.2 shows that Wj E

n m (n),

Pj E

n m - l (n),

v

i~

1.

(2.67)

If n is of class ’6’00 then, by (2.65), Wj E ’6’00 (n),

Pj E ’6’00 (n),

vi»

1.

(2.68)

§3. Discretization of the Stokes equations (I) Section 3.1 deals with the general concept of approximation of a normed space and Section 3.2 contains a general convergence theorem for the approximation of a general variational problem. In the last section and throughout all of Section 4 we describe some particular approximations of the basic space V of the Navier-Stokes equations. We give the corresponding numerical scheme for the Stokes equations and then apply the general convergence theorem to this case. In Section 3.3 we consider the finite difference method. Finite element methods will be treated in the next section (Section 4; 4.1 to 4.5). The approximations of the space V introduced here will be used

40

The steady-state Stokes equations

Ch. I, §3

throughout subsequent chapters, and they will be referred to as (APX1), (APX2), .... 3.1. Approximation of a normed space When computational methods are involved, a normed space Wmust be approximated by a family (Wh)h E Jf of normed spaces Wh . The set X of indices depends on the type of approximation considered: we will consider below the main situations for X, i.e., X = N(= positive integers) for the Galerkin method, X =IIF=l (0, hPJ for finite differ› ences, and X = a set of triangulations of the domain n for finite element methods. The precise form of X need not be known; we need only to know that there exists a filter on X, and we are concerned with passing to the limit through this filter. For the sake of simplicity we will always speak about passage to the limit as "h -. 0", which is, strictly speaking, the correct terminology for finite differences; definit› ions and results can be easily adapted to the other cases. Definition 3.1. An internal approximation of a normed vector space W is a set consisting of a family of triples {Wh , Ph, rh}’ hEX where (i) Wh is a normed vector space; (ii) Ph is a linear continuous operator from Wh into W; (iii) rh is a (perhaps nonlinear) operator from W into Wh. The natural way to compare an element U E Wand an element U in W or to compare uh and ’h U in Wh . The first point of view is certainly more interesting as we make comparisons in a fixed space. Nevertheless comparisons in Wh can also be useful. (See Fig. 3.) Another way to compare an element u E W is to compare a certain image wu of u in some other space F, with a certain image Ph uh of uh

uh E Wh is either to compare Ph uh and

w

W

rh

• Ph

Wh

Fig. 3

F

Discretization of the Stokes equations

Ch. I, § 3

41

in F. This leads to the concept of external approximation of a space W, which contains the concept of internal approximation as a particular case. Definition 3.2. An external approximation of a normed space Wis a set consisting of (i) a normed space F and an isomorphism w of Winto F. (ii) a family of triples {Wh , Ph’ rh hE J{’ , in which, for each h, - Wh is a normed space - Ph a linear continuous mapping of Wh into F - rh a (perhaps nonlinear) mapping of Winto Wh . When F = Wand W = identity, we get of course an internal approxi› mation of W. It is easy to specialize what follows to internal approxi› mations. In most cases, Wh are finite dimensional spaces; rather often the operators Ph are injective. In some cases the operators rh are linear, or linear only on some subspace of W, but there is no need to impose this condition in the general case; also, no continuity property of the rh is required. The operators Ph and rh are called prolongation and restriction operators, respectively. When the spaces Wand F are Hilbert spaces, and when the spaces Wh are likewise Hilbert spaces, the approximation is said to be a Hilbert approximation. Definition 3.3. For given h, U E W, uh E Wh , we say that (i) Ilwu-Phuh II F is the error between U and uh’ (ii) lIuh - rhullwh is the discrete error between u and uh’ (iii) IIwu-PJ,rhuIIF is the truncation error ofu. We now defi e stable and convergent approximations. Definition 3.4. The prolongation operators Ph are said to be stable if their norms IIPh II

=

Sup

uhEWh lIuhllWh= 1

IIPh uh IIF

can be majorized independently of h.

42

The steady-state Stokes equations

Ch. I, § 3

The approximation of the space W is said to be stable if the prolong› ation operators are stable. Let us now consider what happens when "h -+ 0". Definition 3.5. We will say that a family uh converges strongly (or weakly) to U if Ph Uh converges to when h -+ 0 in the strong (or weak) topology of F. We will say that the family uh converges discretely to U if

wu

lim

h-+O

lIuh -rh

ull w

h

= O.

Definition 3.6. We will say that an external approximation of a normed space Wis convergent if the two following conditions hold: (Cl) for all u E W

lim

h-+O

Phrhu

= WU

in the strong topology of F. (C2) for each sequence uh’ of elements of Wh" (h’-+O), such that Ph,uh’ converges to some element rJ> in the weak topology of F, we have, rJ> E w W; i.e., rJ> = WU for some u E W. 0

w

Remark 3.1. Condition (C2) disappears when is surjective and especially in the case of internal approximations. 0 The following proposition shows that condition (CI) can in some sense be weakened for internal and external approximations. Proposition 3.1. Let there be given a stable external approximation of a space W which is convergent in the following restrictive sense: the operators rh are defined only on a dense subset if"of Wand condition (Cl) in Definition 3.6 holds only for the u belonging to if"(condition (C2) remains unchanged). Then it is possible to extend the definition of the restriction operators rh to the whole space Wso that condition (Cl) is valid for each u E Wand hence the approximation of Wis stable and convergent without any restriction.

Discretization of the Stokes equations

Ch. I, §3

43

Proof. Let u E W,u $."#/; we must define in some way ’hu E Wh ,V h, so that Ph’hu -+ WU as h -+ O. This element u can be approximated by elements in "#/, and these elements in turn can be approximated by elements in the space Ph Wh ; we have only to suitably combine these two approximations. For each integer n ~ I, there exists un E "#/such that lIu n - ullw ~ I In and hence (3.1 )

where Co is the norm of the isomorphism w. For each fixed integer n, Phrhun converges to wu" in F as h -+ O. Thus there exists some TIn > 0, such that Ihl ~ TIn implies -

IIPh’hu" -wun II F

I

~ -.

n

We can suppose that TIn is less than both TIn -1 and I In so that the form a strictly decreasing sequence converging to 0:

o < Tln+1 < ... < TIl; Let us define

’h u

TIn -+

TIn

O.

by

It is clear that for Tln+1

< Ih I ~

TIn’

and consequently

for Ihl ~ TIn’ This implies the convergence of Ph’hU to completes the proof.

wu as h-+O and

44

The steady-state Stokes equations

Ch. I, §3

Remark 3.2. If the mappings rh are defined on the whole space Wand condition (Cl) holds for all u E~ Proposition 3.1 shows us that we can modify the value of rhu on the complement ofWso that condition (Cl) is satisfied for all u E W. Galerkin approximation of a normed space.

As a very easy example we can define a Galerkin approximation of a separable normed space W. Let Wh , hEN = ’JC, be an increasing sequence of finite dimensional subspaces of W whose union is dense in W. For each h, let Ph be the canonical injection of Wh into W, and for any u E Who which does not belong to Who-I’ let ’hu = 0 if h ~ h o’ and ’hu = u if h > h o. It is clear that Phrh u -+ u as h -+ 00, for any u E UhEN Wh . The operator rh is defined only on "fI/= UhEN Wh which is dense in W. Since the prolong› ation operators have norm one they are stable, and according to Proposition 3.1 the definition of the operators rh can be extended in some way (which does not matter) to the whole space W so that we get a stable convergent internal approximation of W; this is a Galerkin approximation of W. 3.2. A general convergence theorem

Let us now discuss the approximation of the general variational prob› lem (2.12). W is a Hilbert space, a(u, v) is a coercive bilinear continuous form on WxW, a(u, u) ~ odlulI~,

’1u E W, (a> 0),

(3.2)

and Q is a linear continuous form on W. Let u denote the unique solution in W of a(u, v)

= (Q,V) ,

’Iv E W.

(3.3)

With respect to the approximation of this element u, let there be given an external stable and convergent Hilbert approximation of the space W, say {Wh ,Ph’ rh }hE:JC’ Likewise, for each hE ’JC, let there be given (i) a continuous bilinear form ah (uh ,J?,) on WhxWh which is coercive and which, more precisely, satisfies

Discretization of the Stokes equations

Ch. I, §3 3~o

45

> 0, independent of h, such that

ah (uh’ uh) ~ CXo lIuh I ~,

(3.4)

YUh E Wh

where 1I•lIh stands for the norm in Wh , (ii) a continuous linear form on Wh , Qh E Wh, such that (3.5)

in which 1I•II*h stands for the norm in Wh, and in which {3 is indepen› dent of h. We now associate with equation (3.3) the following family of approx› imate equations: For fixed hE ’Jf, find uh E Wh such that (3.6) By the preceding hypotheses, Theorem 2.2 (in which W,W’,a and Q are replaced by Wh, Wh,ah’ and ~respectively) now asserts that equation (3.6) has a unique solution; we will say that uh is the approximate solution of equation (3.3). A general theorem on the convergence of the approximate solutions uh to the exact solution will be given after defining precisely the mann› er in which the forms ah and Qh are consistent with the forms a and Q. We make the following consistency hypotheses: If the family vh converges weakly to v as h-+O, and if the family wh converges strongly to W as h-+O, then lim ah (vh’ wh) = a(v, w)

h-+O

lim ah (wh,vh) =a(w,v).

(3.7)

If the family vh converges weakly to v as h-+O, then lim (~,vh) = (Q,v).

(3.8)

h-+O

h-+O

The general convergence theorem is then Theorem 3.1. Under the hypotheses (3.2), (3.4), (3.5), (3.7) and (3.8),

the solution h-+O.

Uh

of (3.6) converges strongly to the solution

U

of (3.3), as

46

The steady-state

Proof. Putting Vh

=uh in (3.6) and using (3.4)

Ch. I, § 3

and (3.5), we find

=
ah(uh,uh)

aolluhll~

Stokes equations

(3.9)

~ IItKnll.hlluhllh ~f3l1uhllh;

hence (3.10) As the operators Ph are stable, there exists a constant Co which major› izes the norm of these operators IIPh li

= IIPh llt(Wh’ F) ~

co;

(3.11)

and hence

cof3

IIPhuhIlF~-’

(3.12)

ao

Under these conditions, there exists some if> E F and a sequence h’ converging to 0, such that

l~

Ph’uh’

= if>

in the weak topology of F; according to hypothesis (e2) in Definition 3.6, if> E w W, whence if> = wu. for some u: E W: lim Ph’uh’

h’-+O

= wu.,

(weak topology of F).

(3.13)

Let us show that u; = u. For a fixed v E W, we write (3.6) with vh = ’hv and then take the limit with the sequence h’ which gives, by using (3.7), (3.8), and (3.13): ah(uh"h v)

=
Finally a(u.,v)

=
and because v E W is arbitrary, U. is a solution of (3.3) and thus U. = u. One may show in exactly the same way, that from every subsequence

47

Discretization of the Stokes equations

Ch. I §3

of Phuh’ one can extract a subsequence which converges in the weak topology of F to wu. This proves that Phuh as a whole converges to in the weak topology, as h -+ O.

wu

Proof of the Strong Convergence. Let us consider the expression

or

By (3.7), (3.8), and (3.9), lim 0h(uh,uh)

h-+O

= (Q,u)

Finally lim X h = -ctu, u) + (Q, u) = 0,

h-+O

(3.14)

according to (3.3) (when v = u). With (3.4) and (3.11) we now get

o ~aolluh-rhulI~

~Xh’

o~

F ~

II Phuh-Phrhuli

2

whence 2 Co

-

ao

Xh-+O.

Using now condition (Cl) of Definition 3.6 and

we see that this converges to 0 as h-+O. The theorem is proved. Remark 3.3. As announced in Remark 2.2, point (ii), Theorem 3.1 is applicable to the Galerkin approximation of (3.3) used in the proof of Theorem 2.2. One takes Wh = Wm , V h = mE X, X =J!I: and as in the

48

Ch. I, § 3

The steady-state Stokes equations

example at the end of Section 3.2 one obtains a Galerkin approximation of W. With

Theorem 3.1 is applicable and shows that um converges to strong topology of W as m~oo.

U

in the

Remark 3.4. If {Wih h" i" N(h) constitutes a basis of Wh , then the approximate problem (3.6) is equivalent to a regular linear system for the components of uh in this basis; i.e., if N(h)

uh

= Fl ~ ~ih

wih

N(h)

~ ~mah(wih,Wjh)=(~,W]1,>’

1=1

1 <.;<.N(h).

(3.15)

The solution of (3.15) is obtained by the usual methods for algebraic linear systems. When a basis of Wh cannot be easily constructed ( and this happens sometimes for the Stokes problem), some special method must be found to actually solve (3.6).

3.3. Approximation by finite differences We study the approximation by finite differences of the space Hb(n), then the same for the space V, and finally the approximation of Stokes problem by the corresponding scheme. The approximation of V considered here will be denoted by (APXl).

3.3.1. Notation When working with fmite differences, h denotes the vector-mesh, h= (hI’ ... , h,,) where hi is the mesh in the xi direction and thus

for some strictly positive numbers hP; hence

49

Discretization of the Stokes equations

Ch. I, §3

n

J( =

l’ (0, h9J• 1=1

(3.16)

We are interested in passing to the limit h~O. For all h E J( we define: (i) 11; is the vector hie;, where the i th coordinate of e i is 5i/ = the Kronecker delta. (ii) Rh is the set of points of tRn of the form i l hI + ... + inhn, in which the i; are integers of arbitrary sign Vi E~). (iii) 0h (M), M = (Ill’ ... , Iln ), is the set n IT i=1

(

hi

Il’ - 1

2’

~hi

Il’ + 1

2

)

and is called a block. (iv) 0h (M, r) is the set U 1< i < n -r
a

0h(M+- h;);

2

of course 0h (M, 0) = 0h (M). (v) whM is the characteristic function of the block 0h (M). (vi) 5ih (or 5; if no confusion can arise) is the finite difference operator (3.17) Ifi = VI ,... , in) E N"is a multi-index, then dh (or simply 51) will denote the operator s::i

U

= s::iII U

s::in

... Un .

(3.18)

(vii) With each open set 0 of tRn and each non-negative integer r we associate the following point sets o

0" = {ME Rh , 0h(M, r) C O}

(3.19)

50

The steady-state Stokes equations

Ch. I, §3

(3.20) (viii) Sometimes we will use other finite difference operators such as ’Vjh and Vjh(also denoted ’Vj and Vj);

=

rp(x+h j ) - rp(x) h.

=

rp(x) - rp(x-h h.

r = 0

r= I

’Vjh rp(x) ’Vjh rp(x)

(3.21)

I

j)

(3.22)

I

Examples of Sets uh(M, r) in the Plane.

3.3.2. External approximation of HA(O)

HA

Let 0 be a Lipschitz open bounded set in lH,n. Let W = (0), F = £2 (o)n + 1 equipped with the natural Hilbert scalar product, and let be the mapping

w

(3.23) from Winto F. It is clear that IIwu IIF

= [u]H~(n)

so that w is an isomorphism from Winto F. Space Wh : With the preceding notation, Wh will be the space of step functions uh(x)

=

L:0

1

Meoh

uh(M)whM(X), uh(M) E lH,n. o

(3.24)

The functions whM for M E 0h are linearly independent and span the whole space Wh; they form a basis of Who The dimension of Wh is n

Discretization of the Stokes equations

Ch. I, §3

51

times the number N(h) of points ME n1; Wh is finite dimensional. This space is provided with the scalar product [uh’ vh]h

=

J

n uh(x)vh(x)dx

+ (3.25)

which makes it a Hilbert space. The functions Uh and 8iuh’ I ~ i ~ n, have compact supports in n, by the definition of Wh and the set nl. Hence they will be considered as vector functions defined on n or on \Rn . Operators Ph’ The prolongation operators Ph are the discrete analogue ofw:

(3.26) The norm of Ph is exactly one,

and they are stable. Operators rh : As a consequence of Proposition 3.1 we need only define the operator rh on if’" =!fj (n) which is a dense subspace of Hb (n); we put (rhu)(M)

= u(M),

o

V M E nl,

V u E!fj(n)

(3.27)

which completely defines rhu E Wh . Proposition 3.2. The preceding ex ternal approximation of stable and convergent.

Hb (n) is

Proof. The approximation is stable since the prolongation operators are stable. We must now check conditions (Cl) and (C2) of Definition 3.6. Lemma 3.1. Condition (Cl ) is satisfied: VuE !fj(n), rh u -+ u in L2 (n),

(3.28)

52

The steady-state

Stokes equations

Ch. I, §3

2

(3.29)

0i’h U ~ DiU in L (0),

as h rr I),

Proof. Let U E ~(O) and let h be sufficiently small for the support of to be included in the set

U

(3.30) For any M E uh

=’h u )

o

01,

for any x E

IUh (x) - u(x)1

= lu(M) -

0h (M),

u(x)1

the Taylor formula gives (with

~

cl (u)IM -xl

~

cI (u)

2

Ihl

where Cl (u)

= Sup

Ihl= (

) 1/2

n

~h~ i=1

(3.31)

[grad u(x)1

xEn

I

(3.32)

.

Then

Ct(u)

x

Sup IUh (x)-u(x) I ~ --Ihl . n (h) 2

(3.33)

E

On the set O-O(h), lu(x)1

~ Cl (u)d(x,

IUh(X)-u(x)1 ~Ct

I’) , (u)d(O(h), f).

(3.34)

Hence Sup IUh (x)-u(x)1 ~

XEn

cl (u) {1ftl+d(O(h),

2

I’j}

(3.35)

which converges to 0 as h ~ 0; uh converges to u in L" (0) and then in L2 (0) since 0 is bounded. o To prove (3.29), we use again Taylor’s formula: V ME V x E aheM),

n1,

Discretization ofthe Stokes equations

Ch. I, §3

S3

(3.36)

16iUh (X) - DiU (x)1

= I! h; [uh(M+

!h,)-Uh(M - -.!.h.)] - D•u(x)1

2

2

I

I

I

~ c2(u)lhl,

where c2 (u) depends only on the maximum norm of the second derivatives of u. On the set O-O(h),

and then on the whole set 0 16 i Uh (x) - Diu(x)1 ~

G2 (u)lh 1+ ci (u)d(O(h), I’).

(3.37)

This converges to 0 as h -. 0, and shows the convergence of 6iuh to Diu in the uniform and L 2 norms. Lemma 3.2. Condition (e2) is satisfied.

Proof. Let there be given a sequence uh’ E Wh " h’-. 0, such that Ph’ Uh’ converges to

lim uh’

h’-+O

=
in the weak topology of L 2 (0);

¢O

lim v-»

=

h’-+O

O’Uh’ I

¢i’

1 «:

~

n,

(3.39)

in the weak topology of L2(~n); heregmeans the function equal tog in 0 and equal to 0 in the complement of O.

Ch. I, §3

The steady-state Stokes equations

54

A discrete integration by parts formula gives (3.40) for each o E !?d(tR n ). As h’~ 0, the left-hand side of (3.40) converges to

J

(j)i (X)O(X) dx;

/Rn

the right-hand side converges to

since 5ih 0 converges to Dio in L 2 (tRn ) as shown in Lemma 3.1. So

J

(j)i(X)O(x) dx

/Rn

=- (

J/Rn

(j)o(x)Dio(x)dx, V 0

E~(tRn)

which amounts to saying that (j).= D•A. o I ~ I 1'1',

i

~

n'

(3.41)

in the distribution sense. It is clear now that ¢o E HI (tRn ) , and since ~o vanishes in the com› (n). Thus (j) E W; plement of n, (j)o belongs to

Hb

w

(3.42)

3.3.3. Discrete Poincare inequality The following discrete Poincare inequality (see (l.~)) will allow us to equip the space Wh in (3.24) with another scalar product ))h’ the discrete analogue of the scalar product «.,.)) (see (l.ll)).

«.,.

Proposition 3.3. Let n be a set bounded in the xi direction, and let uh be a scalar step function of type (3.24) (with uh(M) E tR). Then

Ch. r, § 3

ss

Discretization of the Stokes equations

(3.43) where Qis the width of n in this direction.

Proof. For the sake of simplicity we take i support, for any M E Rh ,

= I. Since uh

has a compact

00

uh(M)2 =

L

{[uh(M - jh r)]2 - [uh(M - (j

j=O

+ l)hd 2}

00

=h l

L

j=O

[OrhUh(M-(j+!)hr][Uh(M-jilr)+

uh(M - (j+ I)h r ) ] +00

uh(M)2

~I

= hi j=~oo

IOlhuh(M - (j +

I

"2 )h l ) ! [Iuh (M

- jhl)1

+ IUh(M - (j+ l)h l ) ! ] .

(3.44)

The sums are actually finite. Now for any i E Z, uh (M+ih l)2 is major› ized by exactly the same expression I. There are less than Q/h 1 values of i such that uh(M+ih r) =1= 0 since the xl -width of n is less than Q. Hence +00

L

i=-oo

Uh(M+ihl)2~-.i1

LetYh (M) denote the tube (3.45) as follows:

J

uh (x)2dx

(3.45)

hi

u7=":.

= (hi’"

00

ah (M+ih l). We can interpret +-

hn )

:Fh (/tf)

L

i= -

uh (M+ih l)2 GO

~ Q(h l ... h n ) .L hi

= QJ IUh

:Fh (M)

(x-~hl

IOlhUh(X)I{IUh(X+~hl)l+ )I}dx.

56

The steady-state Stokes equations

Ch. I, § 3

We take summations of the last inequality for all tubes obtain

J

~n

Uh(x)2dx:e;;;; Q

f

J~n

18 1h uh(x)1

(IUh(X+~hl IUh

(X-~hl

§h (M)

and

)1+ )I)dx.

Applying Schwarz’s inequality, we obtain

lu hl 2 = (

J~n

Uh(x)2dx

and (3.43) follows. Proposition 3.4. Let n be a bounded Lipschitz set. If we equip the space Wh with the scalar product (3.46) we have again a stable convergent approximation of H~(n).

Proof. The prolongation operators are stable as a consequence of Proposition 3.3. Remark 3.5. Using the difference operators Vih’ or Vih’ or even any "reasonable" approximation of the differentiation operator a/axj> one can define many other similar approximations of the space H~(n). The modifications arise then in (3.25), where 8ih is replaced by Vih , ... , in the set of points nh which must be suitably defined and in some points of the proof of Lemma 3.1 and 3.2. The same Poincare inequality is valid for the operators V ih and Vih but not for more general operators.

Discretization of the Stokes equations

Ch. I, § 3

57

Remark 3.6. When n is unbounded, one can define an external approximation of Hij(n) with a space Wh consisting of either: AMwhM’ which have c9mpact support (we - step functions !:’MEn~ restrict the sum to a finite number of points M E n~), - or step functions !:,AM whM for the M in the intersection of n~ with some "large" ball: Ixl :s;;;;;. p(h), where p(h)-++oo as h-+O. In the second case Wh is finite dimensional but not in the first case. Without any modification for the other elements of the approximat› ion, it is clear that we obtain a stable convergent approximation of Hij(n) for an unbounded locally Lipschitzian set n. The discrete Poincare inequality is available if n is bounded in one of the directions xl’ . . , X n . 0

3.3.4. Approximation of the space V (APX1).

Let n be a Lipschitzian bounded set in lRn and let ’Ybe the usual space (1.12) and V its closure in Hij(n). We define now an approximation of V using finite differences (which will be denoted by (APXI )). Let F = L2 (n)n+l equipped with the natural Hilbertian scalar product and let us define the mapping w E.?( V, F):

It is clear that

so that wis an isomorphism from V into F. Space Vh We take in the following Vh as the space Wh: Vh is the space of step functions (3.47)

which are discretely divergence free in the following sense: n ’"’

LJ ’ilih Uih(M) = i=l

0, ¥

M E

nh1 0

(3,48)

The steady-state Stokes equations

58

Ch. I, §3

which amounts to saying n

Li=1 Vih Uih(X) = 0,

Vx E

o (h).

(3.49)

No basis of Vh is available; it is clear that Vh is a finite dimensional space with dimension less than or equal to nN(h)-N(h) = (n-l )N(h) since all the functions in (3.47) form a space of dimension nN(h) and there are at most N(h) independent linear constraints in (3.48); it is not clear whether the constraints (3.48) are always linearly independent so that Vh is not necessarily of dimension (n - I)N(h). The space ~ is equipped with one of the scalar products «Uh’ vh ))h

[u" "" I, =

=~ ( 1=1

J

In

OjUh (x)

n u,(x)v,(x)dx

OjUh(x)dx

(3.50)

+ «u" v,»"

(3.51 )

Because of Proposition 3.3 (discrete Poincare inequality), with either one of these scalar products is a Hilbert space. Operators Ph’ These are the discrete analogue of w:

v"

equipped

(3.52) These operators are stable since by (3.43) (3.53) Operators rho We define rh V M = (ml hi’ ... , m"hn ) E Uih

(M)

U

only for u E f, a dense subspace of V;

nl, (rhu)(M) is defined by

= ;th component of uh

= the average value of ui on the face 1 xi = (mi - 2" )h i of 0h (M).

(3.54)

This complicated definition of rhu is necessary if we want uh to belong to ~; actually

59

Discretization of the Stokes equations

Ch.I,§3

Lemma 3.3. For each u Ef, rh u E Ji},. Proof. We have V,.u,. (M) = hi .1. h,

{l

E, u,(x)dx

-

~ Ei u,<x)dx

}

where ~j and ~’j respectively are the faces x, = (m, + 1/2)h j and x, = (m, - 1/2)hj of uh (M). This gives also "Vjh Ujh (M)

=

1

hI’"

n;

l

,u .vdr,

};jU };j

where v stands for the unit vector normal to the boundary of uh (M) and pointing in the outward direction. Then, for each M E

–

j=1

"Vjh uih (M)

n1,

=

1 hI’"

hn

f Ja

u ,

= (by

Stokes formula)

=

1

hI’"

n;

vdr

0h(M)

1

div u dx

= 0,

since div u

= O.

h(M)

Conditions (3.48) are met. Proposition 3.5. The preceding external approximation of V is stable and convergent. Proof. Stability has been shown already. The proof of condition (Cl) is very similar to the proof of Lemma 3.1 and we do not repeat all the details; for example, for x E uh (M), 0 MEUh1 , IUjh (x) - Uj(x)1

where

~

= !Ujh (~)

is some point of the face

- Uj(x)l, Xj

= (m,

- 1/2)h j of uh (M), and hence

60

The steady-state Stokes equations

Ch. 1, §3

and (3.35) is replaced by Sup 1Uh(X)-U(X) I ~

xEn

C’l

(u){lhl + d(n(h), f)}.

(3.55)

The proof for condition (C2) is similar to the proof of Lemma 3.2; more precisely, the same proof as for Lemma 3.2 shows that if Ph,Uh’ -+ rP in the weak topology of F,

as h’ -+ 0, then rP = wu = (u, D 1 U, ... , Dnu), where U E HA (n). Because of Theorem 1.6, proving that U E V now amounts to proving that div U = O. This follows from (3.49) as we will now show. Let a be any test function in ~(n), and let us suppose that h is small enough for the support of a to be included in n(h); then (3.49) shows that

f (i In

(VihUih)(X»)

1=1

or

a(x)dx

=0

(3.56) It is easy to check the discrete integration by parts formula

f

J~n

(V ih O)(x).a(x)dx

=-

f

J~n

O(x)(Vih a)(x)dx;

(3.57)

= O.

(3.58)

then (3.56) becomes

Jf ~n

i

i=l

[Uih(X).(Viha)(x)]dx

With a proof similar to that of Lemma 3.1 (based on Taylor’s formula) we see that (3.59)

in the (uniform and) £2 norm. Since uih converges to ui for the weak topology of £2 (lR n ), letting h -+ 0 in (3.58) gives the result

( [i

J ~n

i=l

Ui(X).Dia(X)] dx

= o.

(3.60)

en.

I, §3

61

Discretization of the Stokes equations

The equality (3.60), true for any a E then ¢J = WU, with u E V.

~(n),

implies that div u

= 0 and

Remark 3.7. The Remark 3.5 can be extended to the present case with, however, one restriction: condition (3.48) - (3.49) in the definition of the spaces Vh cannot be replaced by similar relationships involving other finite difference operators; for example, it seems impossible to replace (3.49) by n

L: BO h uOh (x) = 0,

i=1

I

I

(3.61)

since (3.61) requires many more algebraic relations than (3.49) and probably too many relations (in which case ~ = {O}).

Remark 3.8. When Q is unbounded one can define, using the methods mentioned in Remark 3.5, a stable and convergent external approximat› ion of the space W introduced in (2.31). 3.3.5. Approximation o[ Stokes problem

Using the above approximation of V and the results of Section 3.2, we can propose a finite difference scheme for the approximation of Stokes’ problem. Let us take, for (3.6), (3.62) (3.63) where Vh and

«•,•)hare the space and scalar product just defined, and

u and [ are given as in .section 2.1.

The approximate problem corresponding to (2.6) is then: To find uh E Vh such that

(3.64)

Proposition 3.6. For all h E jf the solution uh o[ (3.64) exists and is unique; moreover as h-+-O the solution uh 0[(3.64) converges to the

62

The steady-state Stokes equations

Ch. I, §3

solution u of (2.6) in the following sense:

(3.65) (3.66) Proof. We have only to check that Theorem 3.1 is applicable. Condition (3.4) is obvious (ao = 1); for (3.5) we notice that I(Qh’ vh>1 =

l(f, vh)1

~

lflolvh I

~ (by the discrete Poincare inequality)

Hence

(3.67) and (3.5) is satisfied. For (3.7) - (3.8) we notice that Ph Vh -+ WV weakly (respectively Ph Wh -+ WW strongly)

means Vh -+ v, and fJih vh -+ D;v in L2 (0) weakly,

(respectively

and it is clear that this implies,

Approximation of the Pressure. We want to present the "approximate" pressure which is implicitly

Ch. I, §3

Discretization of Stokes equations

63

contained in (3.64) as well as the exact pressure p is implicitly contained

in (2.6).

The space Vh in (3.47) is a subspace of the space Wh in (3.24); namely, the space of vh E Wh satisfying the linear constraints (3.48). The form vh -+ v«uh’ vh))h - (f, vh) appears as a linear form defined on Wh which vanishes on Vh . Hence introducing the Lagrange multi› pliers corresponding. to the linear constraints (3.48) we find, with the aid of a classical theorem of linear algebra, that there exist numbers "’lw E dt, M E o.k, such that the equation n

V«uh,vh ))h - (f, vh)

= MELoAM ~ oj, 1=1

(Vih Vih (M)),

(3.68)

holds for each vh E Wh . Let us now introduce the operator Dh E £(Wh , L2 (Q ) ) : n

DhVh(X)=~

1=1

(3.69)

Vihvih(x), V VhEWh’

its adjoint Dh’ E £(L 2 ( Q ), Wh ) is defined by (Dh’0’ vh)

= (0,

Dhvh)’ V vhE Wh, VO E L 2 (Q ).

Let 7Th be the step function which vanishes outside Q(h) = and which satisfies

(3.70) UMEnl 0h(M), h

(3.71 )

Then (3.68) can be written as

or equivalently, (3.72)

Taking successively vh (3.72) as

= whMej

o

for ME Q}" j

= 1, ... , n, we

can interpret

Ch. I, §3

The steady-state Stokes equations

64

n

(3.73) where

Vh (ll’h (M)) is the vector (V 1 h ll’h (M), ... , V nh ll’h (M)) and fh(M)

=h

1h 1’"

( n

(3.74)

f(x)dx.

JUh(M)

The equations (3.73), and (3.75) are the discrete form of the equations (2.7) - (2.8); - Dhll’h is the "approximation" of grad p, - DC is a discrete gradient operator. Remark 3.9. As indicated in Remark 3.4, the solution of (3.64) is not easy since we do not know any simple basis of Vh One possibility for solving (3.64) would be to solve the system (3.73), (3.75), which is a linear system with unknowns U1h (M), ... , Unh (M), ll’h (M), ME

o 1 cu:

This system has a unique solution up to an additive constant for the ll’h (M); this non-uniqueness makes the resolution of this linear system difficult; moreover, the matrix of the system is ill-conditioned. More efficient ways for actually computing the approximate solution will be given in Section 5.

The Error. Let us suppose that the exact solution satisfies u E~3 p E
(Q) and

n

(3.76) where rhu is the function of Vh defined by (3.54) and where h (M) is a "small" vector:

Ch. I, §4

Discretization ofStokes equations (II)

65

(3.77)

depending only on the maximum norms of third derivatives of u and second derivatives of p. Let us denote by 1T’h the function ’EMEn~ P(M)whM’ Then, the equality (3.76) is equivalent to

c(U, p)

(3.78) for each vh E Wh (space (3.24)) and implies (3.79) for each vh E Vh . Subtracting this equality from (3.64) we obtain (3.80) and then taking vh = uh - rhu, we see that

Hence we find the following estimates for the discrete error: (3.81 ) (3.82) §4. Discretization of Stokes equations (II)

We study here the discretization of Stokes’ equations by means of finite element methods. The results are less general here than in the previous section and vary according to the dimension. We successively consider conforming finite elements which are: piecewise polynomials of degree two in the two-dimensional case (Section 4.2), piecewise polynomials of degree three in the three-dimensional case (Section 4.3), and piecewise polynomials of degree four in the two-dimensional case (Section 4.4). Finally we consider an external approximation by non-

66

The steady-state Stokes equations

Ch. I, §4

conforming finite elements (any dimension) in Section 4.5. 4.1. Preliminary results

We will have to work with piecewise polynomial functions defined on n-simplices, For that purpose, we recall here some definitions and introduce some notations adapted to the situation. Barycentric coordinates Let there be given in lRn , (n+ 1) points AI’ ... , A n + 1 ,o ) with co› 1 ~ i ~ n+ 1, which do not lie in the same hyperplane; ordinates al [s this amounts saying that the n vectors A 1 A 2’ ... , Al A n+ 1 are independent, or that the matrix :»>

... ~,n+l

a2,1 ~,2

.xl

=

(4.1)

1

is non-singular. Given any point P E lRn , with coordinates xl’ ... , x n ’ there exist (n+ 1) real numbers

such that

n+l

(4.2) n+l

Li=1

A•= 1 I

,

(4.3)

(1) In this section dealing with finite elements, the capital letters A, B, M, P, ... , (sometimes with subscripts) will denote points of the affine space Rn . Couples of such letters, like AB, ... , denote the vector of R n with origin A and terminal point B.

67

Discretization ofStokes equations (II)

Ch. I, §4

where 0 is the origin of IR n. To see this it suffices to remark that (4.2) and (4.3) are equivalent to the linear system Al

d An

xl

=

(4.4) Xn

An+l

which has a unique solution since the matrixd is non-singular, by hypothesis. The quantities Ai are called the barycentric coordinates of P, with respect to the (n+ 1) points AI> ... , A n+ l . As a consequence of (4.4), the numbers Ai appear as linear, generally non-homogeneous functions of the coordinates xI> ... , x n of P: n

Ai=? bijxJ.+b i n+l’ I <.i<.n+ I, J=l’ ,

(4.5)

where the matrix.9J = (b i j ) is the inverse of the matrix.s/. It is easy to see that the point 0 in (4.2) can be replaced by another point of IRn without changing the value of the barycentric coordinates; hence n+l

L: i=l

A•PA• = O. I I

(4.6)

Clearly the barycentric coordinates are also independent of the choice of a basis in IRn. The convex hull of the (n+ I) points Ai is exactly the set of points of IRn with barycentric coordinates satisfying the conditions: (4.7)

This convex hull.v’Is the n-simplex generated by the points Ai, which are called the vertices of the n-simplex. The barycenter G of.9’ is the point of 9’whose barycentric coordinates are all equal and hence equal 1/n+ I. An m-dtmensional face of9’is any m-simplex (I <. m <. n -I) generated by m+ I of the vertices of 9’(of course these vertices do not

The steady-state Stokes equations

68

Ch. I, §4

lie in an (m-l )-dimensional subspace of dln ) . A l-dtmensional face is an edge. In the two-dimensional case (n = 2) the 2-simplices are triangles; the vertices and edges of the simplex are simply the vertices and edges of the triangle. In the three-dimensional case, the 3-simplices are tetrahedrons, the two-faces are the four triangles which form its boundary. An interpolation result

Proposition4.1. LetA 1 , ... ,A n+ 1 be (n+l) points ofdln which are not included in a hyperplane. Given (n+ I ) real numbers cx1’ , CXn+ l’ there exists one and only one linear function u such that u(A i ) = CXj, I <. t <. n+ I, and n+1 U (P)

=

Lj=l

CXjAj(P), V P E dln ,

(4.8)

where the Aj(P) are the barycentric coordinates of P with respect to A 1 , ... ,A n+ 1

Proof. Let n

= /=1 L ~/xi + ~n+1’

u(x)

be this function. The unknowns are ~1’ . , ~n+1 following equations asserting that u(A j ) = CXj:

which satisfy the

n

L ~/:L

1=1

j+

II’

~n+1 = I <. i <. n+l.

The matrix of the system is the transposed matrix t.91 of .91 and thus the function u exists and is unique. It remains to see that (4.8) is the required function; actually

since A{(AI) = 6{1’ the Kronecker delta, for each i and i.

Ch. I, §4

69

Discretization ofStokes equations (II)

Remark 4.1. Higher order interpolation formulas using the bary› centric coordinates will be given later (see Sections 4.2, 4.3, 4.4). Differential properties We give some differential properties of the Ai considered as funct› ions of the cartesian coordinates xb ... , x n ’ of P; here we denote by D the gradient operator D = (Db"" Dn ).

Lemma 4.1. n+l

2, DA;=

0

DAi (P)

PA

i=l

(4.9) j

= 5i/- Ai (P),

I <, i, i <, n+ 1.

(4.10)

Proof. The identity (4.3) immediately implies (4.9). According to (4.5)

ex.

1

aXk

-

--

b i.k

1, ’

I <, l <, n+ I,

I <, k <, n,

and then

n

=k=l :E b, ’ kak , j+ = 5 i/ -

b, n+l - Ai

Ai;

for the last equality we note that~=d-l. Lemma 4.2. Let !/be an n-simplex with vertices AI’ ... ,An+ l and let p’ be the least upper bound of the diameters ofall balls included in

Thesteady-state Stokes equations

70

Ch. I, §4

!/. Then (4.11)

where IDAil is the Euclidian norm of the constant vector DAi’ Proof. We have (4.12)

IDAil = DAi• x,

where x is the unit vector parallel to DAi; but we may write x=lPQ pi ,

whereP and Q belong tog; denoting by 1J.1’ , IJ. n+ 1’ the barycentric coordinates of Q with respect to A l’ ... , A n+ 1 , we have, because of (4.2) - (4.3), n+1

PQ=

L

/=1

IJ.;PA 1,

n+1

L

1=1

IJ.I= 1.

Then

n+1

=

.L p’

L

1=1

IJ..DA . PA 1 J I

= (according

to (4.10))

n+l

= ~pI 1=1 L

1J./(6.’j- Ai)

I

= .L (IJ..p’ I

Ai)’

Since P and Q belong to g, 0 <;; Ai<;; I, 0 EO;; lJ.i <;; I for each t,

I

i

~

~

71

Discretization of Stokes equations (II)

Ch.l, §4

n+ I, and then -I

~

u i-X i ~ I, so that

IDX•.xl ~- 1

p’

I

and (4.11) follows.

Norms ofsome linear transformations Let.y’ andY be two n-simplices with vertices AI’ ... , A n + 1 , and AI"’" A n + 1 We denote by p (resp. p’) the diameter of the smallest ball containing S"’{resp. the diameter of the largest ball contained in Y); p, p, has a similar meaning. We can suppose that, up to a translation, Al = Al = 0, the origin in tR n , and we denote then by A the linear mapping in tR n such that (4.13) The norms of A and A-I can be majorized as follows in terms of ,

-

AI

p, p,p, p :

Lemma 4.3.

Proof. As in the proof of Lemma 4.2, let x be any vector in tR n with norm I. Then

x=~PQ, p

,

where P and Q belong to //. It is clear that Ax= ~PQ, p

,

where ji= AP, (J = AQ. But

P and Qbelong to 9’ too, since

n+l

OP=

~1=1

XiOA i’ 0

~ Xi~

1,

The steady-state Stokes equations

72

implies

§4

n+1

~ Ai

ADP =

1=1

DAb

so that the barycentric coordinates of P with respect to AI’ ... , An + 1 are the same as the barycentric coordinates of P with respect to AI, ... , A n + 1 Hence !PQI E:;; p, and

IAxl E:;; .E. . p’ The first inequality (4.14) is proved. The second inequality is obvious when interchanging the role of9 andY. 0 When handling divergence free vector functions, the following lemma will be useful: Lemma 4.4. Let x-+u(x) be a divergence free vector function defined on 9 (or on 6l~) and let i-+u(i) be defined on!? by

(4.15) Then Uis a divergence free vector function too. Proof. Let aUi

aXi

(cx ii) -

(x)

and

(f3k2)

= ~a"c: aXi

=2

2

CX

i2

2,k

denote the elements of A and A-I. Then CX i 2 u 2

(A- 1 X)

aU 2 a:k

aXk

aXi

and

aUk

=2 k

aXk

(A- 1i

) = O.

73

Discretization ofStokes equations (In

Ch. I, §4

Regular triangulations ofan open set n Let n be an open bounded set in tR n LetJ;; be a family of n-simplices; such a family will be called an admissible triangulation of n if the following conditions are satisfied:

=

n(h)

U .9’ en

[/E

(4.16)

9"h

o

0

If9’and.9"E9h, then9’n 9"=,p, (where9’is the interior ofY’(1» and, eitherY’nY" is empty orY’nY" is exactly a whole m-face for both.v’ andY" (any m, 0 OS;;; m OS;;; n-l).

(4.17)

We will denote by {Jh}hE:K the family of all admissible triangulations of n; with each admissible triangulation.!Jj; we associate the following three numbers: (4.18) (4.19)

a(h)=

Sup

(4.20)

Py

YE..’Th p~

where, as before, p = py is the diameter of the smallest ball containing .9’, and p’ = p’y is the diameter of the greatest ball contained in.9’. For finite element methods, we are concerned with passage to the limit, p(h )-+0. It will appear later that some restrictions on a(h) are necessary to obtain convergent approximations. A sub-family of the family of admissible triangulations {g;;} E ;K will be called a regular triangulation of n if a(h) remains bounded as p(h)-+O, a(h) OS;;; a

< +00,

p(h)-+O

(4.21 )

and n(h) converges to n in the following sense: (l) i.e., the points of.’l with barycentric coordinates, with respect to the vertices of 9’, satisfying

0< Ai< 1, 1 «t

« n+l.

74

The steady-state Stokes equations

For each compact set K CO, there exists 6 = 6 (K) > 0 such that p (h) <; 6 (K)" O(h) ::> K.

Ch. I, §4

(4.22)

JCa will denote the set of admissible triangulations of 0 satisfying (4.21) and (4.22).

Remark 4.2. In the two dimensional case the 2-simplex is a triangle and it is known that 1

2 tan ~ 2 where 8 is the smallest angle of..9". The condition (4.21) thus amounts to saying that the smallest angle of all the triangles..9"E5; remains bounded from below: (4.23) Our purpose now will be to associate to a regular family of triangula› tions {9’j; }h E;c a of 0, various types of approximations of the function spaces with which we are concerned. 4.2. Finite elements of degree 2 (n = 2) Let 0 be a Lipschitzian open bounded set in lR n We describe an internal approximation of Ha(O) (any n) and then an external approx› imation of V (n = 2 only). The approximate functions are piecewise polynomials of degree 2. 4.2.1. Approximation of Ha (0) Let!lh be any admissible triangulation of O. Space Who This is the space of continuous vector functions, which vanish out› side O(h) O(h)

= U

YE!Tj,

.9’

(4.24)

Discretization ofStokes equations

Ch. I, §4

7S

(II)

and whose components are polynomials of degree two(l) on each sim› plex f/E!Th ’ This space Wh is a finite dimensional subspace of nb(o). We equip it with the scalar product induced by nb (0): (4.25)

A basis of~. Iff/is an n-simplex we denote, as before, the vertices off/ by AI, ... , A n + 1 ;we denote also by A ij the mid-point of AiA j. Firstly we have Lemma 4.5. A polynomial of degree less than or equal to two is unique› ly defined by its values at the points Ai’ Ai;, 1 ~ i, j ~ n+ 1 (the vertices and the mid-points of the edges ofan n-simplex J’). Moreover, this polynomial is given in terms of the barycentric co› ordinates with respect to A I’ ... , A n + l , by the formula: n+1 q,(x)

=~

/=1

(2(Ai(X))2 - Ai(x))q,(A i) n+1

+ 2 .~

/,1=1

Ai(X)Aj(X)q,(A il,).

(4.26)

i*j

Proof. Let us show first that (4.26) satisfies the requirements. The function on the right-hand side of (4.26) is a polynomial of degree two since the Ai (x) are linear non-homogeneous functions of XI’ .. , x n (see (4.5)). Besides, if 1/1 (x) denotes this function, I/I(Ak )

1/1 (A k2)

= q,(A k) since

Ai(A k )

= q, (A k2) since

= ~ik

Ai(A k2)

= ~’k/ 2+ 8’2/

Thus 1/1 is as required. (1) Roughly speaking, a polynomial of degree two means a polynomial of degree less than or equal to two.

Ch. I, §4

The steady-state Stokes equations

76

Now, a polynomial of degree two has the form n

~(x)=ao+~(IY•x.+~.x7)+i=l

n -"

"

~

i,/=l

i
(4.27)

a••x•x•, ’/ , /

ao,

and ~ is defined by (n+ I )(n+2)/2 unknown coefficients 04, ~i’ 04/• There are (n+l) points At, n(n+l)/2 points A i/ , and hence the con› ditions on~: ~(Ai)

= given,

~(Ai/)

= given,

(4.28)

are (n+l)(n+2)/2linear equations for the unknown coefficients. Accord› ing to (4.26) this system has a solution for any set of data in (4.28); thus the linear system is a regular system(l), and the solution found in (4.26) is unique. 0 Now let us denote by ’lIh the set of 1;ertices and mid-edges of the n-simplices !/E 9’h. We denote also by dIJ h those points of dIJ h which belong to the interior of fl,(h). According to the preceding lemma there is at mostoone function aj, in Wh which takes given values at the points A E !/h’ Actually we have more. Lemma 4.6. There exists one and onlY one function "h in Wh which takes given values at the points ME 1/h’

Proof. We saw that such a function is necessarily unique. Now, by Lemma 4.5, there exists a function uh whose components are piecewise of degree two, which takes given values 0 at the points polynomials o M EdlJ h and which vanishes at the points M EdlJ h -dIJ h and outside fl,(h). Wejust have to check that this function is continuous. On each (n - 1)› face f/’ of a simplex !/E $it’ each component uih of uh is a polynomial of degree two which has two (perhaps different) values uth and "th’ But uth and uih are polynomials of degree less than or equal to two in (n - I) variables, which are equal at the vertices and the mid-points of the edges of f/’; Lemma 4.5 applied to an (n - I)-dimensional simplex shows that uth = Uih on f/’. Therefore uh is continuous, and uh belongs to Wh: 0 (1) We use the well-known property that, in finite dimensional spaces, the linear operators which are onto, are one-to-one and onto.

Discretization ofStokes equations (II)

Ch. I, §4

77

Repeating the argument of the preceding proof, we see that there exists a unique scalar continuous function, which is a polynomial of degree two on eacho simplex g’Ec1h, and which takes on given values at the points M El7It h, and which vanishes outside f1,(h). Let us denote by whM the function of this type defined by o

0

V PE’l/h’ P*M, (MEl/h)’

(4.29) Finally, we have o

Lemma 4.7. The functions whMei’ M El7It h, i = I, ... , n, form a basis of Wh , an~ the dimension of Wh is nN(h) where N(h) is the number of points in l/h’

Proof. These functions are linearly independent and, clearly, each function aj, E Wh can be written uh(x)=

2:

n 0

ME 9th

~uih(M)eiWhM(X)

1=1

or (4.30) Operator Ph .

The prolongation operator Ph is the identity, (4.31)

The Ph’s have norm one and thus are stable. Operator rh .

We define rh" for u E !’J(n); we set: (4.32)

b

Proposition 4.2. The preceding internal approximation ofH (n) is stable and convergent, provided h belongs to a regular triangulation j{Q ofn.

Thesteady-state Stokes equations

78

Proof. We only have to prove that for each u E

Ch. I, §4

!!} (n),

as p(h)-+O, hE XC/If h is sufficiently small, n(h) contains the support of u, and then as shown by the next lemma

(4.33) and the result follows. Lemma 4.8. Let 57 be an n-simplex,
for 1 <:i,j<:n+1. Then, we have

Sup I
(4.34)

xEg>

Sup I ~ (x) -

XE.’!’

oXi

where c (
...

.1t (x)] <: c(
3

pg p~

on the maximum norm of the third derivatives

(4.35) of
This lemma is a particular case of general theorems concerning poly› nomial interpolation on a simplex in connection with finite elements.

Polynomial Interpolation on a Simplex.

Let9’be an n-simplex and let t! be a finite set of points of 9’having the following property: for any family of given numbers ’YM E IR, MEg’, there exists a unique polynomial p of degree less than or equal to k such that p(M) = ’YM, V M Eft.

(4.36)

Such a set 8 is called k-unisolvent by Ciarlet-Raviart [1] ; for example, according to Proposition 4.1 and Lemma 4.5, the vertices AI’ ... , A n + 1

79

Discretization ofStokes equations (II)

Ch. I, §4

of Yare l-unisolvent, the points Ai’ A ii, 1 :E;;; if i E;; n+ 1, are 2-unisol› vent. Let us denote by Pi the polynomial of degree k such that

(4.37) Then the polynomial pin (4.36) can be written as

(4.38) Now let us suppose that a function if> is given, if> E ~k+ 1 (9’), and let J be the interpolating polynomial of degree k defined by ~(M)

= if>(M), V MEg’

(4.39)

i.e.,

(4.40) Using Taylor’s formula it is proved that for any multi-index i=(h, .. •,in)with[j) =h +"’+in:E;;;k,onehas Di’i(P) = Diif>(p)

(4.41)

where Pi is some point of the open interval (Mi , P), ll, Dlln’M p lI _ ell, ""lin D ll -- D 1’" n’ i - 11’" "’nt

for MiP = (Eli’ ... , En;), £ = (£} , ... , £n)’ ’The error between if> and ’iis majorized on9’by

(4.42)

Thesteady-state Stokes equations

80

for [j] 4.1 (1).

=it + ... in = m <; k, where P.9’ and p~are

Ch.l, §4

defined in Section

This is a consequence of (4.41) and the following estimation of Pi

.

c

Sup ID’p;(x) I <; -,- for U] = m <; k.

PJ’

XEY

For the proofs of (4.41) and (4.43), the reader is referred to Ciarlet› Raviart [1], Raviart [4] and Strang-Fix [1] ; for the particular case of Lemma 4.8, see also Ciarlet-Wagshall [1] . D 4.2.2. Approximation of V (APX 2) Here n is an open bounded set in 6?2; we shall define an external approximation of the space V. Space F, Operator w. The space F is H~(n)

and W is the identity (4.44)

wu=u,V u E V;

wis an isomorphism from V into F.

Let ~ be any admissible triangulation of

n.

Space Vh’ Vh is a subspace of the space Wh previously defined. It is the space of continuous vector functions which vanish outside n(h) = U

YE!1j,

9’

(4.45)

and whose components are polynomials of degree two on each simplex 9’E~ and such that

L

div u.!Ix

=0, V YEJJ;;.

The condition (4.46) is a discrete form of the condition div u

(4.46)

= O.

(1)

The supremum in x is taken on9’; elsewhere when using this notation, the supremum is understood on the whole support of t/J.

Discretization ofStokes equations (II)

Ch. I, §4

81

The functions uh E Ji}, belong to HA (n), but not to V, Vh 4: V. We do not have a simple basis of Vh ; according to Lemma 4.7, any function uh E Vh can be written as uh =

2:.

M E’l/h

uh(M)whM

but the functions whMdo not belong to also that dim v" ~ 2N(h) - N’(h),

v".

Lemma 4.7 and (4.46) show

o

where N(h) is the number of points in 94 and N’(h) is the number of triangles .!7 E,.Jh. We provide the space Vh with the scalar product of HA (n) (as Wh ) : «uh’ Ph))h

= «Uh’

(4.47)

Ph))’

Operator Ph’

The operator Ph is the identity (recall that Vh C HA (n)). The prolon› gation operators have norm one and are thus stable. Operator rho

The restriction operators are more difficult to define because of condition (4.46) which must be satisfied by rhu, Let U be an element ofr; we set rhu=uh=U~+U~, (4.48)

where u~ and u~ belong separately to Wh ; u~ is defined as in (4.32) by 1

uh (M) = u(M),

0

V ME ’lIh .

(4.49)

There is no reason for u~ to belong to ~, and actually u~ will be a "small corre~tor" so that u1 + u~ E fh. We define u~ by its values at the points ME ’lIh ; if M = Ai is the vertex of a triangle then u~(Aj) = 0; if M = A q is the mid-point of an edge, then, letting lJij denote one of the two unit vectors orthogonal to AjAj, we set: u~ (Aij) .AjA j= 0

u~ (A ij ) • "tt I I = - {u(Aij) + "4u(Aj) + "4u(Aj) }’Vjj

+

~

J:

u(tA,+ (l-t)Aj)ov"dt

(4.50)

The steady-state Stokes equations

82

Lemma 4.9.

uh

Ch. 1, §4

defined by (4.48) - (4.50) belongs to Vh

Proof. The main idea in (4.50) was to choose u~ so that (4.51 ) If we show that (4.51) is satisfied, we will then have for any triangle.9":

since u E f (v toff).

= unit ’vector normal to off pointing outward with respect

Let us prove (4.51). The function u~ is equal to (4.52)

On the segment A;A j , the function WhA; is the only function whM in the preceding sum which is not identicalry equal to O. By the definition of WhA;j one easily checks that WhA;j

(tA;+ (I-f)Aj )

= 4t(I-t),

0
< 1.

(4.53)

Likewise

where the only functions whM which do not vanish on AiA j are ~A.’ easily shown that / ] /]

whA.’ whA .. • It is

Then

WhA;

(tA i+ (I-t)At)= (t-1)(2t-l)

WhAt

(tA i+ (I-t)A j )

= t(2t-1).

(4.54)

83

Discretization ofStokes equations 01)

01. I, §4

1 IAiA;1

= (by 4.50»

:=

JI

0

u(tA.+(l-t)Aj)"p.jdt= _I_ I I IA.AI ’;

The lemma is proved. Proposition 4.3. The preceding external approximation of V is stable and convergent. provided h belongs to a regular triangulation JfO/ of n. Proof. Let us check first the condition (C2) of Definition 3.6. We have to show that if a sequence Ph’Uh’, uh ,E Vh’, converges weakly to t/J in F, then t/J = U E V. According to Theorem 1.6, we need only to show that div u = O. Let (J be any function of ~(n);

J

(div

(4.55) by (4.46), we have

U. )9.dx = 0,

(4.56)

n

where (Jh is the step function defined above, which is equal on each filE fJj, to the average value of (J on f/, and which vanishes outside n(h). It is easy to see that when support (J c n(h), SUP!(Jh(X) - (J(x)1 ~ c«(J)p(h),

XEn

so that (J h converges to (J in the L the limit in (4.56) and obtain

00

and L 2 norms; thus we can pass to

84

The steady-state Stokes equations

~

divu’6dx = O.

Ch. I, §4

V6 E!»(n).

n

This proves (4.55). The condition (Cl ) of Definition 3.6 is lim

h-+O

Ph’hU

= WU, VuE r.

(4.57)

This is equivalent to lim lIu -

h-+O

’h U II

= 0, VuE

r.

(4.58)

Let us suppose that p(h) is sufficiently small for O(h) to contain the support ofu. Because of Lemma 4.8 and (4.42), on each triangle f/E ff,,: Suplu(x) - u~(x)1

xeg

SupIDiU(X) xeg

~c7h(u)P;’

DiU~(x)1

~c’h(u)

+. p3

(4.59)

Pfl’

By the proof of Lemma 4.9,

(Aj [u(x) - u~ (x)] "Vijdl

)

I 1

=

(4.60)

Ai

(u - u1 XfA,

+ (I

- flAl)•"ldf.

o

Hence, with (4.60) estimated by (4.59),

IU~(Aij)1

= lu&(Aij) "vijl ~C773(U)P~.

(4.61)

8S

Discretization ofStokes equations (II)

Ch. I, §4

Now by (4.43), we obtain

(4.62) Next, combining (4.61)-(4.62) with (4.52), we get Suplu~(x)1

xE,/

Sup IDiU~(x)1 XEy

E;;; C113(U)P~ E;;; C’Ih (U)

3

’!,. Py

(4.63)

Finally, combining (4.59) and these last inequalities, it follows that Sup Iu(x) xe.n

Uh (x)] E;;; C113 (u)p(h)3

(4.64)

The proof is completed.

Remark 4.3. If 0 is a polygon, it is possible to choose the triangulation

51, such that O(h) = 0, and this is usually done in practical computations. In this case we can extend the preceding computation to any E (0) n C 3 (0) and we find

U

H6

(4.65)

4.2.3. Approximation of Stokes problem Using the preceding approximation of V and Section 3.2 we can pro› pose a finite element scheme for the approximation of a two› dimensional Stokes problem. Let us take in (3.6)

(4.66)

The steady-state Stokes equations

86

Ch. I, §4

where v and f are given as in Section 2.1 (see Theorem 2.1). The approximate problem (3.6) is then (4.67) The solution uh of (4.67) exists and is unique; moreover, we have Proposition 4.4. If p(h) -+ 0 with a(h) "" ex. (t.e., h E Je Ol ) , the solution (n) norm uh of (4.67) converges to the solution u of (2.6) in the

HA

Proof. It is easy to see that Theorem 3.1 is applicable, and the con› clusion gives exactly the convergence result announced. 0

Approximation of the pressure We introduce the approximation of the pressure, as in Section 3.3. The form

is a linear form on Wh, which vanishes on Vh. Since Vh is character› ized by the set of linear constraints (4.46), we know that there exists a family of numbers Ay, .9’E!1h, which are the Lagrange multipliers associated with the constraints (4.46), such that

(4.68)

Let ’Xhy denote the characteristic function of .9’and let the step function ’lrh

=YE.9h L

’lrh

denote

’lrh ([/)x"y

AY

v» (,9’) =(meas f/)

(4.69)

87

Discretization ofStokes equations (II)

Ch. I, §4

We then have (4.70) which is the discrete analogue of equation v«u, v)) - (P, div v)

= (f,

v),

"V

E Ha(n).

(4.71)

Remark 4.4. Since no basis of Vh is available, the solution of (4.67) is not easy. The computation can be effected by the algorithms studied in Section 5.

The error between u and uh

Let us suppose that n has a polygonal boundary (n c tR 2 ) and that u E ~3 (n) and p Et6’l (n). Then according to Remark 4.3, (4.72)

We can take v = Vh =Uh - rhu in (4.69) and (4.70); subtracting then (4.71) from (4.72) there remains (4.73) Let 11’;’ denote the step function

11’;’

=

L

YEYj,

1 (meas9")

(J,l(X)dx) J

Then the right-hand side of (4.73) is equal to

and is majorized by

Hence

x"f/’

(4.74)

88

The ,tetldy•,tate Stoke, eqUiltion,

vlluh - Thull2 E;;;

I

lIuh - Thull E;;; -

V

{11I’h ,

11I’h -

pi pi

+ vllu - Thullllluh - Thull

+ lIu - Thull.

It is easy to see that 11I’h - pi is majorized by have at least

lIuh -

ull E;;; c(u.

01. I. §4

(4.75) C’Ill (P)p(h)

and then we

(4.76)

p)p(h)

and therefore, using (4.64), lIuh -

ull E;;; c1h (u,

(4.77)

p)p(h).

When the boundary of U is not a polygon, an additional error of order p(h) appears as usual, in the right-hand side of (4.76)-(4.77).

Remark 4.5. The estimation (4.77) is not satisfying because it indicates that the error between u and uh is of order p(h) (in the (U) norm), while the distance between u and Vh is of order p(h)2 since this distance is majorized by lIu - Thull (see (4.64». We do not know if this is due to the fact that the estimation (4.76) is not optimal, or if indeed, the error lIu - Uh II is of order p(h). The pur› pose of the next subsection is to give an improvement to the algorithm which allows us to obtain an approximate solution for which the (U) norm error is surely of order p (h)2 .

HA

HA

4.2.4. Utilisation of the bulb functions We will now give another approximation of the spaces nA(fl.) and V, which will lead to an algorithm of approximation of Stokes problem slightly different from (4.67) and for which the error will be of the optimal order (cf. Remark 4.5). The corresponding approximation of V will be denoted by (APX2’), the approximate spaces will be denoted Wh, Vh , ... , leaving the notation Wh, Vh, ... , for the spaces introduced in the study of Approximation (APX2).

Discretization of Stokes equations (II)

Ch. I, §4

Approximation of HA (n). Let 5h be any admissible triangulation of

cR 2 .

n, n open

89

bounded set of

The space Wh is the space of continuous vector functions which vanish outside n(h), and whose components are equal on each triangle S/E5j" to the sum of a polynomial of degree 2 and of a so-called bulb function: the bulb function on Yis the function

where the Xi are the barycentric coordinates with respect to the vertices ofY We observe that ~(x) = 0 on ag; and that ~(x) > 0 onY. The graph of ~ looks like a bulb attached to the boundary ofS/. For eachS/E5i" we denote by ~h.9’ the continuous real valued func› tion on n, equal to the bulb function on f/ and to 0 outside.v. Let {3h be the space of functions of type (4.78)

Then the space

Wh can be written as (4.79)

where Wh is Qte spl!ge introduced in section 4.2.1. Since Wh n {JJh any element vh of Wh admits a unique decomposition of the form

= {OJ, (4.80)

We provide the space induced by (0)

HA

%(included in HA (0)) with the scalar product

«Uh, Vh)) = «Uh, Vh)),

V

Uh’ Vh E Who

Let us set Ph = the identity operator, and let us simply define E !’i)(0) by setting

’h U, U

rhu E Wh (C

Wh ) , o

’hu(M) = u(M), V ME Wh.

(4.81)

90

The steady-state Stokes equations

Ch. I. §4

Then the proof of Proposition 4.2, immediately shows the following Proposition 4.5. The preceding internal approximation of HA(G) is stable and convergent. provided h belongs to a regular triangulation XC! o[G. Approximation of V(APX2’). As before the space F is HA(G) and w is the identity, VuE V; w is an isomorphism from V into F. Let 5h be an admissible triangulation of G.

WU

= U,

-

Space Vh. ~ is a subspace of the space W h , more precisely it is the space of functions Uh E Wh such that (c.f. (4.46»:

Is, q

div u;. dx = 0, V.9’ E!Tj" V linear function q.

(4.82)

While the ~ace Wh is clearly im.£edded in W h , the space ~ is not in› cluded in Vh; the functions in Vh are of a more general type (because of the bulb functions) but they satisfy the algebraic relations (4.82) which are more restrictive than (4.46). We will point out, later on, some very special relations between the spaces Vh and Vh: and we will show in particular that dim Vh = dim

v,;,

iT".

We equip the space included in HA (G), with the scalar product (4.81). Operator Ph. The operator Ph is the identity. The prolongation operators have norm one and are thus stable. Operator rh . The restriction operators denoted;" are of course rather difficult to construct.

Let u be an element ofr"; we set (4.83) where Uh + th is the decomposition (4.80) of;"u; now Uh is defined as

91

Discretization ofStokes equations (II)

Ch. I, §4

(4.84) being the restriction operator of Approximation (APX2) (see section 4.2.2). It remains to choose th E&Wh, so that uh + t h E Vh . Next lemma shows that the conditions (4.82) define a unique th, and that

rh

th

=Y’E.’7h L

~

= area[/

Uh/9’)

4.10. Lemma ,.., Vh•

Uh

I!

Uh (f/) Ph’/ ,

Xi(Uh •v)

dr›

a9’

f

J:/

Uhi

dx

I,

L L L

L

div

Uh

u;, dx =

div Uh dx +

dx vanishes since

f div

J9’

th

i

= I,

2 (4.86)

defined by the relations (4.83) to (4.86) belongs to

Proof. We must show that the relations (4.82) hold for q(x) x2’ and foreachg’E.Jh. Forq = I, we observe that e1iv

(4.85)

dx

=

f

j a9’

Uh

»r»

th’V

U

E

e1iv

Vh,

th

= I, Xl’

dx;

and

er

vanishes for any bulb function th E!;Wh, since th = 0 on ag: Let us now examine the condition (4.82) for q = Xi, t = 1,2. It can be written as 0=

io? Xi div Uh dx = J

or, since

Ph9’

Ja

vanishes outside.v;

J

Xi

div Uh dx +

J

9’

Xi

div th dx,

(4.87)

92

Ch. I. §4

The steady-state Stokes equations

In order to transform the left-hand side of the last inequality we observe that

\ .9’

Xi

a~hy dx = ~ -a a (Xi~hy -aXI

9

- 8/1 By Green’s formula and since

:.g

)

XI

~ ~h.9’

dx

dx (8/1 = the Kronecker delta)

= 0 on a..9: there remains

and according to the next lemma, this quantity is equal to

Finally, the relations are equivalent to the relations area.9’ 60

0h (9’)

I

=(

J

3,7"

xi(Uh’ v)dx -

J (

9

Uhi

dx,

j

= 1,2

93

Discretization ofStokes equations (II)

Ch. I, §4

which are exactly the relations (4.86).

0

In the last proof we used the following result. Lemma 4.11.

~

area 9’

Y

(4.88)

(3hY(X) dx =~.

Proof. For x E 9’, {3hY (x) = Al (x) A2 (x) A3 (x), where the Ai are the barycentric coordinates with respect to the vertices of 9’. We consider the linear transformation A in ~2,

which maps 9’ into the triangle

We have dx = J d.X, J denoting the J aco bian of A,

f’/ dx area 9’ J= - - = --_-= 2 area 9’. fydX area9’ Then,

The computation of the integral

~

xI x,

(I -

xl -

X,)


Y’

is elementary; the value of this integral is 1/ 120 and (4.88) follows.

94

The steady-state Stokes equations

Ch. I, §4

Proposition 4.6. The preceding external approximation of V is stable and convergent provided h belongs to a regular triangulation JC a of n Proof. The condition (C2) of Definition 3.6 is proved by exactly the same arguments as in Proposition 4.3. The condition (Cl) of Definition 3.6 is proved if we establish that lim lIu - ’hull

h-+O

= 0,

VuE r.

Due to (4.58) and the definition of"" it suffices to show that lim 11th 11= 0,

(4.89)

h-+O

th

being defined by (4.85) (4.86). Since div U = 0,

f

Xj(u,v)dr=

Ja’/

f.

JI

ujdx,

diV(XjU)dx=!, .I

and this gives an alternative (equivalent) expression of the

0hj:

(4.90) The majorations contained in the proof of Proposition 4.3, imply that SUPIUh(X) - u(x)1 ~C’113(u)

XE’J’

p3 ,

(4.91)

./

and combining this with (4.90), we get

.

( smce

4 p;

1r,

~ area y’ ) ~c

p,5 113(U):h

/II

P’I

Discretization ofStokes equations (II)

Ch. I, §4

95

(4.92) One can now estimate IIthll. Ong’,

tx, = 0h (.9’) [DAt" A2" A3 + AI" DA2• A3 + AI" A2" DA3] and since IA;(X)I E;;; I and IDAil E;;; I/p~ IDth I E;;;

C 01.

2 ’173 (u)

J

n IDr> I’ dx’;;

+py p3

E;;;

C 01.

c’ of> ~j(u)

(see (4.11», 3 ’173 (u) p(hi

p(h)4 (area n).

(4.93)

Finally, (4.94) and (4.89) is proved. Remark 4.6. From (4.83), and the majorations (4.59) (4.63) (4.93) we infer that Sup lu(x) -

xEO

r" u(x)1

E;;; C ’173 (u)(

Sup IDiU(X) - D, r"u(x)11lll;

XEO

= I,

C

1 + 01.3 )p(h)3

’173 (u) 0I.(l

+ 01.3 )p(h)2,

(4.95)

or.

..., n, VuE h sufficiently small. If n = n(h), the relations (4.95) are valid for each u E V n ~3 (0) and we deduce in particular

i

(4.96)

Approximation of Stokes problem. Using the preceding approximation of V and Section 3.2 we can pro› pose another finite element scheme for the approximation of a two› dimensional Stokes problem. We take in (3.6) (4.97)

96

The steady-state Stokes equations

Ch. I, §4

(4.98) The approximate problem (3.6) is then

To find v« E

iT" such that v((U’h, Vh» = (f, Vh), V Vh E Vh •

(4.99)

The solution u" of (4.99) exists and is unique; moreover by application of Theorem 3.1 we obtain the convergence result.

Proposition 4.7. I[ p(h) ~ 0, with a(h) ~ ex (i.e. hE 3(01)’ the solution 0[(4.99) converges to the solution U 0[(2.6) in theHA(n) norm.

uh

The advantage of this scheme on scheme (4.67) will appear after we have described the approximation of the pressure and given the esti› mation of the error.

Approximation o[ the Pressure. We introduce the approximation of the pressure as in (4.70). The approximate pressure 1rh will now be a function piecewise linear on each triangle.v", The form

VIz ~v((u",

VIz» -

(f,

VIz),

is a linear form on Wh , which vanishes on Vh. Since ~ is characterised by the set of linear constraints (4.82) with q = I, xl’ x 2’ there exists a family of Lagrange multipliers, A~, t = 0, I, 2, g’EYh, such that

Let 7Th denote the scalar function on n, which vanishes outside n(h) and is a.e. equal to A~ + x 1 A~ + x 2 A;" on each Y"E5h. Then the right hand side of the last relation is equal to

l

n

if;, div" dx =(if;" div")

97

Discretization ofStokes equations (II)

Ch. I, §4

and we have (4.100) and this relation is again the discrete analogue of (4.71).

The error between u and iih . Assume that n c cR 2 has a polygonal boundary, n = n(h), and that u E ~3 (n) p E ~2 (n). Then, according to Remark 4.6 (4.101)

lIu - ;;;ull ~ c(u, a) p(h)2.

Let us set v =Vh = iih -;;; u in (4.100) and (4.71). Subtracting (4.71 ) from (4.100) now gives v((iih - u, iih - ;;;u))

Let

1I’h

= (i’/, -

p, div(U/r - ;;;u)).

(4.102)

denote the piecewise linear function, equal on each triangle s" to peG)

+ (x - G)oVp(G),

where G represents the barycenter of f/. By Taylor formula

SUpl1Th(X) - p(x)1 ~c(P)

XEY

p;,

Sup 11Th (x) - p(x)1 ~ c(P) p(h)2.

xEn

(4.103)

However, the right-hand side of (4.102) is also equal to

(1rh -

p, div(U/r - ;;;u))

and because of (4.103) we can majorize the absolute value of this expression by

Hence vlliih - ;;;ull ~ {11Th - pi + vllu - ;;;ulllllU/r - ;;;ulI

98

Thesteady-state Stokes equations

IlUh

-

’hull ~

~ c(P) V

p(h)2

Ch. I, §4

+ C(U, p) p(hY..

Finally, (4.104)

and we get an error of the optimal order i.e. of the order of the distance between u and ~. D Algebraic relation between Vh and

iT;, .

We would like to exhibit now a simple algebraic relation between Vh and iT;,: we will show that there exists a simple isomorphism Ah from onto Vh. This implies that Vh and have exactly the same dimen› sion and that the scheme (4.99) can be interpreted as a scheme in Vh. In other words, we finally exhibit a scheme in Vh giving in some sense the optimal order error(l). It is also interesting to observe that (4.99) is equivalent to a scheme in Vh: a scheme in Vh is simpler to solve than a scheme in iT;, , since we have one instead of three linear constraints on eachgE5;; «4.46) and,.£4.82)). __ Any u" belonging to Wh can be uniquely written as Uh + !.it’ Uh E Wh, The mapping u" ~ uh is a linear projection from Wh onto Wh. th E~h. We denote as Ah the restriction of this projection to iT;,.

v;,

v;,

Lemma 4.12. AJ, is an isomorphism from

iT;, onto Vh-

Proof. We see first that AhUh = uh =Uh - th E Vh ifuh E Vh. Indeed, Oneach 9 E..9h :

and this vanishes since

u" E iT;, and we already

observed that

(1) We get lIu - Uhll <; c(u. p) p(h)2. but apparently not lIu - All Ukll <; c(u. p) p(h)2. We do not know if IlAir Ub - Uhll <; c p(h)2.

Discretization ofStokes equations (II)

Ch. I, §4

(

JY’

div th dx

= 0, V th

99

E.5fJh.

As a projection operator, Ah is one to one. There remains to show that is onto: given Uh E Vh, we have to find th = th (Uh) E.9Jh such that Uh E iT;, and Uh = Uh + th (Uh), i.e. Ah Uh = Uh. This is proved as in Lemma 4.10 and the relations giving th are exactly the same as (4.85), (4.86). D Ah

As a consequence of Lemma 4.12, we write the scheme (4.99) in the following form:

To find Uh E V((Uh

Vh

+ th(Uh),

such that

Vh

+ th(Vh))) = (f,

Vh

+ th(Vh)),YVh

E Vh (4.105)

where t h (uh ) E ai’h is given in terms of uh by Lemma 4.11 and formulas (4.85), (4.86). The solution Uh of (4.105) differs from the solution Uh of (4.67) and the improvement lies in the fact that lIu - A}, Uh II is of optimal order (in the usual case, O(h) = 0 and U E re 3 (0), p E 2 (n)).

4.3. Finite Elements of Degree 3(n

= 3)

Let n be a Lipschitz open bounded set in 1R3. We first describe an in› (n) and then an external approximation of ternal approximation of V. The approximate functions are piecewise polynomials of degree 3.

HA

4.3.1. Approximation of HA (n)

Let3h be an admissible triangulation of n and let (4.106)

If [f is a 2-simplex (i.e., a tetrahedron) we denote by Al ’ ... , A 4 , the vertices of 9’and by B I , ... , B4 , the barycenter of the 2-faces 9"1’ ... ,9’4’ We denote by &J, the set of vertices of the simplices9’E.1j; and by the set of barycenters of the 2-faces of the simplices9’belonging to fTh ;

&r

&h

= &}, u &~.

We first prove the following result.

Lemma 4.13. A polynomial of degree three in 1R3 is uniquely defined

Thesteady-state Stokes equations

100

Ch. I, §4

by its values at the points Ai, Bi’ 1 <; i <; 4, and the values of its first derivatives at the points At. Moreover, the polynomial is given in terms of the barycentric coordinates with respect to Al , ... , A 4 , by the formula 4>(X)

=

4

L1

[1-2 (Ai(X»3 + 3 (At(x»2 ] 4>(A i)

i=

~ £..

1 +-

i= 1

6

Al (x) ... A4 (x) [ Ai(X)

274>(Bi)-

7

4

L

01=1

OI¢i

4

+

L

i,l= 1 i¢j

(At(X»2 Aj(x)[D4>(A i) OAiAj]

4

L

i= 1

(4.107)

Proof. The proof is exactly the same as the proof of Lemma 4.5. The coefficients of 4> are the solutions of a linear system with as many equa› tions as unknowns; we just have to check that the polynomial on the right-hand side of (4.107) fulfills all the required conditions for any set of given datas 4> (Ai)’ 4> (Bi), D4>(A i)• 0 It follows from this lemma that a scalar function 4>h which is defined on O(h) and is a polynomial of degree three on each simplexY’E5h is completely known if the values of 4>h are given at the points Ai Eg’ ~ and Bi E g’~ and also the values of D4>h are given at the points Ai E g’~. Such a function 4>h is differentiable on each f/, f/ E !Jh, but there is no reason for such a function to be differentiable or even continuous in all of O(h). Actually, this function 4>h is at least continuous: on a two face Y" of a tetrahedron Y’ E.9J" 4>h has two - perhaps different - values 4>;’ and 4>h ; but 4>~ and 4>h are polynomials of degree three which take the same values at the vertices and at the barycenter ofY’; the first de› rivatives of 4>~ and 4>h at the vertices of f/’ are also equal (these are the derivatives of tPh which are tangential with respect tof/’). Now, it can be proved exactly as in Lemma 4.5 and 4.6 that 4>1; = 4>’h. We denote then by whM’ ME g’h’ the scalar function which is a piece› wise polynomial of degree three on O(h) with

Ch. I, §4

101

Discretization of Stokes equations (II)

WhM(M)

= 1, whM(P) = 0, V PE

DWhM(P)

&h’ P=I=M

= 0, VP E &1.

(4.108)

wxlt

For ME &h’ i = 1, 2, 3, is the scalar function which is a piecewise polynomial of degree three on O(h) such that

= 0, V PE &h’ DwZ1(p) = 0, V PE &1, P =l=M, WZ1(p)

DwZ1(M)

= ei’ i = 1,2,3.

(4.109)

All of the functions whM’ wZ1 are continuous on O(h). Space

Wh

The space Wh is the space of continuous vector functions uh from 0 into tR 3 , of type 3

2:

i= 1

DiUh (M)wZ1, C4.110)

which vanish outside O(h). It is clear that uh(M) = 0 for any ME 8>h u aOCh); but since the tan› gential derivatives of uh vanish on the faces of the tetrahedrons .9’which are included in OCh), the derivatives Diuh CM), ME 8>1 u aOCh) are not independent. The space Wh is a finite dimensional subspace of HACO); we provide it with the scalar product induced by Hb(O)

CCUh, vh»h

= CCuh’ vh»,Vuh’

C4.111)

vh E Who

Operator Ph .

The prolongation operator Ph is the identity; the Ph are stable. I

Operator rh . For U ~CO),

we define uh = rh U by uh = if the support of U is not included in OCh), and if the support is included in OCh),

= uCM), VME 8>h’ Duh CM) = DuCM) , V M E 8>h. uhCM)

C4.112)

102

The steady-state Stokes equations

Proposition 4.8. The preceding internal approximation 01

Proof. We have only to prove that for each

= rhu

-+ u

~4

HA (12) is

provided h belongs to a regular triangulation ’JfQt

stable and convergent,

0112.

uh

Ch. I,

U E~(n),

in HA(n)

as p(h) -+ 0, hE ’JfQt.

This is proved in a similar fashion to Proposition 4.2. The analogue of (4.42) for Hermite type interpolation polynomials (see Ciarlet & Raviart [ I ] ) shows that Sup Iu(x) - uh(x)1 ~ c 714(u) p~

4

XE.’/’

Sup IDu(x) - Duh(x)1

XE.’I’

~c

714(U)

~

py

.

(4.113)

Hence Sup Iu(x) - uh(x)1 ~ c(u) p(h)4

xEO

Sup IDu(x) - Duh(x)1 ~c(u)

xEO

p(h)3 0(h)

(4.114)

and in particular

lIu -

uh II ~ ac(u) p(h)3

(4.115)

provided supp u C n(h). Approximation 01 V (APX 3).

We recall that 12 is a bounded set in tR 3

Space F, Operator

w.

HA

The space F is (12) and Wis the identity. Let fIj, be an admissible triangulation of n.

Space Vh.

Vh is a subspace of the previous space Wh ; it is the space of

L

uh E Wh such that

div". dx = 0,V .9’Ey’;.

(4.116)

Discretization ofStokes equations (II)

Ch. I, §4

103

We provide the space Vh with the scalar product (4.111) induced by

Hij(n) and Wh . Operator Ph .

This operator is the identity (recall that Vh C

HA (n)).

Operator rh .

The construction of rh is based on the same principle as in Section 4.2. Let U be an element ofr; we set (4.117) where uh and u~ separately belong to Wh ; uh is defined exactly as in (4.112)

= u(M), "M E &h’ Dul(M) = Du(M), "M E s].

uh(M)

(4.118)

The corrector u~ is defined by = 0, Du~(M)

u~(M)

= 0, V ME

&h

(4.119)

and at the points M E &~, the component of u~ (M) which is tangent to the faceg" whose M is the barycenter, is zero; the normal component u~(M)•JJ is characterized by the condition that (4.120)

One can prove (

J9"

u~

0)

that there exists some constant d such that

(x). vdr

=d (area Y’)u~

(M). v

(1) The principle of the proof is similar to that of Lemma 4.9.

104

The steady-state Stokes equations

Ch. I, §4

and (4.120) means that (4.121)

It is clear then that uh belongs to Vh since, for each f/ E

f, div u.
t

tY’

uhvdr =

rty

u’vdr =

rL~ ~iv

9h ’ U

dx =

o.

Proposition 4.9. The preceding ex ternal approximation of V is stable and convergent, provided h belongs to a set of regular triangulations ’;1{a

ofn.

The proof of this proposition follows the same lines as the proof of Proposition 4.3. The approximation of Stokes problem can then be studied exactly as in Section 4.2. 4.4. An internal approximation of V We assume here that n is an open bounded subset of lR 2 with a Lipschitz boundary. We suppose for simplicity that n is simply con› nected; for the multi-connected case see Remark 4.7. In the two-dimensional case, the condition div U = 0 is (4.122) and implies that there exists a function 1/1 (the stream function), such that (4.123)

DiBcretization ofStokeB equation, (II)

Ch. I, §4

lOS

The function I/J is defined locally for any set n, and globally for a simply connected set n. In the present case we can associate with each function u in V the corresponding stream function I/J. The condition u = 0 on an amounts to saying that the tangential and normal derivatives of I/J on an vanish. Then I/J is constant on an and since I/J is only definedup to an additive constant, we can suppose that I/J = 0 on r and hence I/J E (n). Therefore the mapping

H5

(4.124) onto V. is an isomorphism from H~(n) Our purpose now is to construct an approximation of H~ (n) by piece› wise polynomial functions of degree 5 and then to obtain with the iso› morphism (4.124) an internal approximation of V. Internal Approximation of H~(n). Let5h be an admissible triangulation of n, and let n(h)=

u

’/E!Th

9

(4.125)

A 2-simplex is a triangle. Let!Jl be some triangle with vertices AI’ A 2 , A 3 ; we denote by Bl’ B 2 , B 3 , or by A 23 , A 13 , A 12 , the mid-points of the edges A 2A 3 , Al A 3 and A 1 A 2; 1Vij denotes one of the unit vectors normal to the edge AiA j, I ~ i, j ~ 3. We first notice the following result:

Lemma 4.14. A polynomial (/J of degree 5 in lR 2 is uniquely defined by the following values of (/J and its derivatives: (4.126) (4.127) where the Ai are the vertices of a triangle.i/’ and the A ij are the mid› points of the edges.

106

The steady-state Stokes equations

Ch. I, §4

Principle of the Proof We see that there are as many unknowns (21 coefficients for 4» as linear equations for these unknowns (the 21 conditions corresponding to (4.126)-(4.127)) . . As in Lemmas 4.5 and 4.13 it is then sufficient to show that a solu› tion does in fact exist for any set of data in (4.126)-(4.127) and this can be proved by an explicit construction of 4> leading to a formula similar to (4.26) or (4.107). We omit the very technical proof of this point which can be found in A. Zenisek [1] or M. Zlamal [1] .(1) It follows from this lemma that a scalar function which is de› fined on n(h) and is a polynomial of degree five on each triangle are given at the points S"’E5h, is completely known if the values of Ai E &l, B j E &~, and if also the values of the first and second deriva› tives are given at the points Ai E &l, where

"’h

"’h

&~ = set of vertices of the trianglesS"’E-.9h, belonging to the interior of n(h) &~

= set

&h

= &l u &~.

of mid-points of the edges of the trianglesS"’E5;;, belonging to the interior of n(h) (4.128)

but Such a function 4>h is infinitely differentiable on eachy,S"’E~, there is no reason for such a function to be as smooth in all of n (h). Actually, the function 4>h is continuously differentiable in n(h). Let 4>~ and 4>" denote the values of 4>h on two sides of the edge A 1 A 2 of a triangle see !1h ; 4>~ , 4>"h are polynomials of degree less than or equal to five on A 1 A 2 and they are equal together with their first and second derivatives at the points Al and A 2 (six independent conditions) and hence 4>~ = 4>". The tangential derivatives o4>~ lOT and o4>"h lOT, . T = A 1 A 2 IIA 1 A 21 are also necessarily equal. Let us show then that the normal derivatives o4>~/ovI2 and o4>"hlovI2 are equal onA 1 A 2 ~ese derivatives are polynomials of degree less than or equal to four on (1) The principle of the construction is the following: let ~1’ ~2’ ~3 denote the barycentric co› ordinates with respect to A l , A 2, A 3 The affine mapping x -+ y (~l (x), ~2(x», maps the

triangle .Y, on the triangleY:

The construction of ct>(x(y» obtain the function ct>(x).

=

on,Pis elementary; then using the inverse mappingy

-+X,

we

107

Discretization ofStokes equations (II)

Ch. I, §4

Al A 2 ; they are equal at A 1 and A 2 together with their first derivatives, and they are equal at A 12’ Therefore they are equal on A 1 A 2’ This shows that ~h is continuously differentiable on n(h). To each point M E &~ we associate the function 1/J2 M, which is a piecewise polynomial of degree five on n(h), such that,

a

~ 1/J2M(M) uV

a:

al/JO M

(P)

= 1 and all the other nodal values . are zero, l.e.,

= 0,

DCiI/J2M(P)

VP E &~,

= 0,

(4.129)

P =l=M,

&1,

VPE

of 1/J2 M

[a] ~ 2.

To each point M E &~, we associate the six functions 1/J1 M, ... , 1/J2M defined as follows: they are piecewise polynomials of degree five on n(h) and

I/J1M (M) = I, all the other nodal values of I/J1M are zero (4.130) for i = 1 or .2, Dj 1/J~:,j(M) values of I/Ji,:,j are zero

= [)ij’ and all the other nodal

Dil/J:M(M) = I, DID21/J~M(M)=

1, D~1/J2M(M)=

1,

and all the other nodal values of I/J~M and of I/J~M

and I/J~M

(4.131)

(4.132)

respectively are zero.

All these functions are continuously differentiable on n(h). Space X h .

The space X h is the space of continuously differentiable scalar func› tions on n (or \R2) of type: (4.133)

These functions vanish outside n(h), and since they are continuously differentiable in n,

The steady-state Stokes equations

108

= 0, VME&~

DQ1/Ih(M) a1/l h(M)

av

= 0,

[a]

naO(h),

-c r, (3.134)

VM E &h2 n aO(h).

The space X h is a finite dimensional subspace of H~(O); it with the scalar product induced by H5(0): «1/Ih’(/>h))h = «1/Ih’(/>h))

2

"0(0.)

Ch. I, §4

,V1/Ih’(/>h EX h•

we provide (4.135)

Operator Ph’ Ph = the identity as X h C H5 (0). Operator rho For 1/1 E ’@(O) (a dense subspace of H5 (0)), we define rh 1/1 1/Ih by its nodal values

=

=D

DQ 1/Ih (M) a1/lh

-

av..II

_

(A ij )

Q

1/I(M), VM E &~,

a1/l

av..II

- -

[a] ~ 2 2

(4.136)

(A ..), VA .. E &h’ II

II

Proposition 4.10. (Xh, Ph’ r h) defines a stable and convergent internal approximation of H~(O), provided h belongs to a regular triangulation 3(01 of O. Proof. We only have to prove that, for each 1/1 E9(O), rh1/l~1/IinHij(0),

asp(h)~O.

(4.137)

This follows from an analog of (4.42) and (4.82), for Hermite type polynomial interpolation (see Ciarlet & Raviart [1] , G. Strang and G. Fix [1], A. Zenisek [1], cf. M. Zhimal [1]): Sup 11/Ih(x) - 1/1 (x) 1~C716(1/I)P~

(4.138)

Sup

(4.139)

Sup

(4.140)

xEg

xES"

xES"

Therefore H1/Ih - 1/111

2 "0(0. (h))

~ c( 1/1 )a 2 p(h)3

(4.141)

Discretization of Stokes equations (II)

Ch. I, §4

109

and it is clear by (4.22) that 111/111

2

B(f..O - O(h»

-+-

0 as p(h)

-+-

(4.142)

O.

Intemal approximation of V (APX4).

We recall that S1 is a bounded simply connected set of (fl2. We define an intemal approximation of V, using the preceding approximation of Hij(S1) and the isomorphism (4.93). Let there be given an admissible triangulation !lJ, of S1. We associate with fIh, the space Vh , and the operators Ph- rh, as follows.

Vh . It is the space of continuous vector functions uh defined on S1 (or (fl2), of type

Space

uh =

1al/laX2

h,-

al/l h

ax}

l,

(4.143)

~

I/Ih belonging to the previous space X h . It is clear that uh vanishes outside S1(h) and is continuous since I/Ih is continuously differentiable, and that div uh = O. Therefore uh E V, and Vh is a finite dimensional subspace of We provide it with the

v.

scalar product induced by V

(4.144) In particular, (4.145) Operator Ph: the identity. Operator rh’

or,

Let U belong to and let 1/1 denote the corresponding stream function (see (4.124)); clearly, 1/1 E ’@(S1) and we can define I/Ih E X h by (4.136). Then we set U h

=r

h

U

=

(al/l h , - al/l h )

ax 2

ax }

E

V . h

(4.146)

The steady-state Stokes equations

110

Ch. I, §4

Proposition 4.11. The preceding internal approximation 01 V is stable and convergent il p(h) -+ 0, with h belonging to a regular triangulation Jell! o/n. Proof. We have only to show that

(4.147) According to (4.124), (4.145), (4.145), we have

lIu h - ull ~ IIl}Jh - I}JII H5(.o

)•

The convergence (4.147) follows then from (4.141) and (4.142).

Approximation 01 Stokes Problem. We take for (3.6), ah (Uh’ Ph) (~,

Ph>

= V «Uh ’ Ph »h = V «Uh ’ Ph»

= (f,

(4.148) (4.149)

Ph>’

where v and I are given as in Section 2.1 (see Theorem 2.1). The approximate problem associated with (2.6) is

To find uh E Vh such that V «uh , Ph»

= (t. Ph>’

VPh E Vh •

(4.150)

The solution uh of (4.150) exists and is unique and, according to Theorem 3.1 and Proposition 4.8,

uh -+ U in V strongly if p(h) -+ 0, h E Jell!’

(4.150)

The Error between Uand uh’ Let us suppose that n has a polygonal boundary, so that we can choose triangulations.Jh such that n(h) = n. Let us suppose that the solution U of Stokes problem is so smooth that U E ~s (U); then, by (4.141) and (4.145),(1) lIu - ’hull ~ c(u, a) p(h)3.

. (4.152)

(1) For the sake of simplicity (4.141) was proved for I/iE~ (0); the proof is valid for any I/i E’6" (n) n H~ (0).

Discretization ofStokes equations (II)

Ch. I, §4

111

The equations

V«u, v)

= (f, v>,

V«Uh’ Vh»

= (f,

V v E V,

Vh>’ V Vh E V,

give

= O.

V«U - Uh’ ’hU - Uh»

Therefore,

lIu - uhll2 = «u - uh’ U - lj,u» ~ lIu - uhllllu - ’hull lIu - Uhll ~ lIu - ’hull,

and by (4.152), we obtain (4.153)

lIu - uhll ~ c(u, ex) p(h)3.

Remark 4.7. If n is a multi-connected open subset of 61. 2 , and if U belongs to V, then according to Lemma 2.4 there exists a function 1/1 satisfying (4.123). Since U vanishes on an, the function 1/1 is necessarily single valued, al/l/av vanishes on an, 1/1 vanishes on r o the external part of an and is constant on the internal components of an, r 1 , r 2 , .... The mapping (4.124),

gives here an isomorphism between the space

x = {I/I E H2 (n), .1. ’I’

1/1

= 0 on r o,

al/l av

I = const. on the fjs, -

= 0 on

an}

(4.154)

and V. An easy adaption of Proposition 4.10, leads to an approximation of X, and, as in Proposition 4.11 we deduce from this an approximation of the space V in the multi-connected case. Remark 4.8. (i) An internal approximation of V with functions piece› wise polynomials of degree 6 is constructed in F. Thomasset [I].

The steady-state Stokes equations

112

Ch. I, §4

(ii) Internal approximations of V are not available if n = 3.

4.5. Non-conforming finite elements Because of the condition div u = 0, it is not possible to approximate V by the most simple finite elements, the piecewise linear continuous functions. This was shown by M. Fortin [2]. Our purpose here is to describe an approximation of V by linear, non-conforming finite elements, which in this case, means, piecewise linear but discontinuous functions. This leads to the approximation of V denoted by (APX 5). Then we associate with this approximation of Va new approximation scheme for Stokes problem.

HA

4.5.1. Approximation of (0). We suppose that 0 is a bounded Lipschitz open set in 6{n and in (0) by non-conforming piecewise this section we will approximate linear finite elements. LetJ’;; denote an admissible triangulation of O. If .9’E§h, we denote by AI’ ... , An+ 1 its vertices, by 9i the (n - I )-face which does not con› tain Ai’ and by B, the barycentre of the face fJ’t. If G denotes the barycenter of.9’, then since the barycentric coordinates of Bt with respect to the Aj , j =1= i, are equal to I In we have

HA

GB.= ~ GAj I Nt n

or

1

GBt= - - GAt,

n

n+l

=~

j=1

GAj _ GAt

n

n

(4.155)

(4.156)

since "£7=V GAj= 0 (the barycentric coordinates of G with respect to AI’ ... , An+ 1 , are equal to I/(n+ I )). We deduce from this, that

(4.157) and therefore the vectors B 1 Bj , j = 2, ... , n+ I, are linearly independent like the vectors Al A j , ; = 2, ... , n+ 1. Because of this, the barycentric coordinates of a point P, with respect to B 1 , , Bn+ 1 , can be defined, and we denote by III’ ... , 1Jn+ l ’ these coordinates. We remark also that for each given set of (n+ 1) numbers 131 , , I3n+ 1 , there exists one and

Discretization ofStokes equations (In

Ch. I, §4

only one linear function taking on at the points B 1 , , f3 n+ l ’ and this function U is

f3 1 ,

113 ... ,

B n + 1 , the values

n+l U (P)

= ~1=1 f3i l.l.i (P );

(4.158)

(see Proposition 4.1).

Space Who Wh is the space of vector-functions uh which are linear on eachY’E5j;, vanish outside .Q(h)(l) and are such that the value of uh at the barycenter B, of some (n - 1)-dimensional face f/i of a simplex Y’ E.!Jh is zero, if this face belongs to the boundary of .Q(h); if this face intersects the interior of .Q(h) then the values of uh at B, are the same when B, is considered as a point of the two different adjacent simplices. Let ~ denote the set of points B, which are barycenters of an (n - 1) dimensional face of a simplex Y’E!J’i, and which belong to the interior of .Q(h). A function uh E Wh is completely characterized by its values at the points B, E ’l/h . We denote by whB’ B being a point of ’l/h’ the scalar function which is linear on each simplexY’E3h, satisfies all the boundary and matching conditions that the functions of Wh satisfy and, moreover, WhB(B)

= 1,

whB(M)=Q VME’l/h’ M=I=B.

(4.159)

Such a function whB has a support equal to the two simplices which are adjacent to B.

A,

Two adjacent triangles (n (1) As before, o(h)

= U

.9’E9j,

9’.

= 2).

Ch. I, §4

The steady-state Stokes equations

114

Lemma 4.15. The functions WhBej ofWh, where B E’lih and 1 ~i~n,

form a basis of Who Hence the dimension of Wh is nN(h), N(h) being the number of points in’llho

Proof. It is clear that these functions are linearly independent and that they span the whole space Wh: by Proposition 4.1, any uh E Wh can be written as (4.160)

Ha

The space Wh is not included in (0); actually the derivative DjUh of some function uh E Wh is the sum of Dirac distributions located on the faces of the simplices and of a step function Dih uh defined almost everywhere by (4.161)

DihUh(X) = DjUh(x), V x Ef/, V .9’Eff".

Since uh is linear on ~ Dih uh is constant on each simplex. We equip Wh with the following scalar product: n

[uh,vh]h=(uh,vh)+

~1=1

(4.162)

(DihUh,DihVh)

which is the discrete analogue of the scalar product of Ha(O): n

[u, v]

= (u,

Space F, operators

v)

+L

j=1

(4.163)

(Dju, Djv).

W, Ph .

We take, as in Section 3.3, F = L2 (o)n + 1, and for UE

wthe isomorphism

Hb(O) 1-+ Wu = (u, D1 u, ... , Dnu) E F.

(4.164)

Similarly, the operator Ph is defined by uh E Wh

1-+

Phuh

’" = (uh’ D1huh’

’" ... ,Dnhuh) EF.

The operators Ph each have norm equal to 1 and are stable.

(4.165)

115

Discretization ofStokes equations (II)

Ch. I, §4

Operator rho

We define rhu uh (B)

= uh’

=u(B),

for U E!B(n), by (4.166)

VB E Wh .

Proposition 4.12. Ifh belongs to a regular triangulation XC! of n, the preceding approximation of

HA (n) is stable

and convergent.

Proof. We have to check the conditions (Cl ) and (C2) of Definition 3.6. For condition (C2), we have to prove that, for each U E!B(n), uh -+- U in L2 (n) as p(h) -+- 0,

(4.167)

Dihuh -+-Dju in L2 (n ) as p(h)-+-

O.

(4.168)

On each simplex f/we can apply the result (4.42) to each component of u; this gives: Sup lu(x)-uh(x)1 ~c172(u)P;"

xEY

p2

Sup IDju(x) -DjUh(x)1 ~C112(u)+,

XEy

(4.169)

py

Therefore

+ [u]

IIPhuh - wuIIF~c(u)o:p(h)

H~

(n-n(h»

(4.170)

and this goes to 0 as p(h)-+-O. To prove the condition (Cl) let us suppose that Ph,uh’ converges weakly in F to cP = (cPo, ... , cP n ); this means that uh’ -+- cPo in L 2 ( n ) weakly,

(4.171)

Djhuh’ -+- cPi in L 2 (n) weakly, 1 ~ i ~ n.

(4.172)

Since the functions have compact supports included in n, (4.171) and (4.172) amount to saying that: u;,,-+-~

DjhUh’-+-~

in L2 (lR n ) weakly,

(4.173)

in L 2 (lR n ) weakly, 1 ~ i ~ n,


n and to 0 in

Lo».

(4.174)

116

Ch. I, §4

The steady-state Stokes equations

If we show that (4.175)

1 <. i <. n,

~t= D;~o’

it will follow that ~ EHI (~n) and hence tPo EHA(O) with tPt = DttPo, which amounts to saying that tP = WU, U =tPo. Let 9 be any test function in9(~n); then:

(

J~n

uh,(Dt9)dx-+ (

J~n

tPo(Dt9)dx,

The equality (4.175) is proved if we show that

or that

(

tP;9dx

=- (

J~’

= (

Uh’ (Dt 9) dx + (

)~n

)~n

)~n

tPo(Di9)dx, V 9 E

tn

(Dth

,q)(~n),

u;,,) 8 dx -+ 0, as p (h’) -+ 0,

for each () E 9(~n). But according to a technical estimate proved in section 4.5.4 (see (4.218)), we have If~1

<. ctn,

0) OI.p(h) 1I()IIH 1 (n ) lIuhllh’

and it is clear that f~ -+0 as p(h )"+0. Discrete Poincare Inequality. The following discrete Poincare inequality will allow us to endow ))h’ the the space Wh described above, with another scalar product discrete analogue of the scalar product « .,. )) of HA(0) (see (l.11)) and Proposition 3.3).

«.,.

Proposition 4.13. Let us suppose that 0 is a bounded set in lR n . Then there exists a constant c(O, 01.) depending only on 0 and the constant Of.

117

Discretization of Stokes equations (II)

Ch. I, §4

in (4.21) such that the inequality n IUhIL2(0) E;;;

c(S1, Q)

i~

ID/hUh IL2(O)’

(4.177)

holds for any scalar function of type (4.160): (4.178) A similar inequality holds for the vector functions of type (4.160): (4.179) where (4.180)

Proof. The inequality (4.179) follows immediately from (4.177). In order to prove (4.177), we will show that

I

!

u.6 dx [ .. c(!1, ,,)[6I L , ( 0 ) 11",,11.,

(4.181)

n

for each uh of type (4.178) and for each () in9(S1); (4.181) implies (4.177) since!~’(S1) is dense in L2(S1). Let us denote by X the solution of the Dirichlet problem 6.X = 0 in S1, X EBb(S1).

The function X is ’if on nand 00

(4.182)

(1) Strictly speaking this inequality is true only if 0 is smooth enough; in the general case (4.182) is valid if we define x by AX = 8, X EH~(O’) where 0’ is smooth and 0’ :::> n•This makes no change in the following.

The steady-state Stokes equations

118

Ch. I, §4

and the Green formula implies

)g

(

Uh

sx dx =

Hence

(

)ag

Uh

aX dI’-

aV

where

It is clear that n

!h= ~ J~,

(4.184)

with

where

aX

x•= ›ax. I

(4.185)

I

and VI’ . , Vn’ are the components of the unit vector v normal to ag: Because of the Green formula,

i.

( . uh

X,., er = r a~,
Jy

119

Discretization ofStokes equations (II)

Ch. I, §4

so that

We have already considered this expression in the proof of Proposition 4.12. Using the majoration (4.176) we get

Ij~1

<;; c(n, 0) a p(h)IIXiIlHl(,{l)lIuhllh’

Ij~1

<;;c(n, 0) a p(h)IIXIIH2(,{l)lIuhllh’

Due to (4.182) and (4.184), we get I~I

<;;c(n, O)ap(h) IOIL2(n)lIuhllh’

(4.186)

The combination of (4.183) and (4.186) gives precisely (4.181) with c(O, a)

= c(O) + c(n,

0) a p(h).

(1)

Proposition 4.14. Let 0 be a bounded Lipschitz set. Let us suppose that we equip the space Wh with the scalar product n

«uh’ vh))h

= ~,=1

(Dihuh’ DihVh),

(4.187)

and leave the other unchanged in the statement of Proposition 4.12. Then this approximation of Hij(0) is again stable and convergent. Proof. The only difference between this and Proposition 4.12 comes from the stability of the operators Ph and this difficulty is completely overcome by Proposition 4.13 and (4.179), which give lIu hllh <;; [uh]h <;; c(O) lIuhllh’ V uh E Who

(4.188)

4.5.2 Approximation of V (APX5). Let 0 be a Lipschitz bounded set in 6ln and let ey be the usual space (1.12) and V its closure in Hij (0). We now defme an approximation of V similar to the preceding approximation of H~(O). (1) p(h) is obviously bounded; for example p(h) <; diameter of n

120

The steady-state Stokes equations

As previously we take F= L2 ( o ) n + 1, and operator U

Space

E V-+wu

wE!f(V,

= {u, Diu, ... ,Dnu}.

Ch.l, §4

F) as the linear

(4.189)

Vh’

Vh is a subspace of the preceding space Wh : n

Vh = {Uh E Wh I L Dih"oJ 1=1

= O}.

(4.190)

The condition in (4.190) concerning the divergence of uh is equivalent to div uh = 0 inS": V gE!Jh.

(4.191)

We equip the space Vh with the scalar product ((uh’ v,,))h induced by

Who

Operator Ph .

As before, Phuh

= {uh’ D1huh’ ... , Dnhuh }.

The operators Ph are stable because of the inequality (4.179) (or (4.188)). Operator rho

We have to define rhu = uh E Vh’ for U E’Yo Since uh must satisfy the condition (4.191), the operator rh used for the approximation of nA(o) does not satisfy all the requirements. We choose the following operator rh instead: uh =rhu is characterized by the values of uh (B), BE Wh ; if B E Wh , B is the barycenter of some (n - 1)-faceg’ of some n-simplex gE5/;, and we set uh(B)

=

1

meas n - 1 (9")

i

[/’

U

df’,

(4.192)

Let us show that uh E Vh; since div uh is constant on each simplex the condition (4.191) is equivalent to

L u. div

dx

= O. V.9’E.9;;.

~

Ch. I, §4

Discretization ofStokes equations (II)

121

Applying the Green formula, we get

where a+ fI is the set of (n - I)-dimensional farces of.9’; by (4.182) this is equal to

rJy,

u’v".dr=

rJay

and this last integral is zero since div U

u.vdr=

rJ~

div u dr,

= O.

Proposition 4.1 S. The previous external approximation of V is stable and convergent, provided h belongs to a regular triangulation 3ta of n. Proof. We have noted already that the Ph are stable. Let us check the condition (C2) of Definition 3.6. For that, let us suppose that Ph,Uh’ ~

rp in F, weakly.

(4.193)

Exactly as in Proposition 4.12. we see that

rp = wu, U EHA(n)

(4.194)

Here, moreover, we must prove that U E V, i.e., div u = O. But (4.193) means in particular n

~ Dih,uih’~div i=1

u in £2(n), weakly,

and since ~7=1 Dih,uih’ is identically zero, div u is zero. Let us check the condition (Cl); if u E’Y’ we denote by uh the function rh u and by Jlh the function of W h defmed by Jlh (B)

= u(B), V B E’l/h .

It was proved in Proposition 4.12 that

122

The steady-state Stokes equations

IIPh)/h - WUIIF ~ c(u)ap(h)

+ [U]OI(n

- n (h»’

cs.t,

§4

(4.195)

It suffices now to show that

II Ph Uh -

Ph)/h IIF

= [Uh -

)/h]h -+0, as p (h)-+O.

(4.196)

Because of the inequality (4.188), it suffices to prove that lIuh - )/hllh-+O, as p(h)-+O.

Each B of 9th is the barycenter of some face.v" of some simplex.y’; we can write n

u(x)

where (13 1 , la(x)1

’" -au =u(B) + L., i=1

(B)•(x•-/3-)

aXi

"

+ a(x),

13n) are the coordinates of Band ~ c (u) P.t. V x E.9’.

(4.197)

,

(4.198)

Integrating (4.197) onY’, we find

since

fg (x i -

I3;) dx = O. Because of (4.1 98},

(4.199) with (4.200) Inside the simplex ..9"with faces..9"I’ ... , 9’n+l’ n+l

uh(x) - )/h(x) =

2:

;=1

h(B;)Ilt(x)

where fJ.l’ , fJ. n , are the barycentric coordinates of x with respect to B 1 , , Bn+ 1 Therefore, in..9", n+l

[grad (uh - )/h)1 ~ c (u)p],.

and by Lemma 4.2 and (4.21),

2:

i=1

[grad fJ.;1

Discretization ofStokes equations (II)

Ch. I, §4

Therefore in all

123

n

l

IDih (Uh - vh)(x)1 ~

c (u) cxp (h).

(4.201 )

and this implies lIuh - vh IIh ~C(Ul

CX l n)p (h)

(4.202)

so that (4.203)

4.5.3. Approximation of the Stokes Problem. Using the preceding approximation of V and the general results of Section 3.2, we can propose another scheme for the Stokes problem. Let/belong to L 2 (n ), and v> O. We set, with the preceding notations, ah(uhl vh)

= v((uh’ vh»h’

($lh,vh)=(f,vh)j

VUhl vh E Vh

VVhEVh’

(4.204) (4.205)

The approximate problem is To fmd uh E Vh, such that V((Uh’ vh»h = (f, vh)’

V vh E Vh•

(4.206)

The solution uh of (4.206) exists and is unique. If p(h)-+(), with h belonging to a regular triangulation ~, then the following convergences hold Uh ~ U in L 2(0)

strongly,

DihUh ~ Diu in L2 ( n )

strongly, l~ i ~ n.

(4.207)

This follows, of course, from Theorem 3.1. We can, as in Section 3.3 and as for other approximations, introduce the discrete pressure. It is a step function 1rh of the type 1rh

=

2: »«

YEy"

(9’) Xh5/ ’

(4.208)

124

The steady-state Stokes equations

where ’Trh (9’) is the value of ’Trh onY’, ’Trh (9) E !H, and Xh~ istic function of g. This function ’Trh is such that

Ch. I, §4

is the character›

(4.209) where divh vh

n

2 DihVih’

=

(4.210)

i=1

The error between U and uh, the solutions of (2.6) and (4.209) respect› ively, can be estimated as in Section 3.3. Let us suppose that il has a polygonal boundary, that il(h) = il, and that U E~3(i2), P E<6"1 (0). We can define an approximation rhu by a formula similar to (4.192), and it is not difficult to see that the estimation (4.203) still holds:

(4.211)

IIp,,rhu - wullF E;;; c(u, ex) p (h).

We will prove later the following lemma.

Lemma 4.16. Let u, p denote the exact solution of (2.6)-(2.8) and let us suppose that u E ~3 (0). Then, (4.212) where

(4.213) If we accept this lemma temporarily, we see that ah(uh-u,Vh)=Qh(Vh), ah (uh - rhu, vh)

Taking vh

=uh -

=ah (u -

VVh E Vh ,

rhu, vh) - Qh (vh)’

rhu and using (4.205) we obtain

vlluh - rhulI~

E;;; vllu - rhullh lIuh - rhullh

+ IQh (uh - rhu)l.

The estimates (4.211) and (4.213) then give vlluh - rhullh E;;;c(u, p, ex, il) p(h).

(4.214)

More precisely, the constant c in (4.214) depends only on the norms of

u in ,,3 (0) and of p in <6"1 (0).

Ch. I, §4

125

Discretization ofStokes equations 01)

Proof of Lemma 4.16. We take the scalar product in L 2(O), of the equation

-v6u + grad p =[,

(4.215)

with vh; since 0 = O(h), we find

Green’s formula applied in each simplex [/ gives

where

The estimate (4.213) of Qh is then proved exactly as in Lemmas 4.17, 4.18 and 4.19 (see section 4.5.4). Remark 4.9. A simple basis for Vh is available in the two-dimensional case. See the work of Crouzeix [1]. Non-conforming finite elements which are piecewise polynomials of degree k > 1 have also been studied for the approximation of, either H6(O), or the space V. M. Fortin [2] has pointed out that the space V cannot be approximated by conforming finite elements of degree one (i.e., piecewise linear functions). For this reason the approximation studied in this section is certainly very useful for Stokes and Navier-Stokes problems. Several numerical computations of viscous incompressible flows, using these elements have been performed by F. Thomasset [2] . (1) The unit vector normal to aY’is denoted constant v > O.

vin this formula, to avoid any confusion with the

126

The steady-state Stokes equations

Ch. I, §4

4.5.4. Auxiliary Estimates. We prove here some auxiliary technical estimates used before and which will be needed also in Chapter II. These estimates concern the expression

Ji =

L~i

<...

~)dx.

(4.217)

uh in W h , i = 1, ...., n, and for different type of functions t/>. The following proofs are straightforward. They can be simplified by using general results on finite elements.

for

Proposition 4.16.

for t/> E

I}kl ~

c(n, n) a p (h) 1It/>lIn’ (Sl)lI uh IIh ,

H1 (n)

and

uh E

(4.218)

Wh .

The proof is given in the following lemmas. Lemma 4.17.

(4.219) where a+g’is the set of the (n-1)-dimensional faces ofg(1), and vi,9" is the i th component of the unit vectorv.) which is normal tog" and is pointing outward with respect to Y. Proof. Since the functions vanish outside n(h), we have

J"

=

~

[u,
(1) There are (n - 1) such faces.

+
127

Discretization ofStokes equations (In

01. I, §4

The Green-Stokes formula gives

Lemma 4.18. With the notations of Lemma 4.17,

Jk = y~.

9"

];,v

J

(u. (x) - u. (B»

:/,

(~(x) - ~(9")

""9/ ar, (4.220)

where B = By, is the barycenter ofY’, and ifJ0I’) is the average value ofifJ on fI’.

Proof. We first show that

j ih --

2:

ffE.<Jh

2:

ff’ E a +y

J

(uh(x) - uh (B)) ifJ(X)vi,:/’

ar

(4.221)

Y’

To prove this equality we just have to show that (4.222) But for a face f/’ belonging to the boundary of O(h), uh(B) = 0 and the contribution of this face to the sum is zero. Iff/’ belongs to the boundary of two adjacent simplices, the face contributes to the sum in two opposite terms: the uh(B) and ifJ(x) are the same and the vi,Y’ are equal but with opposite signs when9" is considered as a part of the boundary of the two simplices. Hence the sum (4.222) is zero.

128

The steady-state Stokes equations

01. I, §4

The equality (4.220) is then easily deduced from (4.221) if we prove that

But to prove (4.223) we simply note that [Uh (x) - Uh (B)]

(

J.’/’,

since 4>(9") and (

i.

vi,Y’

uh(x)

4> (7’)

vi~;I’

ar

= 0,

are constant on.v" and since

ar =uh(B)

r ,ar, Jy

(4.224)

because uh is linear ong’ and B is the barycenter ofY’. Lemma 4.19.

Proof. Since uh(x) - uh(B) is a linear function on.y" which vanishes at x = B, we can write it onYas

where {31’ .. , {3n’ are the coordinates of B. Hence

and

L.

( u. (x) - ... (B)) (’"(x) - ’"(7’))

"t. if ar

129

Discretization ofStokes equations (II)

Ch. I, §4

But, since grad

r

i.

uh

is constant on.v’,

(grad Uh)2 er = meas n _ 1 (9"’) -lgrad

Uh 12

Let us denote by ~ the distance betweenY’ and the opposite vertex of 9: It is well known that measn(g’)

=.!.n ~

meas(n_l)(Y’).

Hence (4.226) since (4.227)

Py<,~.

The reason (4.227) holds is that the largest ball included ing’has a diameter equal to p.0md this ball is included in the set bounded by the hyperplane containingy’ and the parallel hyperplane issuing from the opposite vertex. Therefore

and

I}LI <. c(n) 9’E§h ~

(L

[grad ""I’dx

~..;p:;

p!/

.

)’/2 (LI,(x) _ <J>(9’)I’dr )’ .9" E a+9’

/2 (4.228)

We then obtain (4.225) by using (4.20), (4.21) and applying the Schwarz inequality to (4.228).

The steady-state Stokes equations

130

Lemma 4.20

~(X)

\ ’/

L

’:I’E 9ih

-

~«/’) ’

L

.’I’ E a+.’I

<; c(n) ()( p(h)

ar

~

<; c (n)el’ p(h)

(cj>(x) - cj>(.9’»2

Ch. I, §4

L

(grad

(~)

’dx,

(4.229)

ar

:/’

L

(grad

(4.230)

~)’dx

Proof. The inequality (4.230) follows directly from (4.229). The inequality (4.229) is an obvious consequence of the trace theorems in HI (.9) if we replace the-constant c(n )~2 in the right-hand side of (4.229) by some constant c(.9) depending on the particular simplex.9; the interest of (4.229) is that this inequality is uniformly valid with respect to the simplices.9 in Sj, . To prove (4.229) we make some transformation in the coordinates which maps .9on a fixed simplex .9"and then we apply the trace theorem inequality in if. and come back to .9. For simplicity we suppose that Al = 0, that v" is contained in the hyperplane x n = 0, and that the vertices of!7’ are AI, , An; the referential simplex is the simplex ~with vertices AI’ , All + 1 , Al = 0, and Al A j+ 1 = ej, i = 1, , n. The face corresponding to [f" is the face!]’ with vertices AI’ , An’ Let A denote the linear operator in dln which is defined by

Aj

= AA;. i = 2, ... , n+ 1

and let A’ be the linear mapping in dln -

1,

which is defined by

Aj=A’A j, i=2, ... .n. A change of coordinates for the integral in the left hand side of (4.170) gives

{

2

_

1 )c/U(X) drg , - ldet A’]

(-2

J

y

U (X) dry

Ch. I, §4

131

Discretization ofStokes equations (II)

where (4.231) and a(X)

= a(x),

x= A- 1 x .

(4.232)

For the simplex~, the trace theorem inequality and the Poincare inequality give (recall that a(B) = 0)

L

,,,(X)

ar ’;c(9)

~

y’

We come back to f/ and the coordinates

and hence

f

-

:I

a2(X)dr~c(9’)I A-1112

We write

Xi.

J ( –~ jj

k=1

.; c(9) ldet AIIiA-III’

I

aXk

f".

(Ax) 12

)dx

(grad u)’ dx

We arrive to

In order to prove (4.229), it remains to be shown that -,---ld_et_A....;.IIIA -1112 ~ cexp(h) det IA’I

(4.233)

The steady-state Stokes equations

132

Since

= det(A’)-1 = A

det A det A’

det A-I

Ch. I, §4

(1) nn

and

IAnn11

IIAII,

~

the left-hand side of (4.233) is majorized by

IIAII IIA-111 2 Because of Lemma 4.3, this term is majorized by Pf PfI’

(pr) PfI’

2

~c(~

ex

PfI’~

c

(~ex

P (h),

and (4.233) follows. The proof of Lemma 4.20 is complete. The estimation (4.218) now follows obviously from (4.225) and (4.230). 0 Next inequality proved by similar methods will be useful in Chapter II. Proposition 4.17.

IfAI"c(n, p,

m.,2

( ; ~ 10~ILq,(O»)

(

for E wl,q’(n) and uh E Wh ’ l/q’ = I - l/q, l/q 1

~

)

10 h UhI LP(O)

= lip

(4.234)

- lin for

Starting with the expression (4.220) of Jk, we will proceed essentially as in Lemmas 4.19 and 4.20, with the main difference that Schwarz inequalities for integrals are replaced by Holder inequalities with suitable exponents.

Lemma 4.21

v; I ~

2:

2:

g e5i, g’Eil+g

Py (meas n _ l (S""»I!-r’.

(1) Ann is the (n, n) element of A; note that Ajn= 0, 1 <. i <. n-l, due to our choice of the

coordinate axes.

Discretization ofStokes equations (II)

Ch. I, §4

133

(4.235)

where gradyuh denotes the value of grad uh on 9’(1

~

i

~

n).

Proof. As for Lemma 4.19, we write on the face 9"

and thus IUh (x) - uh (B)I ~ P

’/

[grad uh

I, V

x E

9’.

Let ’Y be some real number, 1 < ’Y < 00, which will be specified later, and let ’Y’ be the conjugate exponent 1h + 1h’ = 1. We have

i

(uh (x) - uh (B))(Xi(x) -

Xi(9")) vi,9’,dr

9"

~ Py [grad uh I• Ixi(x) - Xi(9")/ df ~ (with Holder inequality) .:; Py

1

(meas, _ I (9")) II"’ grad

Uh

1(t,l x,(x)

- x,( 9")

I"ar

and (4.235) follows by summation.

Lemma 4.22. Let ,

’Y =

q(n - 1)

n

Then for 1 ~ i

n

~

(L"X,(Xl~c(n,

p)

’Y’

and ’Y = - - . ’Y’ - 1

X,(9"W

dr)

Ih

(meas (9"))lh n-l /, Py (meas, (9’))1 q

(1

9’

’

[grad xilq dx

q

)1/ , .

t",

The steady-state Stokes equations

134

Ch. I, §4

Proof. We proceed exactly as in Lemma 4.20 and use the same notations. Setting a (X) = l/J (X) - l/J (9"), we have

(L

(LIU(XW dr9’•t’= ldet A’r ’ h

la(XlI’

dr",Y".

We apply Poincare inequality and Sobolev imbedding theorem on the reference simplex!/:

lal

-

L ’Y ([1")

~ (by the Sobolev inequality on ~co(n, ~

~Cl ~

~

p)

!/’ C

din-I)

laI Wl /q , q ’ (9" )

(by a trace theorem, see Lions [I]) (n, p)

lal W1 ’q’› (,/,)

(by the Poincare inequality)

-

c (g’) Cz (n, p)

(I ~

~

J-l

j

aa , I a- (X)lq dX _ X j

)

1/q , (1)

Using a majoration given in Lemma 4.20, we see that

~

n

2: I aXj ~ (AX) Iq’

c’(q’) IIA-111 (

)1/q ,

j=1

We get 11I

lal",g1"c(.9’)c,(n. p) IIA- ( (1) The ci(n, p)’s also depend on 9" but g;’ is fixed.

#,

f

_I ::. (AX)lq’ dx

y

J

)

1~

Ch. I, §4

135

Discretization of Stokes equations (II)

and coming back to the simplex [/,

Finally

(

Ln

/=1

f.


a
)1/ . q

We achieve the proof by observing that det A =

meas n (!l) meas n ([/)’

det A’ =

measn_l ~’) measn_l ([/’)

,

and by Lemma 4.3

"A-lll~P~.

Pi;;

Lemma 4.23.

Ilk I... ctn. P )d’

(~

ID/hUh lL1’lO))

(~lDt~ILq’lO) ’

(4.238)

Proof. In order to prove (4.238), we combine (4.235) and (4.237). We obtain Of’y + If’y’= I):

136

The steady-state Stokes equations

Ch. I, §4

and, with (4.226),

(4.239)

since meas n _ 1 (g’

,

) ~

n

na

t: meas n (fI’) ~ - meas n (fl’). Py Py

The n-dimensional measure of g’ is larger than the n-dimensional meas› therefore ure of a ball of diameter

plt ;

(p ,)n y

~

c (n) meas n (fI’)

and Py ~ ex. p; ~ c (n) ex. (meas, (g’))l/n.

With this majoration, we infer from (4.239) that

~

(

get

/ lgrad

~I"

dx ) 1/.’,

Applying Holder inequality with exponent q and q’ to this sum, we

tiki (

~c(n,p,n)ex.2

~

(

[grad lq’ dx

no (h)

~

yEY’h l /q . )

q

measn([/’!’/Plgradyuhlq .

)l/

(4.240)

The problem of obtaining (4.238) is now reduced to proving the follow› ing inequality:

Ch. I, R4

137

Discretization of Stokes equations (II)

( .r L:

/E.9h

meas n (9’) [grad

Uh IP

) IIp

(4.241 )

since the right-hand side of this inequality is equal to

(" ~h

L,grad IP fP u"

dx

and this is easily bounded by

c(P) (

~

ID/hUhlL1’CoJ.

Up to a modification of notation, the proof of (4.241) is the purpose of the next lemma.

Lemma 4.24. Let ai’ zi’ be N pairs of non negative numbers, and let p, n, q be as before. Then (

N

~ J-l

a?/p zi

)llq

~

( N

~ J= 1

a.zt’ J

) IIp

(4.242)

J

and we observe that the function z -+a!!lpzq

+aq/pz q =a q/p

211221

p-a (

a

1

q /P

zP

2 2

)

+aqlpzq

22

attains its maximum, on the interval [0, (pla2 )llp], at the end points, and this maximum is equal to pq[p . We then proceed by induction on N; assuming that (2.57) is true for N - 1 terms, we write

The steady-state Stokes equations

138

Ch. I, §S

(4.243)

where N

aZ = Zz = ( ~

aj

zf

) 1/(P+l)

Using the inequality previously established for N right-hand side of (4.243) by P _ _ p l/p _ (alzl+aZz Z) -

and (4.242) is proved.

N

( j~ajzj

p

= 2, we majorize

the

)1/P

,

0

§ 5. Numerical Algorithms

We saw that discretization of the Stokes equations does not solve completely the problem of numerical approximation of these equations; for the actual computation of the solution, we must have a basis of the space Vh such that the analogue of (3.6) leads to an algebraic linear system for the components of"h ’ with a sufficiently sparse matrix. This occurs only with the schemes corresponding to (APX4) and (APX5); for the discrete problem associated with (APXl )-(APX3) we do not even have an explicit basis of Y". In Sections 5.1 to 5.3 we will study two algorithms which are very useful for the practical solution of the discretized equations. In Sections 5.1 and 5.2 we consider the continuous case and in Section 5.3 we show briefly how they can be adapted to the discrete problems. The results proved in Section 5.4 are related to this problem but they also show how incompressible fluids can be considered as the limit of "slightly" compressible fluids. 5.1. Uzawa Algorithm

In Proposition 2.1 we interpreted the Stokes problem as a variational problem, an optimization problem with linear constraints. The algorithms

139

Numerical Algorithms

Ch. I, §5

described in this and the following sections are classical algorithms of optimization. We will present these algorithms and study their con› vergence without any direct reference to optimization theory, although the idea of the algorithm and the proof of the convergence result from optimization theory. Let us consider the functions u and p defined by Theorem 2.1 ; we will obtain u, p as limits of sequences u’", p’" which are much easier to compute than u and p. We start the algorithm with an arbitrary element pO,

po E L2 (n).

(5.l)

When pm is known, we define u m + I and conditions

u" +1

E

0), by the

HA (n) and

v((u m +l, v)) - (p’", div v) pm+l

pm+l (m ~

= (f,

v), V v E Hb(n),

(5.2)

EL 2 (n ) and

(pm+l _ pm, q)

+ P (div u m+ 1, q)

=

0, V q E L2 (n).

(5.3)

We suppose that p > 0 is a fixed number; other conditions on p will be given later. The existence and uniqueness of the solution u’" + 1 of (5.2) is very easy, because of the projection theorem (Theorem 2.2). Actually ~+1 is simply a solution of the Dirichlet problem

u m + 1 EHb(n) _v/:,um+l

= grad p’" + fEH-1 (n).

When um + 1 is known, alent to pm+I

= p’" _

p

pm+l

(5.4)

is explicitly given by (5.3) which is equiv›

div u m + 1 E L 2 (n).

(5.5)

Convergence of the Algorithm

Theorem 5.1. If the number p satisfies

o

(5.6)

then as m-+OQ, u m+ 1 converges to u in Hb(n) and pm+l converges to

140

The steady-state Stokes equations

Ch. I, §5

p weakly in L2 (0)/6t.

Proof. The equation (2.7) which is satisfied by u and p is equivalent to v«u, v)) - (P, div v) = (f, v), V v E HA (0). (5.7) Let us take v = um + 1 _ u in equations (5.2) and (5.7) and then let us subtract the resulting equations; this gives: vllu m+1- u

or

liz = (pm - p, div u m+ 1)

vllv m+ 11l Z = tq’", div v m+ 1),

(5.8)

where we have set (5.9)

(5.10)

q’" =pm _po

Taking q

= pm+1

- p in (5.3), we get:

(qm+1 _ q’", qm+1) + p(div vm+ 1 , qm+1)

= 0,

or equivalently Iq m+11Z _

Iqm 12 + Iq m+1 _ qm 12 = -2p(div v m+1, qm+1).

(5.11)

We next multiply equation (5.8) by 2p, and then add equation (5.11), obtaining Iqm+11 2 _lq mI2+ Iqm+1_qmI2+2pvllvm+111Z

(5.12)

= -2p (div v m+ 1, qm+1 _ qm).

We majorize the right-hand side of (5.12) by 2p [div v m+11 Iqm+l _ qm 1

which is less than or equal to 2p IIvm + 111

Iq m+1

_ q" I

since [div vi ~ IIvll, V v EHA(O),

(1)

(5.13)

(1) This inequality easily established if II E (!i’) (n) is valid by a continuity argument for each II E H~ (n).

By the same method one can check that if n = 2 or 3

Numerical Algorithms

Ch. I, §5

141

We can then majorize the last expression by 8l q m+ l _ qm 12 + p; Ilvm+1112,

where 0 < 8 < 1 is arbitrary at the present time. Hence Iqm+11 2 _ Iqm 12 + (1 _ 8)l q m+ l _ q’" 12

+ p(2v _p...) IIvm + 111 2 ~ O.

(5.14)

8

If we add the inequalities (5.14) for m = 0, ... , N, we find N

IqN+11 2 + (l - 8)

L:

I

m=O

2 q m+ l _ qm 1

(5.15)

N

+ (2v

-

~)p

m~o

1 2 Ilvm+ 11 ~ IqOI2.

Because of condition (5.6), there exists some 8 such that

0<~<8<1, 2v

and hence (2v -

P "6) > O.

With such a 8 fixed, the inequality (5.15) shows that I q m+ l

_ q’" 12

IIvm + 111 2

= I p m+ l _

p’" 12 --+- 0 as

m-r»,

= lIum + 1-uI12 --+- Oas m --+- oo

(5.16)

The convergence ofu m + 1 to u is thereby proved. Now by (5.14) we see also that the sequence p’" is bounded in L 2 (Q ). We can then extract from p" a subsequence pm’ converging weakly in L 2 (Q) to some element p*. The equation (5.2) gives in the limit v((u, v)) - (p*, div v)

= if,

v), V v EHa(Q),

and by comparison with (5.7), we get (p - p*, div v)

whence

= 0,

V v E Ha(Q),

142

The steady-state Stokes equations

Ch. I, §S

grad (p - p.) = 0, p; = p + const. From any subsequence of p’", we can extract a subsequence converging weakly in L 2(O) to p + c; hence the sequence pm converges as a whole to p for the weak topology of L2 (O)/6t Remark 5.1. Let us define p by imposing the condition

J

n p(x)dx

=O.

Let us suppose that po in L 2 (0) is chosen so that

J

n pO(x)dx

= O.

Then clearly we have

J

n pm (x)dx = 0,

m ;>1,

and the whole sequence p’" converges to p, weakly, in the space L 2 (0).

o

5.2. Arrow-Hurwicz Algorithm.

In this case too, the functions u and p are the limits of two sequences which are recursively defined. We start the algorithm with arbitrary elements u O, pO, uO EHA(O), po EL 2(O). (5.17)

u’", p"

of

I

When pm and u" are known, we defme pm+l and u m+ 1 as the solutions m 1

u + EHA(O) and ((u m+1 - u’", v»

+

pv((u m, v» - p(pm, div v)

= if, v), V v E HA(O), pm+l EL

2(0)

and

cx(pm+l - p’", q)

+ p(div u m+ 1, q) = 0, V q

E L 2(0).

(5.18)

I

(5.19)

Numerical Algorithm,

Ch. I, §5

143

We suppose that p and (X are two strictly positive numbers; conditions on p and (X will appear later. The existence and uniqueness ofu m + 1 EHA(fl.) satisfying (5.18) is easily established with the projection theorem; u m+ 1 is the solution of the Dirichlet problem

_.6.um+ 1 = -.6.u m + pv.6.um - p grad pm + f.

u m + 1 EHb(fl.)•

(5.20)

Then pm+l is explicitly given by (5.19) which is equivalent to

pm+l = pm _

! ..

(X

div u m+ 1 E £2 (fl.).

(5.21)

Convergence of the Algorithm. Theorem 5.2. If the numbers

(X

and p satisfy

2av O
(5.22)

then, as m-r», u" converges to u in HA (fl.) and »" converges to p weakly in £2 (fl.)/61 Proof. Let

(5.23) (5.24) Equations (5.18) and (5.7) give

«v m+ 1 - v m, v)) + pv«v m, v)) = p(qm, div v), V v E Hb(fl.), and taking v

= v m + 1 we obtain

IIvm+ 111 2 _lIv mll2 + IIv m+ 1 - v mll 2 + 2pvllv m+ 1 11 2

= 2pv«v m+ 1 , vm+1_ vm )) + 2p(qm ,div v m+ 1 ) p2

v2

<; 6I1vm+l_vmIl2+ __ lIvm+1112 6

+ 2p(qm, div v m+ 1 ) ,

(5.25)

Thesteady-state Stokes equations

144

Ch. I, §S

where 6 > 0 is arbitrary at the present time. Equation (5.19), with q = qm+1 can be written as 0:1 qm+11 2-o:lqm 12+0:I qm+l_ qm 12= _2p(qm+l, div u m+ 1 )

=_2p(qm,

div v m+ 1 ) - 2p(qm+l_ qm,div vm +1) <._2p(qm, divvm+ 1)+ 2plqm+l_qm Ildivv m+ 11

<. (by (5.13)) <’-2p(qm,divv m+ 1)+2plqm+l- qm

IlIvm+111.

Finally, with the same I) as before, 0:I qm+11 2_ 0:1q m12+ o:l qm+l- q ml2 <. _2p(qm, divv m+ 1) +0:6I qm+l_ q mI2+

:1) IIv m+ 1112. 2

(5.26)

Adding inequalities (5.25) and (5.26), we get 0:1 qm+11 2 + IIv m+ 1112 _ o:lqm 12 _ IIvm 11 2+ 0:(l-6)l qm+l- qm 12

+(l_I))lIvm+l_vmIl2+p(2V_P~2

-

:6)lI vm+111 2 <. O. (5.27)

If condition (5.22) holds, then

and for some 0 < 6 < I sufficiently close to I, we have again

so that p(2v -

pv 2

6 -

P

0:6) > O.

By adding inequalities (5.27) for m = 0, ... , N, we obtain an inequality of the same type as (5.15), and the proof is completed as for Theorem 5.1.

145

Numerical algorithms

Ch. I, §S

Remark 5.2. It is easy to extend the Remark 5.1 to this algorithm. 5.3. Discrete Form of these Algorithms. We describe the discrete form of these algorithms in the case of finite differences (approximation APXI). In order to actually compute the step functions uh and 1rh which are solutions of (3.64), (3.71) and (3.73), we define two sequences of step functions U’{; , 1r’{;, of the type (5.29)

(5.30) which are recursively defmed by the analogue of one of the preceding algorithms. Uzawa Algorithm. We start with an aribtrary 1r2 of type (5.30). When TrW is known, we defme uW+ 1 and 1r;:,+1 by u;:,+1

E Wh and (5.31 ) (5.32)

where Dh is the discrete divergence operator defined by (3.69). If p satisfies the same condition (5.6), a repetition of the proof of Theorem 5.1 shows that, as m-r» (5.33) 1rW ~ 1rh

up to a constant;

(5.34)

the convergence holds-for any norm on the fmite dimensional spaces considered. Arrow-Hurwicz Algorithm. We start with arbitrary u2, 1r2 of type (5.29) and (5.30) respectively,

146

Ch. I, §S

The steady-state Stokes equations

When u’l: ,1r’f: are known, we defme u’l: + 1 , 1r’f: + 1 by

u’l:+1

Wh and «u’l:+1 - u’l:, vh»h + pv«u’l:, vh))h E

- p(Tr’l:, Dhvh) = if, vh), V vh

(5.35) E

Who

Tr’l:+1 (M) = Tr’l: (M) - !!.. Dhu’l: + 1 (M), V M E n~. a

(5.36)

If p satisfies the condition (5.22), an extension of the proof of Theorem 5.2 gives the convergence (5.33) - (5.34). Discrete Arrow-Hurwicz Algorithm. The problems (5.31) and (5.35) are discrete Dirichlet problems and their solution is easy and quite standard. Nevertheless, it is interesting to notice that in the fmite dimensional case, we can use another form of Arrow-Hurwicz algorithm, for which we do not have any boundary value problem to solve during the iteration process. When u’l:, Tr’l: are known, we define u’l:+1 by

u’l:+1

E

Wh and,

(u’l:+I_ u’l:, vh) + pv«u’l: , vh))h - p(Tr’l:, Dhvh)

(5.37)

and then 1r’f:+l is again defined by (5.36). The variational equation (5.37) is equivalent to the following equations. n

u’l: + 1 (M) = u’l: (M) + pv

?

1=1

(6~ u’l: )(M)

- P (Vh TrJ:’ )(M) + t"(M), V M E

n

(5.38)

1 h,

where Vh and fh were defined in (3.74). The proof of Theorem 5.2 can be extended to this situation as follows. Since Wh is a fmite dimensional space, all the norms defmed on Wh are equivalent, and hence there exists some constant S(h) depend› ing on h, such that (5.39)

Slightly compressible fluids

Ch. I, §6

147

We will compute S(h), and use this remark extensively in Chapter III (S(h) = 2(~7=11/hl)1/2). Now, if p satisfies:

0< < p

2001

(5.40)

Olv 2S2(h) + l ’

then the convergences (5.33) - (5.34) are also true for the algorithm (5.36) - (5.37). The proof is the same as for Theorem 5.2. The inequality (5.25) is just replaced by (vI:’ = ul:’ - uh’ "W = lI’W - lI’h) :

IvW+ 11 2-lvWI 2 + IvW+ 1- vWI + 2pvllvW+ll1~

= 2pv«vW+ 1, vW+ 1- vW»h + 2p ("W,Dh vl:’+l) E;;; 2pvllvW+ 1l1h

IIvW+ 1 - vW IIh + 2p("W, o,,vI:’+1 )

E;;; ’2pvS (h) II vW +Il1h E;;; l)lvW+ 1 _ vl:’12

Iv;:’+1 - vW I + 2p(,,1:’ , D" vI:’+1)

+ p2v2S2(h)IIVW+ll1~ l) (5.41)

The inequality (5.27) is accordingly changed to

0l1"W+ 112 + IvW+ 11 2-0l1"WI 2-lvWI 2 + Ol(l-l)I"W+1 -

+ p(2v _ PV2~2(h)

"I:’ 12 + (l-l)lvW+ 1 _

~)I VI:’+l 1~

E;;;

vW 12

(5.42)

0,

and because of (5.40), inequality (5.42) leads to the same conclusion as (5.27).

§ 6. Slightly Compressible Fluids The stationary linearized equations of slightly compressible fluids are

The steady-state Stokes equations

148

1 . -vl:iu e - -e grad div u e Ue

=0

on

Ch. I, §6

=finn,

(6.1)

an,

(6.2)

where e > 0 is "small". Equations (6.1) - (6.2) are also the stationary Lame equations of elasticity. We will show that equations (6.1) - (6.2) have a unique solution u e for e > 0 fixed, and that u. converges to the solution U of the Stokes equations as e-+O. At first, equations (6.1) - (6.2) were used as "approximate" equat› ions for Stokes equations - one way to overcome the difficulty "div u = 0", was to solve equations (6.1) - (6.2) with e sufficiently small in place of solving the Stokes equations. Nowadays, since many efficient algorithms are known for solving the Stokes equations, and since the discretization of (6.1) - (6.2) leads to a very ill-conditioned matrix for very small e, one can try to do the converse: compute ue for small e by using Stokes equations. In Section 6.1 we show the relation between u, and U and in Section Then in Section 6.2 we give an asymptotic development of ue as ~. 6.3 we show how one can proceed to compute ue ’ for small e, using this asymptotic expansion.

6.1. Convergence of Ue to U. Theorem 6.1. Let .0 be a bounded Lipschitz domain in eRn. For e > 0 fixed, there exists a unique », E (.0) which satisfies (6.1).

HA

When e-+o,

ue -+ U in the norm of HA (.0), -

where

U

divu

_ _e

e

-+

p in the norm of L2 (n ),

(6.3) (6.4)

and p are defined by (2.6)-(2.9) and moreover

In pix) dx = o.

(6.5)

Proof. It is easy to show that the problem (6.1)-(6.2) is equivalent to the following variational problem:

Ch. I, §6

Slightly compressible fluids

To find U e E

149

HA (n) such that

1 v«U e , JI)) + - (div Ue ’ div JI) e

=(f, JI), V JI EHA(n).

(6.6)

Actually, if u, satisfies (6.1)-(6.2) then ue E HA(n) and satisfies By a continuity argument, it satisfies (6.6) too, (6.6) for each v E ~(n). for each v E (n). Conversely, if u, E (n) is a solution of (6.6) then u e satisfies (6.1) in the distribution sense and (6.2) in the sense of trace theorems. The existence and uniqueness ofu e satisfying (6.6) results from the projection theorem: we apply Theorem 2.2 with

HA

HA

I

W= HA (n), a(u, JI) = v«u, JI)) + - (div u, div JI), e


The coercivity of a and the continuity of a and Qare obvious. To prove (6.3) let us subtract (2.7) from (6.1); this gives I

.

-vli (u e - u) - -e grad div Ue

=+ grad p

(6.7)

and thus I v«u e - u, JI)) + -(div ue ’ div v)

e

= -(P,

div JI), V II EHA(n). (6.8)

Equation (6.8) follows easily from (6.7) for v E argument, (6.8) is satisfied for each v E (n). Let us put v = ue - U in (6.8); we obtain

HA

~(n);

by a continuity

1

vllu e - uII2 + -Idiv uel 2

e

= - (P, div ue)

~

Ipl [div uel

1 e ~ -Idivui+ -lpl2

2e

so that

2

1 1111"e- .,11 2+ 2 ldiv ue l2 <; !.-lpI2. e

2

This proves (6.3). Consequently, (6.7) shows that

(6.9)

150

The steady-state Stokes equations

~

aXi

(diV Ue ) -+ _ e

Ch. I, §6

ap

(6.10)

aXi’

in the norm of B-1 (0), i = 1, ... , n (t::.(U e ~ u) converges to 0 in B- 1 (0), because of (6.3». According to the following lemma, n

div u a div u, " Ip + - - eI ~ L. II-a (P + --)IIH e i"’l xi e since, because

L

s,

(P +

vanishes on

d~

", ) dx

I

(0)’

(6.11)

ao and (6.5) holds,

=o.

The convergence (6.4) is proved. Lemma 6.1. Let 0 be a bounded Lipschitz domain in eRn. Then there exists a constant c = c(O) depending only on 0, such that laIL 2(0)<:c(0)

{I J

o

adxl +

t 1-1

lIaaaIlH-I(O)} , X•

(6.12)

I

for every a in L2 (0). Proof. Let us denote by [a] the expression between the brackets on the right-hand side of (6.12); [a] is a norm on L2(0); it is obviously a semi-norm and, if [a] = 0, then a is a constant since aajaxi = 0, i = 1, ... , n, and this constant is zero since 10 a dx = O. It is clear that there exists a constant c’ = c’(O) such that [a] ~c’(0)laIL2(n),

VaEL 2 (fl. ).

(6.13)

If we show that L 2(O) is complete for the norm [a] then, by the closed graph theorem, [a] and lel will be two equivalent norms on L 2(0) and (6.12) will be proved. In order to show that L 2(0) is complete for the norm [a], let us consider a sequence am ’ which is a Cauchy sequence for this norm. Then the integrals 10 am dx form a Cauchy sequence in R and the derivatives aa m jaxi are Cauchy sequences in B-1 (fl.):

Ch. I, §6

L,

Slightly compressible fluids

151

am dx-+ A as m-rw,

aa m -+Xt as m-r»,

-

aXt

1

(6.14)

in H- (il), 1 ~ i

It is clear that (grad am ’ v)

= 0,

~

n.

(6.15)

V v E 1/ and, because of (6.15),

m

By Proposition 1.1, there exists some distribution a such that

aa

XI•= uxt ~

,

l~i~n

Proposition 1.2 shows that a E L2(il). We can choose a so that

L

adx=A

and it is easy to see that the sequence am converges to this element a of

L 2(il) in the norm [a].

Remark 6.1. If il is not connected, (6.12) is true if we replace IIo adxl by

t I J adxl OJ

where the ilj are the connected components of il. For extending Theorem 6.1 to this case we just have to defme p by imposing: (6.16)

6.2. Asymptotic Expansion olu e

From now on we denote by u O and po the solution of Stokes problem (2.6)-(2.9) which satisfies (6.5) (in place ofu and p)

Ch. I, §6

The steady-state Stokes equations

152

We will show that ue has an asymptotic development ue = uO +

U

+ eu 2 + ... + eNuN + ...

H6

where all the u! belong to the space (0). The functions ui and some auxiliary functions defined as follows:

(6.17)

rJ

are recursively

uO, pO, are already known;

(6.18)

when u" -1, p’" -1 are known (m ~ I), we define u" and pm as the solut› ions of the nonhomogeneous Stokes problem

(6.19) -7)

6.u m + grad p’" = 0

(6.20)

div u" = _pm-1

(6.21) (6.22)

The existence and uniqueness of u m and pm follow from Theorem 2.4. The condition (6.22) is useful in two ways:it ensures the complete uniqueness of pm which is otherwise only unique up to the addition of a constant; it also ensures the compatibility condition necessary for the level m + I:

f

In

div u m+ 1 dx

We denote by ,;:,

=

Jr

u m +1 . vdr

=0 =

II:, N;;;;’ I, the quantities

!n

p’" dx.

(6.23)

N

u’! = ~ emum e

II: =

m=O

’

(6.24)

N

~

m=O

empm.

(6.25)

Theorem 6.2. Let 0 be a bounded domain of class ~2 in tRn Then for each m ;;;;, 1, there exist functions u", p", uniquely defined by (6.19)-(6.22).

Ch. I, §6

Slightly compressible fluids

153

For each N ~ 0, as e-+O,

u-tI!

e N e~

e

I

0 in the HA(O) norm,

div U

(6.26)

0 in the £2(0) norm.

(- _ _e _ pN)~ EN e e

(6.27)

Proof. The existence and uniqueness results have been established as previously remarked. We multiply (6.20) by em and add these equations for m = I, ... , N. We then add the resulting equation to the equation satisfied by uO and pO (formerly denoted u and p)

-vb.uo + grad pO =f. After expanding we obtain -vb.~

I e

- - grad div ~

=f - eN grad pN .

By comparison with (6.3) we find

u, - u’j E HA (0),

v" E £2 (0),

(6.28)

I

-vb.(u e - u1j - ; grad div (ue - u’j) = + eN grad pN.

(6.29)

As for (6.8), we show that (6.29) is equivalent to I

v«u e - u’j, v» + - (div (ue - u’j), div v) = e

_eN (pN, div V), V u EHA(O).

Putting v = Ue - u’j in (6.30) we get I

vllue- u’j1l2+ -ldiv(ue-u~)12 e

= - eN (pN, div(u e- ~» ~eNlpNlldiv(ue-

~2e

1

Idiv(ue-u~)12+

u~)1

e2N + 1

-2- lpNI2

(6.30)

154

The steady-state Stokes equations

so that

I 2e IdiV(Ue-~)12~-2-1~12.

vllue-~1I2+

Ch. I, §6

e2N + 1

(6.31)

The inequality (6.31) clearly implies (6.26). This, in turn, implies that 0 in B-1 (n) and hence (6.29) shows that

lIeN b.(Ue - ufj)~

a

II _N+ (:’"

~

uXi

diIV (Ue -

N)~ a~.~

Ue

uXi

10 H-I(n) \HI,

1....-I’....- n.

(6.32)

But N

. UN - -I grad div

e

e

=-

"" em grad div u m -I £..

e

m=l

N

=~ L e

m=l

em grad pm-l

= grad
which along with (6.32) implies that _I

eN

~

aXi

(_ divu e e

_

P’!) e

~

0 inB-I(n)

as e--O. Finally (6.27) results from (6.22), (6.25) and Lemma 6.1. Remark 6.2. Remark 6.1 can be easily adapted to Theorem 6.2: this theorem holds for non-connected sets n, provided we replace condition (6.22) by the conditions:

!

p"’dx=O

(6.33)

nj

on each connected component n j of n. Remark 6.3. A discrete version of the result given by Theorem 6.2 is studied by R. S. Falk [I],

6.3. Numerical Algorithms Let us show briefly how one can extend the algorithms described in Section 5 to the solution of the nonhomogeneous Stokes problems (6.19) - (6.22) which is, at this point, the only difficulty for practical computation of the asymptotic expansion (6.17) of ue

Slightly compressible fluids

Ch. I, §6

ISS

We only describe the adaptation of the Uzawa algorithm. Changing our notation, we write problem (6.19) - (6.22) as the problem: to find v, p such that

v E HA (0), p E -v6.v

+ grad

L 2 (0)

(6.34)

= 0,

p

(6.35) (6.36)

div v = lj),

J

(6.37)

n p(x)dx = 0,

L~(x)dx

where lj) is given with

=O.

(6.38)

The existence and uniqueness of v and p are known. We start the algorithm with any

pO E £’ (il), such that

f

pO (x) dx = 0,

(6.39)

n

When pm is known, we define v m+ l (m ~ 0) by

vm + l EHA(O) and v«v m+ l , w» - (pm, div w)

= 0, V wE HA(O)

(6.40)

pm+l EL 2(0) and (pm+l_

p’", 8) + p(div vm+l_lj), 8) = 0,

8 E L 2(0).

(6.41)

The equation (6.40) is a Dirichlet problem for yn+l:

vm + l EHA(O) -v6.v m+ 1

=-

grad pm E H- 1 (0)

(6.42)

and (6.41) gives pm + I directly as p" +1

= p" _

p(div vm+l_lj) E L 2(0).

(6.43)

The steady-state Stokes equations

156

Ch. I, §6

We notice that

J

n p’" dx = 0, V m .. O.

(6.44)

Exactly as for Theorem 5.1 (see also Remark 5.1), one can prove the following result. Theorem

6.3. If the number p satisfies

O
(6.45) pm+l

CHAPTER II STEADY-STATE NA VIER -STOKES EQUATIONS

Introduction In this chapter we will be concerned with the steady-state Navier-Stokes equations from the same point of view as in the previous chapter, i.e., existence, uniqueness and numerical approximation of the solution. How› ever, there are three important differences with the linear case; these are: - The introduction of the compactness methods. For passing to the limit in the nonlinear term we need strong convergence results; these are obtained by compactness arguments. - Some technical difficulties related to the nonlinear term and connected with the Sobolev inequalities. Their consequence is a treat› ment of the equation which varies slightly according to the dimension of the space. - Thenon-uniqueness ofsolutions, in general. Uniqueness occurs only when "the data are small enough, or the viscosity is large enough". In Section 1 we describe some existence, uniqueness, and regularity results in various situations (0 bounded or not, homogeneous or non-homogeneous equations, ... ). In Section 2 we prove a discrete Sobolev inequality and a discrete compactness theorem for step function spaces considered in the approximation (APX1) of V (approx› imation of V by finite differences) and for nonconforming element function appearing in the approximation (APX5) of V. The similar results for the approximations (APX2), ... , (APX4), are already available as consequences of the theorem in the continuous case. Section 3 deals with the approximation of the stationary problem: discretization and resolution of the discretized problems. The purpose of Section 4 is to show an example of non-uniqueness of solutions for the steady-state Navier-Stokes equations. The proof is based on a topological degree argument; the presentation is essentially self-contained.

§ 1. Existence and Uniqueness Theorems. In this section we study some existence and uniqueness results for the steady-state (nonlinear) Navier-Stokes equations. The existence results are obtained by constructing approximate solutions to the 157

Ste4dy-rtate Navier-Stoke. equlltion.

IS8

Ch. II, § 1

equation by the Galerkin Method, and then passing to the limit, as in the linear case. As we have already said, for passing to the limit we need, in the nonlinear case, some strong convergence properties of the sequence, and these are obtained by compactness methods. In Section 1.1 we recall the Sobolev inequalities and a compactness theorem for the Sobolev spaces; this theorem is of course the basic tool for the compactness method. In Section 1.2 we give a variational formulation of the homogeneous Navier-Stokes equations (i.e., the Navier-Stokes equations with homogeneous boundary conditions); we study some properties of a nonlinear (trilinear) form which occurs in the variational formulation. We then give a general existence theorem and a rather restricted uniqueness result. In Section 1.3 we consider the case where the set n is unbounded and we give regularity results for solutions. Section 1.4 deals with inhomogeneous Navier-Stokes equations. 1.1. Sobolev Inequalities and Compactness

Theorems.

Imbedding Theorems.

We recall the Sobolev imbedding theorems which will be used frequently from now on. Let m be an integer and p any finite number greater than or equal to one, p;> 1; then, if lip - min = llq > 0 the space Wm ,p (lR n ) is included in Lq (lR n ) and the injection is continuous. Ifu E wm,p(lR n) and lip - min = 0 then u belongs to Lq(~) for any bounded set ~ and any q, 1 ~ q < 00. If lip - min < 0 then a function in Wm,p (lRn ) is almost everywhere equal to a continuous function; such a function has also some Holder or Lipschitz continuity properties but these properties will not be used here; ifa function belongs to wm,p(Rn) with lip - min < 0 then the derivatives of order ex belong to Wm-o,P( lR n ) and some embedding results of preceding type hold for these derivatives if lip - (m - a)ln> o. ForuEWm,P(lRn),m;> 1, 1 ~p
1

m 1

1

m

if

P- ’;i = q> 0, luILq(Rn) ~

if

p- ’;i = 0, luILq(eJ) Y bounded set

c(m, p, n)lIu IlWm,p(Rn),

~ c(m, p, n, q, tJ) lIullwm,p(Rn) ~ C

lR n , Y q, 1 ~ q < 00,

(1.1)

Existence and uniqueness theorems

Ch. II, § 1

if

m

1

p - n < 0, IUI~o«(!)

~ ctm, n, p,

V bounded set

159

o ) IIu" wm, P(Rn)

,

o, o C 6ln

If 0 is any open set of 6ln , results similar to (1.1) can usually be obtained if 0 is sufficiently smooth so that: There exists a continuous linear prolongation operator (1.2)

Property (1.2) is satisfied by a locally Lipschitz set O. When (1.2) is satisfied, the properties (1.1) applied to IIu, u E Wm,P(O) give in partic› ular, assuming that u E wm,P(O), m;> 1, 1 < p < 00, and (1.2) holds: if

P- n= q> 0, lulLq(O) ~

if

p- n= 0, lul LQ( l!i) ~

m

1

1

m

1

ctm, p, n, 0) II ull Wm’p(O)’

ctm. p, n, q, o , 0) IIull wm, p(O)’

any q, 1 ~ q < 00, any bounded set () C 0,

if

p- n< 0, lul~o«(!) m

I

~ ctm, p, n, q, 0,

any bounded set o, () C

o

IIullwm,p(o)’

(1.3)

n.

o

When u E Wm,P(O), the function uwhich is equal to u in 0 and to in belongs to wm , p(6ln ), and hence the properties (1.3) are valid without any hypothesis on O. The case of particular interest for us is the case p = 2, m = 1, i.e., the case HACO). Without any regularity property required for 0 we have for u EH5(0)

CO,

n

= 2, luILq(O)~c(q,(),

0) IIuIIH~(O)

V bounded set () CO, V q, 1 ~ q < 00

n« n

3,

lu1L1(O)

~ c(O) IIuIIH~(O)

= 4, lu1 L4(O) ~

n;> 3,

lui

c(O) IIuIIH~(O)

L2n/(n-2) (0) ~ c(O) IIuIIH~(o>,

(1.4)

Ch.U, §1

StudY-Itate Navier-Stokea equattona

160

Compactness Theorems.

Theorem 1.1. Let 0 be any bounded open set of(i{n satisfying (1.2). Then the embedding

(1.5)

is compact for any ql’ 1 E;;;; ql < 00, ifp ’> n, and for any ql’ 1 E;;;; ql (q given by l/p - l/n = l/q) if 1 E;;;;p < n. With the same values of p and ql’ the embedding o

Wl,p(O) C Lql (0)


is compact for any bounded open set O. As a particular case of Theorem 1.1 we notice that for any unbounded set 0, if u E W1 ,P (0 ), then the restriction of u to o, o C (j C 0, ~ bound› ed, belongs to Lql (~) and this restriction mapping is compact (1.7)

(same values of p and Ql)’ For all the preceding properties of Sobolev spaces, the reader is referred to the references mentioned in the first section of Chapter I (see also at the end the comments on Chapter I). 1.2. The Homogeneous Navier-Stokes Equations.

Let 0 be a Lipschitz, bounded open set in (i{n with boundary r, let f E L2(0) be a given vector function. We are looking for a vector function u = (ul’ ... , un) and a scalar function p, representing the velocity and the pressure of the fluid, which are defined in 0 and satisfy the following equations and boundary conditions: n

-vtsu « divu u

=0

2:

1=1

=0 on

u;D;u+gradp=j

(1.8)

inO

in 0

(1.9)

r.

(1.10)

Exactly as in Section 2.1 of Chapter I, if t. u, p are smooth functions satisfying (1.8) - (1.10) then u E V and, for each v E

r.

161

Existence and uniqueness theorems

Ch. II, § 1

v«u,

V» + b (U,

u, V) = if, V)

(1.11)

where (1.12) A continuity argument shows moreover that equation (1.11) is satisfied .by any V E V. Conversely, if u is a smooth function in V such that (1.11) holds for each V E1"; then because of Proposition 1.3, Chapter I, there exists a distribution p such that (1.8) is satisfied, and (1.9) - (1.10) are satisfied since u E V. For u and v in V, the expression b (u, u, v) does not necessarily make sense and then the variational formulation of (1.8) - (1.10) is not exact› ly "to find u E V such that (1.11) holds for each v E V." The variational formulation will be slightly different, and this will be stated after studying some properties of the form b(u, v, w). Let us introduce first the following spaces:

V= the closure

of course

of "/I’inHA(n) n L n (n);

HA (n) n Ln (n) and +

lIuIIH~(n)

(1.13)

Vare equipped with the norm (1.14)

luILn(n)’

In general iTis a subspace of V, different from V but, because of (1.4), iT= V for n = 2, 3 or 4 (and n bounded);

V, = the closure

of j/in HA(n)

it is understood again that the Hilbertian norm

+ lIulI~s(n)}1/2;

{lIulI~~(n)

V, is

n HS(n),

(s ~ 1);

(1.15)

HA (n) n H’(n) and V, are equipped with

(1.16)

included in V.

The Trilinear Form b. The form b is trilinear and continuous on various spaces among the spaces V, V, The most convenient result concerning b is the following result which is independent of any property of n.

v,.

Lemma 1.1. Theform b is defined and trilinear continuous on HA(!?’) X HA(n) X (HA(n) n Ln (n», n bounded or unbounded, any

Ch.U, §1

SteadY-Itate Navier-Stoke, equation,

162

dimension ofspace tRn Proof. Ifu, v E V and Uj E

W

E

v: then because of (1.4)

(n;> 3):

L2n/(n- 2)(0 ), D j vj E L2 (0 ), wjELn(O), 1
i
By the Holder inequality, U;(Djvj)wj belongs to Ll (0) and

I

f

u,

U,ll,’I"i dx] <; I IL2_/(_ -

.0

2)(0)

( 1.17)

Ill,’/1 L’ (0) IwIIL"to).

Then b(u, v, w) is well defined and Ib(u, v,

w)1 <. c(n) lIullHH.n ) I vI H~(.n)

lIwilH~

(.0)

n Ln(.nf

(1.18)

The form b is obviously trilinear and (1.18) ensures the continuity of b. When n 2, we have the same result, (HA(O) n L2(0) =HA(O», but (1.17) must be replaced by

=

I

t

U,ll,’I"i dx I<; Iu,IL’(o) ID,•h•(0) IwIIL’(n)’

(1.19)

In particular one has

Lemma 1.2. For any open set 0, b is a trilinear continuous form on V X V X Y.’ If 0 is boundedand n <. 4, b is trilinear continuous on VX VX

v.

We will prove, when needed, other properties of b similar to those given before; the proof will be always the same as in Lemma 1.1 (use of Holder’s inequality and the embedding theorem (1.4». W~ denote by Btu; v), u, v EHMO), the linear continuous fonn on V defined by (B(u, v),

W) --

1" b(u, v, W), u; v EHo(O), V W E" V.

For u =v, we write B(u)=B(u,u),

uEHA(O).

Another fundamental property of b is the following:

(1.20) (1.21)

Existence and uniqueness theorems

Ch. II, § 1

Lemma 1.3. For any open set

n,

b(u, v, v) = 0, VuE V, v EHA(n) b(u, v, w)

= -b(u,

163

n Ln(n)

w, v), VuE V, v, wEHA(n)

(1.22)

n Ln(rl.). (1.23)

Proof. Property (1.23) is a consequence of (1.22) when we replace v by

v + w, and we use the multilinear properties of b.

In order to prove (1.22), it suffices to show this equality for u E "f/’ and v E!’I(rl.). But for such u and v

L:

btu, v, v) = - -1 n 2 f=1

1 n

div

U(Vj)2

dx =

o.

(1.24)

Variational Formulation. For n bounded, and n arbitrary, we associate with (1.8) - (1.10) the problem To find u E V such that v«u, v)) + btu, u, v)

= (f, v),

Vv E

(1.25)

iT:

if given in L2(n)). It is clear from

(1.11) and (1.13) that if u and pare smooth functions satisfying (1.8) - (1.10), then u satisfies (1.25). Conversely if u E V satisfies (1. 25), then (-vl:iu +

L:i

uiDiu -

t. v) = 0, V v E

"f/’

(1.26)

l:iuEH-l(rl.),fEL2(n),anduiDiuELn’(n)(lln’= I-lIn), since ui E L 2n/n- 2(n) by (1.4) and Diu E L2(rl.). Now, according to Proposit› ion 1.1 and 1.2, Chapter I, there exists a distribution p E Lloc(rl.)(l), (1) Proposition L1.1lhows the existence of p as a distribution, p E~
L\oc<Sl).

SteDdy•state NavieroStokes eqU/ltions

164

Ch. II, § 1

such that (1.8) is satisfied in the distribution sense, then (1.9) and (1.10) are satisfied respectively in the distribution and the trace theorem senses. Theorem 1.2. Let 0 be a bounded set in Rn and let / be given in H-l (0). Then Problem (1.25) has at least one solution u E V and there exists a distribution p E Lkc(O) such that (1.8) - (1.9) are satisfied. Proof. We have only to prove the existence of u; the existence of p and the interpretation of (1.8) - (1.9) have already been shown. The existence ofu is proved by the Galerkin method: we construct an approximate solution of (1.25) and then pass to the limit. The space Vis separable as a subspace of (0). Because of (1.13) there exists a sequenceJVl’ ... , wm ’ , of linearly independent elements of j"’which is total in V. This sequence is also free and total in V. For each fixed integer m ;> 1, we would like to define an approximate solution um of (1.25) by

HA

m

=!

Um . 1=1’ ~t mWt’ ~I .. mER

(1.27)

1I«Um, Wk)) + b(Um, Um’ Wk) =

’

k = 1, ... , m.

(1.28)

The equations (1.27) - (1.28) are a system of nonlinear equations for

~ I, m’ .. , ~m. m’ and the existence of a solution of this system is not

obvious, but follows from the next lemma.

Lemma 1.4. Let X be a finite dimensional Hilbert space with scalar product [.,.] and norm [.] and let P be a continuous mapping from X into itself such that [P(~),

~]

> 0 for

Then there exists P(~)

~ E

=k

[~]

X,

[~]

~

> O.

(1.29)

k, such that

= O.

(1.30)

The proof of Lemma 1.4 follows the proof of Theorem 1.2. We apply this lemma for proving the existence of Um, as follows: X = the space spanned by wI, .. , wm ; the scalar product on X is the scalar product « .,.)) induced by V; and P = Pm is defmed by [Pm (U), v] = «Pm (u), v)) = J.’«U, v)) + b(u, u, v)-

if, v), Yu, v EX.

Ch.n, §1

Existence and uniqueness theorems

165

The continuity of the mapping Pm is obvious; let us show (1.29). [Pm (U), U]

= vlluII2 + b(u,

u, U) -

(I, U)

= (by (1.22» = vlluII2 -

(I, U)

~ vlluII2 -lltllprlluII,

(1.31) IIuII (vllull - Iltllv’). It follows that [Pmu, u] > 0 for IIull = k, and k sufficiently large: more precisely, k > ltv IltllV" The hypotheses of Lemma 1.4 are satisfied and [Pm (U), u] ;>

there exists a solution um of (1.27) - (1.28). Passage to the Limit. We multiply (1.28) by k = 1, ... , m; this gives

m

~k,

and add corresponding equalities for

vII"", 11 2 + b("""""" um) = (f, um)

or, because of (1.24), vII"", 11 2

=(I, um) ~

Iltllv,IIu", II.

We obtain then the a priori estimate: I

II"",II ~ ;- IttlIV’.

(1.32)

Since the sequence um remains bounded in V, there exists some u in V and a subsequence m’~oo such that u",,-"u for the weak topology of V.

(1.33)

The compactness theorem 1.2 shows in particular that the injection of V into L 2 (.0) is compact, so we have also um’ ~ u in the norm of L 2 (.0).

(1.34)

Let us admit for a short time the following lemma. Lemma 1.5.

then

n», converges

b(u lJ , ulJ ’

v)~b(u,

U,

to u in V weakly and in L2 (n ) strongly, v), V v E "f’:

(1.35)

Steady-atate Navier-8tokel equatioss

166

Ch. II, § 1

Then we can pass to the limit in (1.28) with the subsequence m’-+oo. From (1.33), (1.34), (1.35) we find that v«u, v»

+ b(u, u, v) = (f, v)

(1.36)

for any v = wI’ ... , wm Equation (1.36) is also true for any v which is a linear combination of wI’ ... , wm ’ Since these combinations are dense in a continuity argument finally shows that (1.36) holds for each v E Vand that u is a solution of (1.25).

v:

Proof of Lemma 1.4. This is an easy consequence of the Brouwer fixed point theorem. Suppose that P has no zero in the ball D of X centered at 0 and with radius k, Then the following application

=-

~-+S(~)

k

P(~)

[P(~)]

maps D into itself and is continuous. The Brouwer theorem implies then that S has a fixed point in D: there exists ~o ED, such that [~o]

k P(~o) [P(~o)]

- I! - ~o•

If we take the norm of both sides of this equation we see that = k, and if we take the scalar product of each side with ~o, we find

[~ ]2

o

= k2 =_ k [P(~o),

[P(~o)]

~o]

This equality contradicts (1.29) and thus P(~)

ofD.

must vanish at some point

Proof of Lemma 1.5. It is easy to show, as for (1.22) - (1.23), that b(up., up.’ v) = - b(up., v, Up.)

But Up.i converges to "t in L 2 (n ) strongly; since Div,E L"" (n), it is easy to check that

J

n u ....,V,ljdx-+

f

n "",D,vj dx.

Existence and uniqueness theorems

Ch. II, § 1

Hence

b(u",~

u"’) converges to b(u~

v~

u)

v~

=-

b(u~

167

u, v).

Uniqueness. FOI uniqueness we only have the following result:

Theorem 1.3. If n ~ 4 and if v is sufficiently large or f "sufficiently small" so that (1.37) then there exists a unique solution u of(1.25).

The constant c(n) in (1.37) is the constant c(n) in (l.18); its estimation is connected with the estimation of the constants in (104) and this is given for instance in Lions [I]. Proof of Theorem 1.3. We can take v n E;;; 4; we obtain with (1.22)

= u in (1.25) since V= V for (1.38)

so that any solution u of (1.25) satisfies 1

lIuli E;;; - Iltllv’ . v

(1.39)

Now let u; and u be two different solutions of (1.25) and let u = u; - u .We subtract the equations (1.25) corresponding to u; and u and we obtain v«u, v) + b (u.~

We take v

vllull

u, v) + btu, u., v)

V v E V.

(lAO)

= u in (lAO) and use again (1.22); hence: 2

=-

b(u, u., u).

With (1.18) and (1.39) this gives (for u V

= 0,

= u.)

lIull2 E;;; c(n) lIull2l1u.1I E;;;

c(n)

c(n)

(v - -

v

2

-lltllv,lIuli , v IIfllv ’) lIuli

2

~

O.

Because of (1.37) this inequality implies lIu’l = 0, which means u.. =u.....

Steady•state Navie,..Stokes equations

168

Ch. II, § 1

Remark 1.1. The solution of (1.25) is probably not unique if (1.37) is not satisfied or at least for v small enough if fixed). A non-uniqueness result for v small will be proved in Section 4 for a problem very similar to (1.25). Remark 1.2. For n > 4, Theorem 1.2 shows the existence of solutions of (1.25) satisfying (1.39): the maiorations (1.32) and (1.33) give indeed:

U

.

I

lIuli <; lim lIum,lI <; -llfll v’. m’.... v

(1.41)

Nevertheless the proof of Theorem 1.3 cannot be extended to this case; (1.40) holds for each v E iT and it is not possible to take v = u.

1.3. The Homogeneous Navier-Stokes Equations (continued). The Unbounded Case.

n

We can study the case unbounded by introducing the same spaces as in Section 2.3, Chapter I. Let us recall that

Y = the completed space of ’Yfor the norm 11•11. Let us consider also the space

Y= the closure Yn

(1.42)

Y

of o/’in the space

(1.43)

Ln (n) equipped with the norm:

lIuli + lIuliLn(n) (1) .

(1.44)

We recall that because of Lemma 2.3, Chapter I, we have the con› tinuous injection Y C {u E LCIt(n), DiU E L 2(n), 1 <; i <; n}

(1.45)

for n ~ 3, where

2n n-2

(1.46)

a=-.

(1) For a smooth open set

n,

Y is probably equal to Y n Ln(n), but this result is not proved.

Existence and uniqueness theorems

Ch. II, § 1

169

Since n is unbounded the spaces L’Y (n) are not decreasing when ’Y increases as in the bounded case and Y=1= Y even for n :E; ; 4. Lemma 1.2 cannot be extended to the unbounded case; however, we have: Lemma 1.6. For n ~ 3, the form b is defined and trilinear continuous on Y X YX Yand

b(u, v, v)

= 0,

b(u, v, w)

= -b(u, w, v),

Vu E Y, v E ~

(1.47)

VuE Y, v, w E Y.

Proof. The inequality (1.17) (n ~ 3) is valid; for u, v E Y, w E then have

(1.48)

Y we

so that

Ib(u, v, w)l:E;;; c(n) lIullyllvllyllwllf.

(1.49)

The relations (1.47) and (l.48) are proved exactly as (1.22) and (1.23): we prove them for u, v, wE fand then pass to the limit. The variational formulation of Problem (1.8) - (1.10) for n unbound› ed and n ~ 3 is set as follows:

To find u E Y such that v((u, v)) + b(u, u, v)

=(f, v),

V vE

Y.

(1.50)

Theorem 1.4. Let n be an open set in ~n, n ~ 3, and let f be given in the dual space of Y. Then there exists at least one u in Y which satisfies (1.50).

Y:

Proof. The proof is very similar to the proof of Theorem 1.2 (the bound› ed case). There exis,ts a sequence wI’ .. , w m ’ . , of elements of f which is free and total in Y and hence in Y; this sequence is not necessarily the same sequence as before. We define an approximate solution u m by

Steady-state Navier-Stokes equations

170

Ch.II, §1

m 1.1.

-m

= i~= I~i ’ m Wi’ ~i

,

(1.51)

m E tH,

(1.52) The existence ofu m satisfying (1.51) - (1.52) is proved exactly as before, using Lemma 104. We have then an a priori estimate analogous to (1.32): 1

1I"m II ~ - IIflIy’ (II-II = the norm in II

(1.53)

Y).

There exists therefore a subsequence m’-+oo and an element such that u m -+u weakly in Y.

U

E

Y (1.54)

The proof finishes as in the bounded case, except for the passage to the limit in the nonlinear term b(u m " "m’, v); it is not true that urn’ converges to u in L 2 (n ) strongly since u does not even belong to L 2 (n ) in general (Y C L2 (n )). Nevertheless, we have Lemma 1.7. b(u~,

converges to u in Y weakly, then,

Ifu~

u~,

v)-+b(u, u, v), V v E ’K

(1.55)

Proof. We can show that u~ -+

u strongly in Lroc(n),

(1.56)

which means that u~ -+ u in L 2 «(!}),

(1.57)

for each bounded set (!) C n. Actually, let 1/1 E9(tHn ) , 1/1 = 1 on (!) and let 0.’ be a bounded subset of 0. containing the support of 1/1. Then the functions 1/Iu~ belong to HA (0.’) and since u~ converges to u weakly in Y, 1/Iu~ -+ 1/Iu, weakly in HA(n’).

Hence 1/Iu~ -+ 1/Iu strongly in L 2 (n ); in particular

J,u~-uI2dx~f ~

and (1.57) follows.

1/I2Iu~-uI2dx-+(), 0’

Ch. II, § 1

171

Existence and uniqueness theorems

Since up converges to u for the L 2 norm, on the support of v, the convergence (1.55) is now proved as in the bounded case: b (up, up’ v)

= -b(u p, v,u p) -+ -b(u,

v, u)

= btu, u, v) 0

Remark 1.3. For n = 2, an element u of Y does not belong in general to any space [)(n). For this reason the proof of Lemma 1.6 fails and b is not defmed on Y X Y X Y. We can replace (1.50) by the problem: to find u E Y such that v«u, v))

+ b(u, u, v)

= (f,

v>, V v E

r:

The same proof as for Theorem 1.4 shows that such a u always exists provided f is given in Y’. 0 Remark 1.4. Since Y* Y (for any n), we cannot set v = u in (1.50). Therefore the proof of Theorem 1.2 cannot be extended to the un› bounded case even for n EO;; 4. 0

Regularity of the Solution. If the dimension n of the space is less than or equal to three, we can obtain some information about any solution u of (1.25) or (1.50) by reiterating the following simple procedure: the information we have on u gives us some regularity property of the nonlinear term n

~

i=1

",Diu.

We then write (1.8) - (1.10) as n

-v6.u

divu

+ grad

= 0,

in

u = 0 on I’:

p =f

n,

- ~

i=1

",Diu, in

n,

(1.58) (1.59) (1.60)

using the’ available regularity properties of f and Proposition 2.2 of Chapter I we obtain new informations on the regularity of u. If the properties of u thus obtained are better than before, we can reiterate the procedure. Let us show, for example, the following result:

Steady-state Navier-Stokes equations

172

Proposition 1.1. Let 0 be an open set of class ~fS)

f be given in ~ ... (n).

Ch.n, §l

in lfi2 or lfi3 and let

Then any solution {u. p} of (1.8)-( 1.10) belongs to ~ ... (n) X ~fS)(n).

Proof. Let us begin with the case where 0 is bounded. The nonlinear term ~7=1 ujDju is also equal to ~7=1 Dj("’ju), because of (1.59). If n = 2, Uj belongs to LCl! (0) for any a, 1 <; a < +00 (by (1.4», and then UjUt belongs to LCl!(O) for any such a, and Dj(ujut) belongs to W-l,Cl!(O) for any such a. Proposition 2.2, Chapter I, shows us that U belongs then to W1,Cl!(0), and p belongs to LCl!(O), for any a. For a> 2, Wl,Cll(O) C L (0) because of (1.3); hence ujDjuELCl!(O) for any a. Then, Proposition 1.2.2 shows us that U E W2•Cl!(0), P E W1•Cl!(0) for any a. It is easy to check that ujDjUE WI, Cl!(0), so that U E W3, Cl! (0 ). Repeating this procedure we find in particular that OO

U

E H’" (0), p E H" (0), for any m ~ I.

(1.61)

The same properties hold for any derivative of U or p; (1.3) implies therefore that any derivative ofu or p belongs to~(n), and this is the property announced. For n = 3, we notice that Uj E L 6 (0) (by (1.4» and then ujDjut E L 3 /2(0).

Proposition 1.2.2 implies that U E W2 , 3/2(0); but (1.3) shows us that U E LCl!(O) for any a, I <; a < +00 (p = 3/2, m = 2, n = 3). Therefore Dj(ujut) E W-l,Cl!(O),

for any a, and at this point we only need to repeat the proof given for n = 2. If 0 is unbounded, we obtain the same regularity on any compact subset of by applying the preceding technique to 1/Iu where 1/1 is a cut-off function, 1/1 Ef» lfin), 1/1 = I on the compact subset of O.

n

Remark 1.4. (i) It is clear that we can assume less regularity for f and obtain less regularity for U and p. (ii) The same technique fails when n ~ 4. For instance, for 0 bounded and n = 4, if we write the nonlinear term as Dj(uju), we just have UiUj E L 2(0), Di(uju) EH-l (0), so that U EHij(O); if we write the non› linear terms as ujDju, we have UiDjU E L 4 / 3 (0), so that

Existence and uniqueness theorems

Ch. II, § 1

173

E W2 , 4/3(0); but this gives nothing more than Uj E £4 (0), 2(0), DjUjE L which was known before. U

1.4. The Non-homogeneous Navier-Stokes Equations.

Let 0 be an open bounded set in
-vllu +

2:i=1

UiDjU

+ grad p = t, in 0,

divu=O, in 0, U

= ¢,

(1.62) (1.63)

on I’.

(1.64)

We will suppose that 0 is of class ~2, that t is given in H-1 (0) and that ¢ is given in the following slightly restrictive form: ¢

= rot r

(1.65)

where (1.66)

and rot denotes the usual rotational operator for n = 2, 3; for n ~ 4, rot denotes a linear differential operator with constant coefficients, such that div (rot == 0 rn

n

Theorem 1.5. Under the above hypotheses, there exists at least one H 1 (0), and a distribution p on 0, such that (1.62) - (1.64) hold.

UE

Proof. Let 1/1 be any vector function belonging to H1 (0) n L n (0) such that I/; EH 1(0) n Ln(O), div 1/1 I/; = ¢ on I’,

Let us set

" u=u -I/;.

= (1.67)

Steady-state Navier-Stokes equations

174

Ch. II, § 1

Then u belongs to HI (n) and satisfies (1.63); (1.64) amounts to saying that

u" E V.

(1.68)

Equation (1.62) is equivalent to

(1.69) where

; =f + vf). t/J -

n

2:

i=l

t/JiDi t/J.

We remark that

" fEH-l(n),

which we show as follows. It is clear thatf + vf).t/J EH-l (n); we notice moreover that t/JiDiErri (n), l/ol + I/Ol = 1, Ol = 2n/(n-2) if n;> 3, any Ol> 2 if n = 2; since HA(n) C La (n), t/JiDi t/J belongs to H-l(n) too. As in Section 1.2 we can show that Problem (1.68) - (1.69) is solved if we find a;; in V such that

" v) v((u,

" " v) + b(u," u, v) +" b(u, t/J, v) + b(t/J, u, " ..... =
(1.70)

The existence of a u" E V satisfying (1.70) can be proved exactly as in Theorem 1.2, provided there exists some 13 > 0, such that

vllvll 2 + b(v, v, v) + b(v, t/J, v) + b(t/J, v, v);> 13 IIvll2, V v or because of (1.22) vllvIl 2+b(v,

t/J, v)~l3l1vIl2,

V vE

v:

E

V, (1.71)

Now (1.71) will certainly be satisfied if we can find t/J which satisfies (1.67) and

(1.72) In order to show this, we will prove the following lemma: Lemma 1.8. For any

7> 0, there

exists some t/J = t/J(7) satisfying (1.67)

175

Existence and uniqueness theorems

Ch. II, § 1

and (1.73) Before this we prove two other lemmas. Lemma 1.9. Let p(x) = d(x, f) = the distance from x to f. For any > 0, there exists a function e E ee 2 (0) such that 0e= I in some neighbourhood off (which depends on e). (1.74) 0e= 0

26(e), 6(e)

ifp(x)~

IDkOe(x)I~-ifp(x)~26(e),

= exp

e

p(x)

I (- -) e

(1.75)

k= l, ... ,n.

(1.76)

Proof. Let us consider with E. Hopf [2], the function for X ~O by

A-+~e(A)

defined

I if A < 6(e)2

~e(A)

=

e log

(6~e)

) if 6(ep

< A < 6(e)

(1.77)

o if A > 6(e) and let us denote by Xe the function Xe(x)

= ~e(p(x».

(1.78)

Since the function p belongs to~2(n), the function Xe satisfies (1.74) - (1. 76) and 0e is obtained by regularization of Xe. Lemma 1.10. There exists a positive constant cl depending only on 0 such that I

I pVIL2(n)~clllvIIH~(n)’

1

V vEHo(O).

(1.79)

Proof. By using a partition of unity subordinated to a covering of I’, and local coordinates near the boundary, we reduce the problem to the same problem with 0 = a half-space = {x = (x n, x’), x n> 0, x’= (Xl’ ... , xn-l) E tRn - 1 } . In this case p (x) = Xn, and it is sufficient to check that

Steady-state Nallier-Stokes equations

176

Ch. II, § 1

(1.80) This inequality is obvious if one proves the following one-dimensional inequality:

f:-1 V~)

I’ds <; 2 (-tV’(9)I’ds’ V v E.’»(O, +<>0).

(1.81)

This is a classical Hardy inequality. In order to prove it, we write s = e", t = eT’ and v(s) =

s

~

s

’f

s

w(t)dt,

v’=w,

o

where rJjf represents the Heaviside function, :P’(a) = I, for a > 0 and = 0 for a < O. By the usual convolution inequality we majorize the last quantity by

rJjf(a)

( J

~~ ~(u)

e-a/’dU).

r~

Iw(e’)I’e’dT = 4J :-Iw
and (1.81) follows. Proof of Lemma 1.8. Let us now show that

1/1 = rot (Bet) satisfies (1.67) and (1.73); (1.67) is obvious because of (1.65) and (1.74), 1/Ij(x) = 0 if p(x)

> 26(e)

177

Existence and uniqueness theorems

Ch. II, § 1

and lI/Jj(x)1

~

where

(_e_1t(X)1 + IDnX)/) if p(x)

C2

p(x)

f

=)

IDt(x)1

t

n

.2:

1,j=1

t

f

x)1 2

ID;t/

~

2<5(e)

(1.82)

1/2

As we supposed that tiE LaG (0), we deduce from (1.82) that

1I/J/x)1

~

c3 ( _e_

p(x)

+ IDP(X)I) ,V t, p(x)

We have therefore

IV;I/JtlL2(n.)~

el

c4 {

~I

L2 +

(J

p<;26 (e)

~

26(e).

’flD~I’
1/’ } (1.83)

But, using Holder’s inequality, we see that

(J

p<;26(e)

1’

’iID~I’dx

\ ’ .. p(e)I’tILa(D) .)

where I/a. = 1/2 - l/n and /J.(e)

={

J

p(x) 0;;;

IDt(x)ln dx 26 (e)

~

l/n ;

)

as ~ since D;~ E Ln (0), 1 ~ i, i ~ n. With this last majorization, (1.4), and Lemma 1.10, (1.83) gives

p(e)~

IV;I/JjIL2(O) ~ Cs (ellvll

+ p(e)lvILCll(O) ~ c6 (e

+ p(e» IIvll, 1 ~

Now it is easy to check (1.73); for each v E b(v,

I/J, v) = -b(v,

v,

I/J)

-r;

i, i ~

n.

(1.84)

Steady-state Navier-Stokes equations

178

Ch.n, §1

(by (1.84))

~

+ J.L(e)) IIv1l 2 .

~c7(e

(1.85)

If e is sufficiently small for c7(e

+ J.L(e))

~’Y,

we obtain (1.73) for each v E "f/" and by continuity, for each v E V. Remark 1.5. (i) For n ~ 3, the conditions (1.66) reduce to

t E H2(O ).

because of the Sobolev imbedding theorems (see (1.3)). (ii) It is easy to write the boundary condition in the form (1.65) for the classical problems of hydrodynamics such as the cavitation problem, the Taylor problem, etc. Remark 1.6. It is easy to extend Proposition 1.1 to non-homogeneous problems. With the hypotheses of this proposition and moreover assuming that r/J E~’" (r) the solution {u. p} of (1.62) - (1.64) belongs to ’if’" (n) X~ ... (n). To prove this we proceed as in Proposition 1.1, directly on the A equations (1.62) - (1.64) (i.e., without introducing u). 0

A uniqueness result similar to Theorem 1.3 holds: for n ~ 4, v "large", and f "small", there is uniqueness: Theorem 1.6. Wesuppose that n ~ 4, that the norm of r/J in Ln (0)(1) is sufficiently small so that Ib(v. r/J.

v)1

~

v

2

illvll , Vv

E V,

(1.86)

and v is sufficiently large so that A

v 2 > 4c (n )11fl1 V’

(1.87)

where c(n) is the constant in (1.18) and

j =f + vl:ir/J (1) For n

= 2 replace

n

-

2:

i=1

(1.88)

r/JPir/J•

Ln(n) by LOt(n) for some Ot

> 2.

179

Existence and uniqueness theorems

Ch.U, §1

Then, there exists a unique solution u, p 0/(1.62) - (1.64)

(1).

Proof. It was proved in Lemmas 1.1, 1.2, 1.3 that b(v,

tP, v) = -b(v,

v, cP),

and Ib(v, v, cP)1 <;; c IIvIl2IcPILn(o)’

Therefore condition (1.86) is satisfied if IcPILn(o> is small enough: this means that (1. 72) is satisfied with 1/1 = cP and we do not need in this case the previous construction 0/1/1. Nevertheless, the proof of existence goes along the same lines, with 1/1 = cPo If"1 is a solution of (1.62) - (1.64), then "1 ="1 - cP is a solution of (1.70) with 1/1 = cPo Taking v ="1 in (1.70) we get "2 " " "" v"2"" vllul11 = -b (ul’ cP, ul) +
by using (1.86), and therefore

" <;; -Iltll 2" ’. lIulll v

(1.89)

v

Let us suppose that UO,"1 are two solutions of (1.62) - (1.64); let

"0 = "0 - cP, "I = ul 1/1

= cP:

flo - Ul; ito and Ul satisfy

cP, U =

" v)) v«"o,

" + b(uo, uo," v) +" b(uo, cP, " v)) + b(ul’ " ul’ " v«ul’ v) +" b(ul’ cP,

(1.70) with

v)

,, " + b(cP, uo, v) =
v)

,, " + b(cP, ul’ v) =
v E V.

We take v = u" in these equations, and subtract the second one from the first one; after expanding and using (1.22) we find:

v lIull2 = -b (u, Because of ( 1.86),

-be;;,

cP,

U) <;;

u1’ U) - b (u,

-i IIUlI

2

cP, U).

.

By (1.18), -b(u,"1’ u) <;; c(n)lIutit lIull2 , (1) A I’ . s a ways, p 15 umque up to a constant.

(1.90)

Steady-Btate Navier-8tokeB eqUiltions

180

Ch. II, §2

and because of (1.89), this is majorized by

2

A

- c(n)llflly,IIGlI2. v We finally arrive at the inequality (

~-~

c(n) Itfliy’ )

I GlI2~

0,

u

and because of (1.87), this implies = O. If Uo = UI’ it is clear that grad Po =grad PI’ and so the difference between Po and PI is constant. Remark 1.7. Non-uniqueness results for Problem (1.62) - (1.64) have been proved, in the two-dimensional case, for certain configurations; c.f, Rabinowitz [2], Velte [I] [2] and Section 4 of this Chapter. §2. Discrete inequalities and compactness theorems. Before going through the numerical approximation of the stationary Navier-Stokes equations, we must introduce new tools: the discrete analogue, for step functions and non-conforming finite elements of the Sobolev inequalities and of the compactness theorem, Theorem 1.1. These and some further Sobolev-type inequalities are the goals set for this section. This section is rather technical and the details of the proofs will not be needed in the sequel. 2.1 Discrete Sobolev Inequalities for Step Functions. The notations are those used for finite differences; see Section 3.3, Chapter I. We recall in particular that Rh is the set of points with co› ordinates ml hi’ ... , mhh n , mi E1l’, h = (hi’ ... , hn ), hi> 0; whM is the characteristic function of the block h (M)

(Ili- i’IJ.; + ~ ), M =

=i~1


Il

. , n ),

(2.1)

and 8th is the difference operator B,lICP(x)

1

= hi

[cp(x +

1 ...

"2 hi) -

cp(x -

1 ...

"2 hi)]

(2.2)

where hi is the vector with i th component hi’ and all other components

O.

Discrete inequalities and compactness theorems

Ch. II, §2

181

Theorem 2.1. Let p denote some number such that I <:,p < n, and let q be defined by l/q = IIp - sln. There exists a constant c = c(n, p) depending only on nand p such that n

!uh ILq(Qn) <; c (n, p)

~

16 th uh 1!J’(a

n) ,

(2.3)

for each step function Uh (2.4)

with compact support.

Proof. (i)

Let us consider the scalar function s .... g(s) = Isl(n-1)p/n-p.

Since (n-l )p/(n-p);;> I, this function is differentiable with derivative g’(s)

= (n -l)p Isl(n (P-2) +p)/(n-p)s. n-p

The Taylor formula can be written g(sl) - g(s2)

= (sl

-"i’~2)gl(AsI

+ (l - A)s2), A E (0, 1)

and gives

(2.5) (ii)

Let M belong to Rh ; we apply (2.5) with sl

=uh (M -

~

rh;), s2

=Uh (M -

(r

~

+ l)h i ) ;

Steady-state Navier-Stokes equations

182

IUh(M - rhi )I(n - l)p/n - P-1Uh(M - (r

...

~ C1 (n, p) 1Uh (M - rh i) - Uh (M - (r

Ch.II,§2

+ I )~)I(n

- l)p/n - P

...

+ I )hi)1

{IUh (M - rhj)ln(p - l)/n - P + 1Uh (M - (r

+ I )hi)ln(p -

l)/n - P}.

Summing these inequalities for r;> 0, we find (the sum is actually finite): +...

1Uh(M)I(n - l)p/n -

P~

c1 (n, p)h j

~

16ihuh(M - (r

+ ~ ) hi) I

{I uh(M - rhj)ln(p - l)/n - P + IUh(M - (r+ l)hi )ln(p-1)/n- p}.

(2.6)

We strengthen inequality (2.6) by replacing the sum on the right-hand side by the sum for r E1t; we can then interpret the sum as an integral and majorize it by

c, (n, p) {

.~,

J__ + ...

(Jl, ~,

I""

16.. u.(II" &)1• +

~ h,)I’(P - 1)/. - d~" p}

, 1Jn) are the coordinates of M and f1t = (1J1’ .,., /oti- b 1J;+ l’ , /otn)• In a similar way we denote by ~ the vector (xl’ ., Xi-1 , Xf+.1’ , xn ) and then write x = (~, xt). For any X E 0h (M), inequality (2.6) gives now

where (/ot1’

1Uh (x)l(n - l)p/n- p= IUh (M)I(n -1)p/n - p

..c, Let us now set

(n, p)

.L~

~

+ ...

__ 16..

,Iu. (,£"

U.(~"

~,+ ~’

(2.7)

&)1

)I•r.-

1)/. -

p} dt,. (2.8)

Discrete inequalities and compactness theorems

Ch.II, §2

Then, IWi(~)ln

-1

183

is majorized by the right-hand side of (2.7), hence

{.~, IU.(~, ~i+ ~i)I-(P-l)/- P}~d~i’ E;;;

<;

(by Holder’s inequality)

c,

(n, P )16,.u.I£P(._)•

(.~

1

L_Iu" (~,

~i+ ahi)~ ~d~YP

-

therefore

Now we have

by

According to the inequality given in the next lemma, this is majorized

n

II

i=1

{J

6ln - 1

n - 1 dX• IWt(x•)l I I

and, because of (2.9), IUh liq(61 n ) E;;; c3 (n, p)

tfr

}

l/n-l

18 th uh

’~~61-N}

!uh

I~~~-; :)/(n-l)p

l)/~

Ch.n, §2

Steady-state Navier-Stokes equations

184

n

<;; Cs (n, p)

1UhILq(Rn)

~

16ihUhILP(Rn)’

Lemma 2.1. Let WI’ . , wn ’ be n measurable bounded functions on ~, with compact supports, with wi independent of Xi’ Then

f

Rn

(

.~ Wi(~i»)

dx

,-1

<; .~

{ ,-1

r IWi(~)ln-l

j

~}

Rn- 1

I/n-l

(2.10)

This is a particular case of an inequality of E. Gagliardo [1] ; see also Lions [1] , page 31. Remark 2.1. For p = n, if the support of uh is included in a bounded set G, then n

1UhILq(Rn)

<;; c(n, q, G)

~1

(2.11)

16thUhILq(Rn),

for each uh of type (2.4), and for any q, 1 <;; q < +00. Actually any such q greater than p can be written as PI n/(n-PI) with 1 <;; PI < n. The in› equality (2.3) is then applicable, with c = c(n, PI) = c’(n, q):

The Holder inequality shows us that

16thu,,1

LPI (R n)

<;; (meas

G’)(llpl)- (lIp)

u".

16 thu ..1

" LP(R n)

where G’ contains the support of 6th Ifwe suppose that Ihl is bound› ed by 1 (or by some constant d), (meas G’) is bounded by (meas G) X Const.; then combining the last two inequalities, we obtain (2.11 ). 0 In the two and three dimensional cases, we prove another related

Ch.lI, §2

Discrete inequalities and compactness theorems

185

inequality which will be useful. Proposition 2.1. Let us suppose that the dimension of the space is two or three. For any step function Uh of type (2.4) with compact support, we have:

(2.12)

(2.13)

Proof. We use the inequality (2.7) with n = 2 and p = 4/3; a more precise analysis of the proof of (2.7) shows that c1 (n, p) = 2 in the present case; actually g(s) =? and for (2.5) we have clearly Ig(s1) -g(s2)1 ~21s1 -s21 {ls 11 + Is21}. We then have, for any x E uh (M), and M E Rh’

1u"(x)1

{},

2

,, 2

[~16ihUh(~’~i)I’

Iu, (~i’~i+

(2.14)

~i+~/’

Since the right-hand side of (2.14) is independent of xi’ we obtain Sup Xi

{},Iu, (~i’

IUh (x)1 2

~

2

J

+00 IcSihUh (xi’

~i)l•

_00

~i+ ~i)1 }d~i’

(1) Similar inequalities are given for the continuous case in Chapter III, Lemma 3.3.

(2.15)

Ch.II,§2

Steady-state Navier-8tokes equations

186

Now we may write, for the two dimensional case,

f

2 R

IUh(X)14dx E;;;

J

R

E;;; (because of

2

[SUpIUh(X)1 2 ] [Sup 1Uh(X)12] dx xl

X2

(2.15))

By the Schwarz inequality and since

f

j R2

IUh

(~1

+ ~1

,

f J

R2

x2)1 2

d~1

dx 2 =

1Uh(x)12dxl dx 2 = IUhli2(R2)’

(2.17)

the last expression is majorized by 36{16lhUhIL2(R2) IUhIL2(R2)} {1~hu"IL2(R2)luhIL2(iR2)

..

18Iu.I~,(o’)

{

i

}

’)} ’ L,(O

16g,,,,,I

Hence (2.12) is proved. In the three dimensional case, using (2.12) and (2.15), we write

Discrete inequalities and compactness theorems

Ch. II, §2

187

(2.18) ~

(because of (2.15»

<; 36 {

i

{1., ;)1 ]

u,

I6,. Ii’(.’) }

[ } , ~,(;"x,+ ~

u, (x,.

x,)I

dx }

(by Schwarz’s inequality) 2

16"

<; 3’2

lu,IL, (. , ) 16"u"IL, (. , ) {

<; 3’22

lu,IL

,<.,){ ~

and (2.13) is proved.

i

16,.u, li , (. , ) }

i,<.,)} ’/2.

’6,.u"l

D

Remark 2.2. The inequalities (2.3), (2.11), and (2.12) can be extended by continuity to classes of step functions with unbounded support. 2.2. A Discrete Compactness Theorem for Step Functions.

We give here a discrete analogue of Theorem 1.1, more precisely of 1,p (n) into Lql (n) is compact if the fact that the injection of W 1 ~ P < nand q1 is any number, such that

1

1 ~ q1 < q, where q

p

= nand q1

1

1 n

= - - -, p

is any number, 1 ~ q1 < +00.

(2.19) (2.20)

Steady-state Navier•Stokes equations

188

Ch. II. §2

Theorem 2.2. Let tfh be a family, may be empty, of step functions of type (2.4) and let

it=

U

Ih I <; CO

(2.21)

g’h’

Let us suppose that the functions

uh

of ithave their supports included in

some fixed bounded subset of lRn , say

n,

(2.22)

< +00.

(2.23)

n

Sup

Uh E I

{IUhILP(/Rn)+

~

1-1

18ihUhILP(/Rn)}

Then if p and ql satisfy conditions (2.19) - (2.20), the family g’is relatively compact in Lql (lR n ) (or Lql (n». Proof. According to a theorem of M. Reisz [I], we must prove the following two properties: (i) For each e > 0, there exists a compact set Ken, such that

J

IUhlql

dx:E;;; e, V uh E ’l.

(2.24)

O-K

(ii)

For each e > 0, there exists 11 > 0 such that, IT12 uh - uhILql(/Rn):E;;;

for any uh E it and any translation operator (T12 cP)

(x)

Q = (Ql’

= cP(x + Q).

(2.25)

e, .

,

Q,.,), with IQI :E;;; 11; T12 denotes the (2.26)

Proof of (i). Because of the Sobolev inequalities (2.3) and (2.11), the family 8 is bounded in J!l (n) where q is given by (2.19) if p < n, and q is some fixed number, q > ql> otherwise (p = n). By the Holder inequality, we then get

Discrete inequalities and compactness theorems

Ch, II, § 2

J

IUhlql dx:S;;; c(meas(n-K)l-(qdq),

189

V Uh E C.

(2.27)

a-K

The right-hand side of (2.27) (and hence the left-hand side) can be made less that e, by choosing the compact K sufficiently large; (i) is proved. . Proof of (ii), First, we show that (2.25) may be replaced by a similar condition on 1T2 uh - uh ILP (condition (2.30) below). Case (a): q1:S;;; p, For any f E H (n) we have f E Lql (n) as well as fE lJ’ (n) since n is bounded and q1 :s;;; p < q. Also, O:S;;; l/q1 - l/p < 1. By the Holder inequality,

ltlLql (a):S;;; (meas

1

n) /ql-l/P.

ltlLP(a) = Const. ltlLP(a)"

Case (b): q1 > p. For any function fE Lq (n) we can write, using the Holder inequality,

J

IfI q, dx

= :s;;;

f

lfI,q’IfI(1-8)q• dx

(J ) (f ltl Ql 6P dx

l/p

ltlql (l - 6)p’ dx

)

lip’

where 8 E (0, 1), p > 1, and as usuaII/p’+ I/p = 1. We can choose 8 and p S9 that q1 8p = p, q1 (1 - 8)p’= q;

this defines p and 8 uniquely, and these numbers belong to the specified intervals, (8ql(q-ql) (q-p) =p(q - ql)’ and p(q - ql) = q - p). Then

(2.28) In particular, for any 2 and

l12 u h

- Uh ILq1(Rn)

uh’

:s;;; IT2Uh - uhliq~R~

IT2Uh - uhl~(Rn)

Steady-state Navier-Stokes equations

190

~

HTl/UhILq(61n)

+ 1UhILq(61n)} 1-

1- 8 :s;: 2 1- 6 1Uh 1Lq(61~

-.

Since the family

Ch. II, §2 6

11Q Uh -Uh l;;’(61 n)

8 ITl/Uh - Uh1lJ’(61n)’

is bounded in Lq (6ln ),

~

I1Q Uh - uh ILql (61 n)

~

(2.29)

cl1Q uh - uh 1;;’(61 n)’

Inequality (2.29) shows us that it suffices to prove condition (ii) with ql replaced by p: V e > 0, 3 11, such that ITl/Uh - uh IL P(61 n) ~

(2.30)

e

for IQI ~ 11 and uh Eg’. The proof of (2.30) follows easily from (2.23) and the next two lemmas. Lemma 2.2.

n

ITl/Uh - uh IL P(61 n)

where

~

Qi

~ ~

denotes the vector ~ht

(2.31)

l’IQ[Uh - uh IL P(61 n)’ ~

Proof. Denoting by I the identity operator, one can check easily the identity n

Tl/-

I=

L i=1

(2.32)

Tt ... Tt (Tt -l). i i-I i

This identity allows us to majorize the norm

ITl/uh- uh I p n L (61 )

by

n

L i=1

ITt .. T2’= (T2’= Uh - uh)l P n’ i 1-1 I L (61 )

We obtain (2.31) recalling that IT(l/lLP(61n)

= lfl LP(61 n)’

for any ex. E 6ln and any functionfE IP(6l n ).

(2.33)

Ch. II, § 2

Discrete inequalities and compactness

191

theorems

Lemma 2.3. ’T2iUh - uh I E;;; LP(6t n)

c(I~1

+ 1~ll/p)

lc5ihUhILP(6tn),

I E;;;iE;;;n. (2.34)

Proof. Since IT2iUh - Uh’LP(6tn)

= IT_2iUh -

uh ILP(6t n

>’

we can suppose that Qi~ 0 and we then set

~ = (~+

Pi) h;, where ai is an integer

and 0 E;;; Pi < I, I E;;; i E;;; n,

We write 011

Tt Uh - Uh 1

~

o,}

(2.35)

-1

= j=O L

T/.t. (Tt. 1

1

u" -

Uh)

+ TOI1 h 1 (Tp1 t.1 Uh -

(2.36)

uh)’

From (2.36) and (2.33), we get the majoration 011 -

IT~

Uh - UhILP(6tn) E;;;

But

"h1 -

I

~

1 ITt. Uh - UhILP(6tn)

+ ITPs~

Uh - UhILP(6tn)"

(2.37)

= hI "It/2 c5 l h 1

and the sum on the right-hand side of (2.37) is equal to alhllc5lhUhILP(6tn);

since al hI E;;; Ql’ we obtain

IT~Uh

- uhILP(6tn) E;;; QlI6lhUhILP(6tn)

..

+ 1’Tp. h. uh

Let us now majorize the norm of Tn" h Uh - Uh. For x E ah(M), x = (xl’ ... , x n ) , and ME Rh , M we have

o if (ml -

- Uh’LP(6t n)’

= (mlh l, ... , mnh n),

I 2")hl <xl «ml -

Jt

hI c5 l h uh (M +~)

2

(2.38)

if (ml -

I

f3t + 2")h l I

f3t + -

I xl
)h < 2 l

Steady-,tate Navier-Stoke, equation,’

192

Ch.n, §2

Hence

J

l1’p,h. Uh -

uh lP dx =

I3thq

uhf){)

f

I6lh Uh (x +

uhf){)

~)IP

dx,

and summing these equations for the different points M of 6lh , we obtain

I1’Pl~’

uh - uh ILP(~n)

= I3l’Phll1’~/2

81h u h ILP(~n)’

Since h is bounded and 131 hI E;;; £1’ this is less than

cQl’PI6lhUhILP(~n) and (2.34) follows for t = I; the proof is the same for i = 2, ... , n.

0

Remark 2.3. In the most common applications of Theorem 2.2, the family il is a sequence of elements Uh m , where hm is converging to zero. Hence ilhm = {Uhm} and Fth is empty if h is not an hm. Remark 2.4. Let us suppose that 0 is bounded and that

Ft

=

U Fth , g’h

Ihl < Co

= {uh E Wh ,

(2.39)

lIuhllh E;;; I}

where Wh is the approximation of HA(O) by finite differences, (APXl). We infer from Theorem 2.2, that 8is a relatively compact set in L 2(0). The following set 8’, which is a subset of 8, is also relatively compact in

L2(0):

v"

’is’ = U \hI <

Co

il}"

if}, = {uh E v",

lIuh IIh E;;; I };

corresponds to the approximation (APX I) of V.

(2.40)

0

2.3. Discrete Sobolev Inequalities for Non-conforming Finite Elements. For the remainder of this section, the notations are those used in Chapter I, Section 4.5, for non-conforming finite elements. We recall in particular that J;;" is a regular triangulation of an open bounded set 0; I¥/h is the set of points B which are barycenters of an (n-l) dimensional face of a simplex.v’ E 9h and which belong to the interior of O(h) (1); whB’ B EI¥/h’ is the scalar function which is linear on each (1) n(h) =

U.9’.

yEY’h

Discrete inequalities and compactness theorems

Ch. II, §2

193

simplex YE3h, vanishes outside n(h), and takes the following nodal values: whB (B) = 1, whB (M) = 0 for all M different from B which are barycenters of an (n - 1) face of an YE5i,. For any function uh of type Uh

= ~

ME IJIIlt

uh (M) whM,

(2.41)

!/

Dihuh is the step function vanishing outside n(h) and such that Dihuh(x) = Diuh(x), V x E

, V!/E5i,.

Theorem 2.3. Let p denote some number such that 1 :E;;; P < n. and let q be defined by l/q = lip - lin. There exists a constant c = c(n, p, 0.) depending only on n, p, 0., such that n

1UhILq(O}:E;;; ctn, p, 0., ex)

~

(2.42)

IDihUhIIP(O}’

for each function of type (2.41). Proof. The proof relies strongly on Proposition 4.17, Chapter I. In order to prove (2.42) we must show that for each (J in9{n),

I

J

n u.

8 dx! "’;;c(n,

where q’ is defined by

p,

nJ 18 1L<’(n I (

11q + 11q’ =

it

1D...... IIP(n»)

(2.43)

1.

We denote by X the solution of the Dirichlet problem D.X

= (J

in 0., X = 0 on an.

The function X is «j on 0. and according to the regularity results for elliptic equations,

IXIW2,q,(0"} :E;;; c(n,

p, 0.) 101Lq(O}

(2.44)

(observe that 1 < q, q’ < 00). Due to the Sobolev inequality the derivat1 ’ , ives X;= DtX of X (I :E;;; i:E;;; n) belong to W ,q (0.), and thus to D (0.), where lip’ + lip = I: IXiILq’(n},:E;;; c(n, p, 0.) 101 LQ’(n},

Xi =DtX, I :E;;; i:E;;; n.

(2.45)

Ch. II, §2

Steady-state Nallier-Stokes equations

194

Using this function I/J we write

where

and

We have

I

L.. i. It. 8 dx 1<;

(2.46)

I

and in order to prove (2.43) we will establish some majorations of type this maioration is (2.43) for the expressions I~I, 0 ~ i ~ n. For easily obtained since the Holder inequality and (2.45) allow us to set

J2

n

1}2 1~ i~

IDihUhILP({l),DiXILP’({l)

<; ctn, p,

For

k, I ~

1v", ... IL1’(O » ,

(2.47)

i ~ n, we use I (4.234) with t/J =x,= DiX, and we obtain

\!L1<;c(n, p,

1I~

mJ8I Lq•(O) ( ,~ n. aj

(i.

1lJ1X, ILq •(o)

1<; c tn, p, n, aj IXI....q,

(~

(i.

1Dp,... 1L1’(o»)

lDp,u.IL1’
Ch.II, §2

Discrete inequalities and compactness theorems ~

~

(by 2.44))

(i

c(n, p, G, a) IOILq’(

1=1

n)

IDl h UhlIJ’ ). (Sl)

195

0

Remark 2.5. In contrast to the usual Sobolev inequalities (see Section 1, and Theorem 2.1, in the discrete case), the inequality (2.42) contains a coefficient c(n, p, G) which depends on G. Indeed, the usual Sobolev inequalities contain coefficients which are independent of the support of the concerned functions. We have lost some information, but this is not very important for what follows. 0 Remark 2.6. The case n = p can be treated as in Remark 2.1. Let p be any number such that 1 ~p < n, and let q be defined by l/q = IIp - lIn. For any function uh of type (2.41), the relation (2.42) gives: n 1UhILq(n)

~

c(n, p, G, a)

~

ID/hUhlIJ’(o)"

According to Holder’s inequality, ID/hUhlIJ’(Sl)

~ (meas

G)I-p/n ID/hUh I Ln(o)"

Hence, with another constant c(n, p, G. a), n IUh ILq (0)

~

c(n, p, G, a)

~

ID/hUhILn(o)"

(2.48)

Changing now the name of the constant c(n, p, G) and observing that q may be any number s- 1, we write (2.48) in the following form n 1UhILq(0)

for each 1 ~q~oo.

uh

~

c(n, q, G, a).

~

ID/hUh’Ln(O)

(2.49)

of type (2.41) with support included in G, and for each q,

0

Remark 2.7. An inequality similar to (2.13) (n = 3) is given in Chapter III. We do not know if an inequality similar to (2.12) (n = 2) is valid for non-conforming finite elements, in the two-dimensional case.

Steady-state Navier-Stokes equations

196

A weaker inequality is given in Chapter III.

Ch. II, §2

0

2.4. A discrete compactness theorem for non-conforming finite elements.

We will now give a discrete analogue of Theorem 1.3, more precisely of the fact that the injection of HA (n) into L 2 (n) is compact for any bounded set n. However, because of a difficulty of a technical nature, the result is proved only in the two-and three-dimensional cases. We denote by {~} h E ~ a regular family of triangulations of n. For each h, we consider the space y" of scalar functions uh of type (2.41), i.e., Uh =

L

ME’Vh

uh(M) whM, uh(M) E fR.

(2.50)

These functions are linear on each simplex g E.!lh, vanish outside n (h) = U f/E5"h [/’, and are continuous at the barycenter of an (n - 1) face of a simplex f/E fIi,. The space Yh is endowed with the scalar product

Now we state the Theorem: Theorem 2.4. Assume that n = 2 or 3 and that 9h , h E ;J(’a.’ is a regular family of triangulations of n. Let g’h be a family which may be empty offunctions of type (2.50), and let ~=

U

p(h) <: Co

ft’h.

(2.52)

Let us suppose that

Sup IIuh IIh < +00.

"hE 8’

(2.53)

Then the family g’is relatively compact in L 2 ( n ).

Proof. The main idea of the proof consists of observing that the space of (piecewise linear) conforming finite element functions is a subspace of the space of (piecewise linear) non-conforming finite element functions. We then obtain the result, using the "nondiscrete" compactness theorem (Theorem 1.1) and the previous a priori estimate.

197

Discrete inequalities and compactness theorems

Ch.l1, §2

The space Yh defined by (2.50) contains the space Xh of continuous piecewise linear functions which vanish on the boundary of n(h) (the confirming elements). Since Xh and Yh are finite dimensional, both are Hilbert spaces for the scalar product "))h (see (2.51)) (1); one can therefore consider the orthogonal projector 1rh from Yh onto Xh , and by definition of such a projector,

«.

« 1rh"h’ vh))h

= «"h’ vh))h’

II1rh"hllh:E;;;; ""h"h’ V"h E

V "h E Yh , V vh E Xh

(2.54)

Yh.

(2.55)

Let us now consider the elements a, of the family g’. Due to (2.55) the family { 1rh"hl"h E g’}

sA

sA

is a bounded family in (0.) since Xh C (0.) and the II•IIh -norm and the (0.) norm (II•II) coincide for conforming elements:

sA

Sup II1rh"h II rFl

Uh E ~

flii

(

n

)

= Sup II1rh"h IIh:E;;;; Sup ""h IIh < +00. Uh E ~

"h E ~

By virtue of Theorem 1.1, the family 1rh"h is relatively compact in

L2 (0.) and therefore there exists a subsequence h’ -+ 0, and an u in

L2 (n ) such that

1rh’"h,-+"inL2(n), as

«-».

The proof will be complete when we show that 1rh’"h’ -"h’ -+ 0, as h’ -+ 0,

(2.56)

and (2.56) follows immediately from the next Lemma.

Lemma 2.9. For each vh in Yh, 11rhVh - vhIL2(rt):E; ; C(~,

n)p(h) IIvhllh’

Proof. As in the proof of Theorem 2.3, let (J be an element of ~(n) let X denote the solution of the Dirichlet problem t::.X

=() in 0.,

X = 0 on

an.

(1) These properties have already been mentioned and used several times.

(2.57)

and

Ch.n, §2

Steady-state Navier-8tokef equation«

198

In order to estimate the norm !Vh - 1rh "tz ILl (n) we oonsider the ex› pression

As in the proof of Theorem 2.3, n

(vh - ’"hvh’ LlX) = « 1rh Vh - Vh’ X))h +

? /~

(2.58)

1=1

Due to Proposition 4.16, Chapter I, (2.59) and with (2.53) and (2.55)

IJ~I

(2.60)

~ c p(h).

In order to estimate «Vh - ’"h’ X)) h» we oonsider the functions Xh characterized by Xh EXh , Xh(M) = X(M),V M..§Cft~ (the functions interpolating X. Note that X is oontinuous in .n, H (.n) C
IIx - ’X.h II ~ ~

c (a) p (h)

Ixi/p (0) ~ (by defmition of X)

c p(h) 18I L 2 ( 0) .

(2.61 )

Using (2.54) we observe that «vh - ’"hv" , X))h = «v" - 1rh v" , X -

Xtz ))h

(1) It is essential to assume that the famDy of triangulation is

resuJar,

uh <

ell

< + -, V h.

(2) See L (4.42), (4.43). These results were stated for L--norms. We use here similar results valicl for L’-norms, see Clarlet &; Raviart [1], Theorem S, po 196.

Ch. II, §3

Approximation of the stationary Navier-Stokes equations

199

and thus 1((Vh - 1rh Vh’ X))h:E;;; IIvh - 1rh Vhllh

IIx -

xhllh:E;;; c P (h).

The relation (2.57) is established. Remark 2.8. In the most common applications of Theorem 2.4, the family 8 is a sequence of elements Uh m’ p(h m ) -+ 0, a(h m ) :E;;; a. Hence 8h m ::: {Uh m} and ~ h = rp if h is not an hm . If lIuhm II hm :E;;; c, the sequence Uhm is relatively compact in L 2 (,Q).

§3. Approximation of the Stationary Navier-Stokes Equations We discuss here the approximation of the stationary Navier-Stokes equations by numerical schemes of the same type as those used for the linear Stokes problem. We give in Section 3.1 a general convergence theorem; we then apply it in Section 3.2 to numerical schemes based on the approximations (APX 1), ..., (APX 5) of the space V. In Section 3.3 we extend to the nonlinear case the numerical algorithms discussed in Section 5, Chapter I. All of this section appears to be an extension to the nonlinear case of the results obtained in the linear case in Sections 3, 4 and 5. Neverthe› less the results are not as strong here as in the linear case due particularly to the nonuniqueness of solutions of the exact problem. 3.1. A General Convergence Theorem Let L2(,Q).

be a bounded Lipschitz open set in ~n and let f be given in By Theorem 1.2 there exists at least one U in V such that

,Q

v((U, v))

+ b(u,

u, v)

= if, v),

Because of Theorem 1.3, this large.

U

Vv E

v:

(3.1 )

is unique if n :E;;; 4 and if v is sufficiently

Steady-state Navier-Stokes equations

200

Ch. II. §3

Our purpose here is to discuss the approximation of Problem 3.1. Let there be given first an external, stable and convergent Hilbert approximation of the space V, say (w, F), (Vh Ph’ ’h )hE’JC’ where the Vh are fmite dimensional; at this point this approximation could be any of the approximations (APXl), ... , (APX5), described in Chapter I. Let us suppose that we are given some consistent approximation of the bilinear form v«u, JI», and of the linear form (j, JI), satisfying the same hypotheses as in Section 3, Chapter I: for each hE;If, ah (uh’ Jlh) is a bilinear continuous form on Vh X Vh , uniformly coercive in the sense

(i)

3: ao > 0 independent of h, such that ah (uh’ Uh) ~

ao

Iluh I ~,

uh E Vh ,

(3.2)

where 1I•lIh stands for the norm in Vh . (ii)

for each h E ;If, Qh is a linear continuous form on Vh , such that IlQh IIv }, ~ (3.

(3.3)

The required consistency hypotheses are: If the family

Jlh

converges weakly to JI as h ... 0 and if the family

wh converges strongly to wash’" 0,(1) then

lim ah (Jlh’ wh) = v«JI, w»,

h-+O

If the family Jlh converges weakly to JI as h ...

lim (Qh,Jlh)=(j,JI)

9, then (3.5)

h-+O

For the approximation of the form b, we suppose that we are given a trilinear continuous form b h (uh’ Jlh’ wh)’ on Vh , such that: bh(Uh, Jlh, Jlh) (l)We recall that this means

PhVh Phwh

= 0, YUh’

wv in F weakly, ~ wW in F strongly.

-+

Jlh E Vh .

(3.6)

Ch.n, §3

Approximation of the stationary Navier-Stokes equations

if the family vh converges weakly to v, as h ~ 0, and if then

W

201

belongs to cr, (3.7)

Sometimes it will be useful to be more precise about the continuity of bh and we shall require Ibh(Uh. vh. wh)1
(3.8)

YUh, Vhf Wh E Vh,

where the constant c = ctn, 0) depends on nand 0 but not on h: an inequality such as (3.8) with c depending on h is obvious, since such an inequality is equivalent to the continuity of the trilinear form bh We can now define an approximate problem for (3.1 ): To find uh E Vh, such that ah (uh’ vh)

+ b h (uh’

Uhf Vh) = ($lh’ Vh>’

(3.9)

We then have Proposition 3.1. For each h, there exists at least one uh in Vhf which is a solution of (3.9). If (3.8) holds and if

aij > c(n,

O)(j,

(3.10)

then, uh is unique.

Proof. The existence of uh follows from Lemma 104. We apply this lemma with X = Vh which is a finite dimensional Hilbert space for the scalar product »h’ We define the operator P from Vh into Vh by,

«’, .

«P(uh)’ vh »h = ah (uh’ vh)

+ b h (uh’

uh’ vh)

- ($lh’ vh>’ YUh’ Vh E Vh.

(3.11)

The operator P is continuous and there remains only to check (1.29); but, with (3.6), «P(uh)’ uh »h

= ah (uh’ ~

~ao

uh) - ($lh’ uh>

(by (3.2)-(3.3»

IIuhll~

- II$lhllvh IIuhllh

~ (ao IIuhIIh - (j) lIuh IIh .

202

Steady-,tate Navier-SttJke, eqUlltio",

Ch. II, §3

Therefore «P(Uh)’ Uh »h

> 0,

provided

(3

lIuhllh =k,andk>-.

ao

Lemma 1.4 gives the existence of at least one uh such that P(Uh)

=0

or «P(Uh), Ph »h

= 0, V Ph

(3.12)

E Vh,

which is exactly equation (3.9). Let us suppose that (3.8) and (3.10) hold and let us show that uh is unique. Iful and ul* are two solutions of (3.9) and ifuh = ul - ul*, then ah (Uh. Ph)

+ bh (ul, ul, Ph) - bh (ul*, ul*, Ph)

= 0, VPh

E Vh ;

taking Ph = uh and using (3.6) we find ah (uh. Uh)

= bh (uh , ul, uh ).

Because of (3.2) and (3.8), we have aolluhll~ "’c"ul"h lIuhll~. If we set J)h = u~ in the equation (3.9) satisfied by

uZ, we find

(3.13)

ah(ul, ul) = (Qh. ut>,

and with (3.2) and (3.3),

ao lIuI I~

’" (3l1ul Ilh ’

I!ulllh’ ’"

1..

(3.14)

ao

Using this majoration and (3.13) we obtain,

(ao- :) lIuhl ~

’" 0;

(3.15)

if (3.10) holds, this shows that uh = O. Theorem 3.1. We assume that conditions (3.2) to (3.7) are satisfied; uh

Ch. 11, §3

Approximation of the stationary Navier-Stokes equations

203

is some solution of (3.9). Ifn <;; 4, the family {PhUh } contains subsequences which are strongly convergent in F Any such subsequence converges to wu, where U is some solution of (3.1). Ifthe solution of (3.1) is unique, the whole family {PhUh} converges to wu. Ifn ~ 5. we have the same conclusions. with only weak convergences in F

Proof. We suppose that n is arbitrary. Putting vh = uh in (3.9), and using (3.2), (3.3) and (3.6) we find ah (uh. Uh)

= ($lh,

uh

>, (3.16)

0:0 IIuh IIh <;; (l

Since the Ph are stable, the family Phuh is bounded in F; therefore there exists some subsequence h’ ~ 0, and some r/J E F such that Ph,uh’ ~ r/J in F weakly.

The condition (C2) for the approximation of a space shows that, necessarily r/J E w V, or r/J = WU, U E V: Ph,uh’ ~ wu in F weakly, h’ ~ 0.

(3.17)

Let v be an element of V and let us write (3.9) with vh =’h v: ah(uh, rh v) + bh(uh, Uh’ ’h v)

= ($lh’ ’hV>.

(3.18)

As h’ ~ 0, according to (3.4), (3.5), (3.7), ah,(uh’. rh’v) ~ v«u, v)), bh,(uh’, uh’, ’h’v) ~ b(u,

u, v),

($lh" rh’v> ~ if, v).

Hence U belongs to V and satisfies v«u. v))

+ btu, u,

v) = if, v), Vv

Er.

(3.19)

If n <;; 4, equation (3.19) holds, by continuity, for each v E V: if n ~ 5, we find, by continuity, that (3.19) is satisfied for each u E V. In both cases, U is a solution of the stationary Navier-Stokes equations. It can be proved by exactly the same method, that any convergent subsequence of Phuh converges to WU, where U is some solution of (3.1). If this solution is unique, the whole family Phuh converges to wu in Fweakly.

Steady-state Navier-Stokes equations

204

Ch. II, §3

let us show the strong convergence when n :E;;;; 4. As in the linear case (see Theorem 1.3.1) we shall consider the expression

Xh

= ah (Uh -

rh u, uh - rh u),

Expanding this expression and using (3.9) with vh = uh we find X h = (!lh’ Uh) - ah (uh’ rh u) - ah (rh u, uh)

+ a(rh

u, rh u),

Because of (2.4), (3.5), and (3.17),

Xh , -+ (f,

u) - a(u, u), as

h’ -+ O.

We take v = U in (3.1) and use (1.22); this gives (f, u)

=a(u,

u),(1)

hence

Xh , -+ 0 as h’ -+ O.

(3.20)

We now finish the proof as in the linear case; the inequality (3.2) shows that

lIu h’ -

rh’ ullh’-+ O.

Since the Ph are stable, this implies IIph,uh’ - ph,rh,ulI F :E;;;; IIph,II.r.Wh"F) lIuh’ - rh,ullh’ -+

O.

Now we write "ph,uh’ - wullF :E;;;; "ph,uh’ - ph,rh,ullF

and the two terms on the right-hand side of this inequality converge ’to oas h’ -+ O. (2) (I)This equation does not hold for n :> S, and this is the reason the proof cannot be extended to those cases. (2) Werecall that for each

UE

P~hu-wu,mF~on~y,uh-O

(see Proposition 3.1).

V, and each h E

;K,

thore exists’h

UE

Vh’ such that

Ch.II, §3

Approximation of the stationary Navier-Stokes equations

205

3.2. Applications

In this section we apply the general convergence theorem to approxi› mation schemes corresponding to the approximations (APXI), ..., (APX5) of V. Approximation (APXI). We choose ah and Qh as in the linear case (see

I. (3.62) and (3.63»),

(3.21 ) (Qh’ vh)

= if, vh)’

Before defining b h , we introduce the trilinear form S(u. v. w)

= b’(u,

b "(u v w)

v. w) + b"(u, v. w)

i

, I b (u .v, w) =-2

t (u, v. w),

i,i= 1

J n

uj(Div,.)w,.dx

= --2I

(3.22)

(3.23)

(3.24)

(3.25)

It is not difficult to see that S(u. u, v)

= btu, u,

v), Vu E V, Vv E~~

(3.26)

but 6 and b are otherwise different. We now define b h as, b h (uh. Vh. wh)

= b h(uh.

Vh’ wh)

+ b};(Uh’ Vh. wh)

(3.27)

(3.28)

I.~

1

f n

"Ih Vlh (6 1h Wlh jdx,

(3.29)

It is clear that b h, b’/" and hence b h are trilinear forms on Vh ; since

Steady-state Navier-Stokes equations

206

Ch.II, §3

Vh has a finite dimension, these forms are continuous.

We have to check (3.6) and (3.7); (3.6) is obvious with our choice of the form bh , and (3.7) is the purpose of the next lemma. Lemma 3.1. I!Phuh converges

to WU, then

b h (Uh’ uh’ rh v) -+ b(u, u, v),

Vv E

r:

(3.30)

Proof. Saying that Phuh converges weakly to wu means that uh -+ U weakly in L 2 ( n )

(3.31 )

5ih uh -+ DiU weakly in L2 (n ), 1 ~ i ~ n.

(3.32)

and The Compactness Theorem 2.2 is applicable and shows that uh -+

U

strongly in L2(n).

(3.33)

We know that if v E r, Ph rh v converges to wv in F strongly; but the proofs of Lemma 1.3.1 and of Proposition 1.3.5 show that actually rh v -+ v in the norm of LOG (0.),

(3.34)

5ih rh v -+ Dtv in the norm of LOG (0.).

(3.35)

If we prove that b,,(Uh’ uh’ rhv)-+b’(u, u, v)

(3.36)

bh(Uh’ uh’ rhv)-+b"(u, u, v),

(3.37)

then, according to (3.27), the proof of (3.7) will be complete. For (3.36) we write Ib,,(Uh’ uh, rhv)- b’(u, u, v)1

~co

n

!

i, j= 1

n

2:

i,j= 1

Ifn

(UihVjh - UiVj) 5th Ujh dx

f ","/ n

(6/h"/h - D"’iJdx ’

+

Ch.II,§3

Approximation of the stationary Nevier-Stokes equations

207

All the preceding integrals converge to 0, and (3.36) is proved; the proof of (3.37) is similar. 0 The convergence result given by Theorem 3.1 is as follows (n ~ 4): in L 2 (n) strongly,

uh’ ~ U

(3.38)

6th , uh’ ~ D, U in L 2 (n) strongly, 1 ~ i ~ n.

(3.39)

Exactly as in the linear case it can be shown that there exists some step function (3.40) such that v«uh’ vh »h

therefore a solution

+ (~7Th’ uh

Vh)= if, Vh)’ V Vh E Wh ;

(3.41 )

of (3.9) is a step function (3.42)

such that n

L

;= I

(VihUih)

(M) = 0, VM E

n~

(3.43)

and n

-v

1

2

L

(3.44)

i= I

n

L

i= I

where 1 fh(M)=--›

hi ... hn

J

f(x)dx.

(3.45)

aheM)

When condition (3.8) and some condition similar to (3.10) are satis-

en,

SteadY-Btate Navier-StokeB equations

208

II, §3

tied, uh and U are unique and the error between U and uh can be esti› and P E<6’2(S1). mated as in the linear case, if, moreover, U E ~3(S1) Using the Taylor formula we can write, n

~

-v

i= 1

I

n

- 2" ~1

5ih ((’ h U)i’h u)(M) + (’VhPXM)

=f(M) +

h(M), VM E

nt,

(3.46)

with (3.47) where c(u, p) depends only on the maximum norms of the second and third derivatives of u, and of the second derivatives of p, Equations (3.46) show that v((’h u, vh»h +bh(’h u, ’h u, vh) - (Vh 1r’h’ vh)

= if +

(3.48)

h’ vh)’

for each vh E Wh where 1r~ is the step function 1rh

=

~Jl.l

MEUh

(3.49)

P(M)whM’

Subtracting (3.48) from (3.9) gives V((uh - ’h u Vh»h

= -bh(Uh’ Uh’ Vh) + bh(’h U, ’h U’ Vh)

+ (eh’ Vh), VVh E Who

We take Vh

=Uh

vlluh -

- rhu and use (3.6) to get:

rhulI~

= -bh(uh

- rh u, Uh’ Uh - rh u)

+ ( h’ Uh - rh u),

and with (3.8) and (3.47), vlluh - rhulI~

~c(n.

S1)lluhllh lIuh - rhulI~

Ihl, S1)lluh IIh lIuh -

+ c(u. p) Iluh - rhullh vlltlh - rhullh "c(n.

rhu11h

+ ct», p) Ihl

Approximation of the stationary Navier-Stokes equations

Ch. II, §3

209

’" (by (3.16)) (j <; - ctn, 0) lIuh - rhu11h + ctu, p) Ihl.

aa

Finally

( v-

~

(3.50)

ctn, 0) ) lIuh - rhullh <; c(U. p) Ihl.

If

JJao

> (jc(n.

0),

(3.51)

(the constant ctn, 0) in (3.8)), this gives the following majoration of the error lIuh - rhul~

c(U. p)

’"

[ v-

:0

Ihl.

0

(3.52)

ctn, 0) ]

Approximation (APX2). In the two-dimensional case we can associate to the approximation (APX2) of V described in Section 1.4.2, a new discretization scheme for the Navier-Stokes equations. We recall that for this approximation j;), is a subspace of (0), and we take, as in the linear case,

HB

(3.53) ($lh. vh )

= (f, vh)’

(3.54)

where « ".)) is the scalar product in j;), and in H6(0). We define the form bh by bh(uh’ Vh’ wh)

= S(Uh’

vs- wh)’ YUh’ Vh’ Wh E Vh ,

(3.55)

where S is defined by (3.23) - (3.25); since Vh is a space of bounded vector functions, the forms S are defined on Vh ; they are trilinear, and hence continuous. Condition (3.6) is obviously satisfied by our choice of bh ; condition (3.7) is the purpose of the next lemma.

Lemma 3.2.

I!PhUh

=uh

converges

to iou, then

bh(uh’ uh’ rhv) ~ btu, u, v), Vv E

r.

(3.56)

Steady-state Navier-Stokes equations

210

§3

Proof. We recall that F =Ha(n) and wand Ph are the identity. Saying that Ph Uh converges weakly in F to wu, amounts to saying that uh --+- u weakly in Hij(n).

(3.57)

The Compactness Theorem 1.1 then shows that uh --+- u strongly in L2(n).

(3.58)

The proof of Lemma 3.2 will be the same as that of Lemma 3.1, if we observe that rh v --+- v in the norm of L" (n)

(3.59)

Dj’h v --+- D, v in the norm of L (n), I E;;;; i E;;;; n,

(3.60)

OO

which was actually proved in Proposition 1.4.3 (see (4.63)).

0

The weak (or strong) convergence result given by Theorem 3.1 is the following one: If p(h) --+- 0, with o(h) E;;;; ex (i. e. h E Ho) ’ then

(3.61 )

Uh’ --+- U in Hij(n) weakly (or strongly).

Exactly as in the linear case (see Section 1.4.2), we can show that there exists a step function 1Th’ 1Th = (Xh

L

(3.62)

1Th(!/)xhY’

.
= the characteristic

function of the simplex !/), such that

v«uh’ Vh)) + bh (uh’ uh’ vh) - (1Th’ div vh)

= if, vh)’ VVh E Wh •

(3.63)

This equation is the discrete analogue of v«u, v)) + b(u, u, v) - (p, div v) = if, v),

V v EHij(n)

n Ln(n).

(3.64)

Since the dimension of the spaces is n = 2, the form b is trilinear continuous on Hij(n) and there exists some constant e such that A

Ib(u, v, w)] E;;;;

cllullllvllllwlI, V u, v, w EHij(n).

(3.65)

Ch.n, §3

Approximation of the stationary Navier-Stokes equations

211

’This can be shown by the same method as (1.18); hence (3.8) holds with ctn. n) = ~. We will get an estimation of the error assuming that U E~3 (ft), P E~2 (ft), and a hypothesis similar to (3.51). We take vh = rh U - uh in (3.63) and v = rh U - uh in (3.64); subtracting these equations we find

+ f)(u,

v((u - uh’ ’h u - uh»

u, ’h u - Uh) - f)(Uh’ uh’ ’h u - uh)

- (p - Trh’ div(rhu - uh»

Since div(rh U

= O.

(3.66)

uh) is a step function which is constant on each simplex

-

5/E !!Jh, we have

(p - Trh’ div(’h u - uh»

= (p -

Trh’ div(rh u - uh»

where Trh is defined by (3.67)

(meas9’) Hence 1(P - Trh’ div(’h u - uh)1 <;

<;

ITrh - pi Iluh -

ITrh - plldiv(rh u -

uh)1

(3.68)

rhull.

We then estimate the difference f)(Uh. uh, ’hu - u) - f)(u,

= f)(Uh - f)(u,

u, rhu - u)

rhu, uh’ ’h u - uh)

u, ’hU - u)

= f)(Uh -

+ f)(’h U, Uhf ’h U -

Uh)

’hU’ Uh’ rh u - Uh)

+ f)(rhU’ rhu - u, rhu - Uh) + f)(rhU - u, u, rhu - Uh)’ The absolute value of this sum can be majorized because of (3.65) by ~lIuhllllrhu

-uhll2 +~{llrhulI+

IIrhu - uh

II.

We recall that vlluh

11

2

=
uh)

and therefore 1

lIuh II <; -

v

1f1 .

IlulI}lIrhu -ull’ (3.69)

212

Steady-state Navier-8tokes equations

Ch. II, §3

Hence the sum (3.69) is less than or equal to ~

- IfI lIuh

- ’huII2

lI’hu - Uh

II .

v

+ ~{II’hull

+ lIulI}

ulI•

lI’hu -

(3.70)

With the majorations (3.68) and (3.70), we get from (3.66) ( v-

f

IfI)

+ 17I’h and finally (v -

f

lIuh - ’h u ll2 <; vlluh - ’hull

pllluh - ’h

IfI)

lIu -

’hull

ull + C{II’h ull + lIull} lIu -

rhull ,

lIuh - ’hull <; vllu - ’hull

+17I’h-pl+t{II’hull+llull}

Ilu-rhull.

(3.71)

Under the assumption

v2 >tlfl,

(3.72)

the inequality (3.71) gives a majoration of the error between Uh and u and uh’

’hu and hence between

Approximation (APX3). As in the linear case, the method here is very

similar to the method used for the approximation (APX2).

Approximation (APX4). We recall that n must be a simply connected open set in tR 2 .

Since (APX4) is an internal approximation of V, the simplest scheme (3.9) associated with this approximation is To find uh E Vh such that v«uh’ vh))

+ b(uh’ uh’ vh)

= (f,

vh)’ VVh E Vh .

(3.73)

Theorem 3.1 is applicable and shows that uh -+ u in ~ as p(h) -+ 0,

(3.74)

provided a(h) <; a (i.e. hE JCa ). The error between u and uh can be estimated as follows (if we have

Approximation of the stationary Navier-Stokes equations

Ch.U, §3

uniqueness): we take v

u,

=U -

v«U, v) + biu, u, v)

213

uh in the variational equation satisfied by

= (f, v),

Vv E V.

We then take v = rhu - uh in (3.73); we subtract these equations and find vlluh - uII2

= v«u

- uh’ U - rhu» + b(uh’ uh’ rh u - uh)

- b(u, u, rhu - uh) .

(3.75)

The difference b(uh’ uh’ rh u - uh) - b(u, u, rh u - uh)

is equal to b(uh - U, uh’ rhu - uh) + btu, uh - U, rh u - uh)

= b(uh

- U, uh’ rh u - u)

+ b(u, uh - U, rhu - u)

+ b(uh

= b(uh

- U, uh’ U - uh) - U, uh’ rhu - u)

+b(uh -u,u,u-uh)+b(u,uh

-u,rhu-u),

Because of (l.18) this is majorized by c{llulI+ Iluhll} lIuh -ullllrhu-ulI+cllullllu-uhIl2.

We recall that

vlluII2

=
IlulI ’" - 1If1l y’ , v and similarly

vlluh 11 2 =
u

Therefore the last expression is rnajorized by 2c Ifllluh

v

-ull lrhu-ulI+~lfl lu-uhIl2.

v

214

Steady-state Navier-Stokes equations

We deduce then from (3.75)

( v-;- Ifl )

IUh

( v-;- IfI)

nUh - ull <:

-ulI’ <:

(v+ 2: IfI)

Ch.U, §3

IIUh

-ullI’h U -ulI,

(v + 2: IfI) "’hu - ul.

(3.76)

If JJ2

>c

"!lI v’,

(3.77)

inequality (3.76) shows that the error lIuh - U II has the same order as IIrh U - U II. Since c is the constant c(n), n = 2, in (1.18), the inequality (3.77) is exactly inequality (1.37) which ensures the uniqueness of the solution U of the exact problem. Approximation (APX5). If we consider the approximation (APX5) of

V, we can set

(3.78) (3.79) (3.80)

(3.81)

b h" (uh’ vh’ wh)

=-

~

-2I L..

i.I"l

f

n

uih vlh (D ih wlh) dx.

(3.82)

It is clear that b", bl: and bh are trilinear forms on Vh and since Vh has a finite dimension, these forms are continuous. We have to check (3.6) and (3.7); (3.6) is obvious with our choice of the form bh , and (3.7) is the purpose of next lemma. lemma 3.3. Assume that n then

< 3.

Ifphuh converges

weakly to

WU,

(3.83)

Approximation of the stationary Navier-Stokes equations

Ch. II, § 3

215

Proof. By definition, we are assuming that uh -+ U in L 2 (n) weakly,

and

D ih uh -+ Diu in L2 (n ) weakly, 1 ~ i ~

(3.84)

n.

(3.85)

The Compactness Theorem 2.4 shows that uh -+ U in L 2 ( n ) strongly.

or.

(3.86)

We know that if v E Ph ’h v converges to wv in F strongly; but the proofs of Propositions 1.4.12 and 1.4.15 show that furthermore ’h v -+ v in the norm of Loa (n), (3.87) Dih’h

v -+ Div in the norm of Loa (n).

(3.88)

The proof of (3.7) will be complete if we prove that b,,(Uh’ uh’ ’hv)-+b’(u, U, v),

(3.89)

b;;(Uh’ uh’ ’h v) -+ b"(u, u, v) .

(3.90)

For (3.89) we write

n

2:

i.]» 1

n

2:

i.]» 1

J

n U, ’jCDih Ujh - D, Uj) dx .

All the preceding integrals converge to 0 and (3.89) follows. The proof of (3.90) is similar. 0 The convergence result given by Theorem 3.1 states that uh ’ -+

u in L2(n) strongly,

Dih, uh ’ -+ Diu in L 2 (n) strongly, 1 E>;; i ~ n

(we recall that n = 2 or 3 only).

(3.91) (3.92)

216

Steady-state

Navier-Stoke« equotions

Ch.n, §3

Exactly as in the linear case it can be shown that there exists some step function 1fh constant on each g, g E 9j, , and vanishing outside n(h) such that v«Uh’ vh ))h + bh (uh’ uh’ vh) - (1fh’ divh vh)

= (f, vh)’

VVh E Wh .

(3.93) When condition (3.8) and some condition similar to (3.10) are satis› fied, uh and U are unique and the error between U and uh can be esti› mated as in the linear case, if moreover U EW 3 (n ) and p E~2(n). Lemma 3.4. Let U and p denote the exact solution of (1.8)-(1.1 0) and let us suppose that U Ei’3(n), p Ei’2(f2), and that 0. = 0. (h). Then v«u, vh ))h + b h (u, U, vh) - (P, div vh) = (f, vh) + Qh (vh)’ (3.94) where (3.95)

Proof. We take the scalar product in L 2 (n ) ofvh E Wh with the equa› tion (1.8) written in the form 1

- v/).u

Since 0.

~

n

+ 2" ~1

= n(h),

(ui Diu - Di(uiu))

+ grad p = f ,

we find

( 1 tv(/).u, Vh)9’ + 2" (Ui Diu - Di(uiu), Vh)9’

+ (grad p ’ Vh)9’ - (f, Vh)9’

t= o.

The Green formula applied several times in each simplex f/ shows that the left-hand side of this relation is equal to {v«u, vh ))h

+ S(u,

u, vh) - (P, divh vh) - (f, vh)

- Qh(vh)} = 0

where

Approximation of the stationary Navier-Stokes

Ch.U, §3

equations

217

The estimation (3.95) of Qh (vh) follows easily from Proposition 1.4.16. 0 We now proceed as for (3.71). We take vh (3.94) and subtract these relations. We get v«u - uh’ uh - ’hu»h

+ b h (u,

=uh

- ’hu in (3.93) and

U, uh - ’hu)

- b h (Uh’ uh’ uh - ’h u) - (p - 1rh’ divh (’h u - uh»

= Qh (vh)

(3.96)

.

Since divh (’hu - uh) vanishes, the corresponding term disappears. We then estimate the difference bh(uh, uh, uh - ’h u) - b h (u, U, uh - ’h u)

= bh(uh - ’h u, uh’ uh - ’hu)+bh(’h U, Uh’"h - ’h U) - bh(u, U, uh - ’hu)

=

= b h (uh - ’h u, "h’"h -’h") + bh(’h u, ’h" - U, uh - ’hu)

+ bh(’h u

- U, U, uh - ’h U),

The absolute value of this sum can be majorized, because of (l.18) and (3.65), by

+ c{ lI’hullh + lIull} "’h u

cIIuh IIl1uh - ’huI12

- ullh•

(3.97)

lIuh - ’hullh .

We recall that vlluh I ~

= (f,

and therefore

Iluh IIh E;;;

I

-

v

uh>

IfI .

Hence the sum (3.97) is less or equal to

=-v IfI lIuh - ’hulI~

+ c{

lI’hulih

+ lIulI} "’h u

- ullh•

lI’hU - Uh II h

(3.98)

With this majoration and (3.95) we get from (3.96)

( v -;- IfI ) lu, -

r,uft~

.. vllu, - r,u!, lIu - r,ull,

+ c{ Il’hulih + lIulI} lIuh - ’hul1h lIu - ’hullh + C(U, p) p(h) lIuh - ’hullh .

(3.99)

Steady-state Navier-Stokes equations

218

Ch.U, §3

Finally

(

- ;- Ifl ) lIu. - ,.ull. ".lIu - ,.uR. + c{

IIrhullh

+ lIull} lIu -

rhullh

+ c(u,

p) p(h) .

(3.100)

With the assumption

v2

> clfl ,

(3.101)

the inequality (3.100) gives a majoration of the error between uh and ’h u and hence between u and uh .

3.3. Numerical Algorithms. The following analysis is restricted to the dimensions n <;; 4. We wish to extend to the nonlinear case the numerical algorithms des› cribed, for the linear case, in Section 5, Chapter I. We observe that the stationary Navier-Stokes equations are not the Euler equations of an optimization problem like the Stokes equations. The following algorithms then are some extension of the Uzawa and Arrow-Hurwicz algorithms classically related to optimization problems. In the remainder of this section we will always use the trilinear form b(u, v, w) defined by (3.23)-(3.25). This is a trilinear continuous form on HA(o.) X HA(o.) X HA(o.) and there exists some constant c = c(n) such that Ib(u, v, (n

~

w)1 ~ ten) lIullllvllllwll, V u, v, w EHA(o.),

4).

(3.102)

We have already noticed that b(u, v, A

w) = b(u, v, w) ifu E V, v, w EHA(n) _

1

b(u, v, v) - 0, u, v EHo(o.).

(3.103) (3.104)

Uzawa Algorithm. In order to approximate the solutions of (1.8)› (1.11) we shall construct, as in Section 1.5, two sequences of elements u" E HA(o.), pm E L2(o.).

(3.105)

This construction is relatively easy because the condition div u = 0 will disappear in the approximate equations.

Approximation of the stationary Navier-Stokes equations

Ch. II, §3

219

We start the algorithm with an arbitrary element pO:

pO E L2(n).

(3.1 06)

When p’" is known (m ;;;;;, 0), we define u m + 1 as some solution of

um + 1 EHA(n), and v«u m+ 1, v» + Scum + 1, U m+ 1, v)

= pm, div v)

+(f, V), VvEHA(n).

(3.107)

We then define pm + 1 by

pm+l EL2(n), and (pm+l_pm, q)+p(divu m+ 1 , q)=O, VqEL2(n).

(3.108)

Later we will give the conditions that the number p > a must satisfy. The existence of um + 1 satisfying (3.107) is not obvious, but can be proved using the Galerkin method, exactly as in Theorem 1.2. There› fore we will skip the proof. It is not difficult to see that tim + 1 is the solution of the following nonlinear Dirichlet problem:

um + 1 EHA(n) n

2:

-vilu m + 1 +

=-

i= 1

rrz + 1 D.um + 1

u,

’

+ ~ (div um+ 1 )u m+1 (3.109) 2

grad pm + fEH-l (n).

The solution of (3.107)-(3.109) is not, in general, unique. When u,m +1 is known, pm + 1 is explicitly given by (3.108) which is equivalent to pm+l =pm _ p divum+1 EL2(n). (3.110) To investigate convergence we will assume that c(n)

v- -

v

Ilfllv’

_

= v > 0;

(3.111)

with (3.102), (3.103) and Theorem 1.3, the condition (3.111) implies the uniqueness of the solution of (1.8)-(1.11); p is unique up to an additive constant; we fix this constant by req uiring that

f

p(x)dx

n

= O.

(3.112)

Ch. II, §3

Steady-state Navier-Stokes equations

220

Proposition 3.2. We assume that n ~ 4 and that condition (3.00) holds.

We suppose also that the number p satisfies 0
Then, as m -+

(3.113)

00,

u" -+ u in the norm of HA (n),

(3.114)

pin L 2 (n), weakly,

(3.115)

p"

-+

where {u, p} is the unique solution of (1.8)-( 1.11) which satisfies (3.112). Proof. We set

=um + 1 qm + 1 = p’" + 1 -

vm + 1

U

P

(3.116)

and we proceed as in the proof of Theorem 1.5.1. We subtract equation (3.107) from the equation

v«u, v» + G(u, u, v) - (p, div v) =
(3.117)

and we take v = vm + 1 to obtain

v IIv m+ 111 2

= (qm , div vm+ 1 ) + G(u, u, vm+ 1 ) _ G(um + 1, um+ 1, vm + 1 )

=_ (qm + 1 -

qm , div vm + 1 )

- (qm + 1, div vm + 1) + G(v m+ 1, u, vm + 1 ) ~ ~

(by (3.102) and (1.39»

_ (qm + 1

_

qm, div vm + 1 )

_

(qm + 1 , div vm + 1 )

A

C

+ - IIfllv’ IIv m+111 2 . v We take q = qm+ 1 in (3.108) and we find,

Iqm + 112

Iqm 12 + Iqm + 1 _ qm /2 = - 2p(div vm+ 1, qm + 1) .

(3.118)

_

We multiply the last inequality in (3.118) by 2p and add it to (3.119); this gives Iqm+11 2 -lq m l2 + Iqm+l _ qml2

+("_ ~

Ilf llv’)

"""+1"’,;

(3.119)

Ch.U, §3

221

Approximation of the stationary Navier-Stokes equations ~

_ 2p(qm + 1 _ qm , div Vm + 1 ) .

(3.120)

This inequality is similar to equation (5.12) in the proof of Theorem 1.5.1 with v replaced by v (see (3.111 )). The proof can be completed exactly as in Theorem 1.5.1. Remark 3.1. In the general case, when uniqueness is not assumed, we can prove weak convergence results for the average values 1

N

- m=1 2:

N

um’

1 -

N

N

2:

m=1

pm .

These sequences are bounded in HA(Q) and L2(Q), and every weakly convergent subsequence converges to a couple {u, p} which is a solu› tion of (1.8)-(1.11).

Arrow -Hurwicz Algorithm. We construct a sequence of couples {u m, pm } defined as follows.

We start the algorithm with arbitrary elements uOEHA(Q),poEL 2(Q).

(3.121)

I

When pm ,u m are known, we define pm + 1, u m + 1 , as solutions of

um + 1 EHA(Q) and ((um+l_um,v))+pv((um,v))+b(um,um+l,v) - p(pm, div v) = p(f, v), V v EH5(Q)

(3.122)

1(3.123)

pm+l EL2(Q) and a(pm+l_ pm,q)+p(divu m+

1,q)=O,

VqEL2(Q).

We suppose that p and a are two strictly positive numbers; conditions on p and a will be given later. The existence and uniqueness of u m+ 1 EH5(Q) satisfying (3.122) is easy with the projection theorem; (3.122) is a linear variational equation equivalent to the Dirichlet problem u m+ 1 EH5(Q) n

- Llu m+

=-

1

+o

2:

i= 1

u’!! Dsu’" + 1 + f!.... (div u m )um + 1 I

I

Llu m + pvLlu m + p grad p’" + f.

2

(3.124)

Ch.n, §3

Steady-state Navier-Stokes equations

222

Hence pm +1 is explicitly given by (3.123) which is equivalent to

pm+1 =pm -~divum+1

a

(3.125)

EL2(O).

Convergence can be proved under stronger conditions than those used in Proposition 3.2. Proposition 3.3. We assume that n

v-

2....

~

4, that

4.... 2

~ litII v ’ - ~ 2 lit I ~. = v* > 0 v v

(3.126)

and that 001*

(3.127)

O
Then, as m ~

+v

00,

u" ~ u in the norm of nA(o),

(3.128)

p" ~ pin L 2(O) weakly,

(3.129)

where {u, p} is the unique solution of (1.8)-( 1.11) which satisfies (3.112). Proof. We use again the notation (3.116). We take v = 2vm +1 in (3.117) and (3.122) and subtract these equations; this gives IIv m + 111 2 -llv m 11 2 + IIv m+ 1 _ vm 11 2 + 2pv IIv m+1 11 2

= 2pv((vm + 1, vm + 1 _ vm » + 2p(qm , div vm +1) + 2b(um, um + 1, vm + 1) - 2b(u, u, vm +1) ~

1

_ Ilvm + 1 _ vm 112 + 4 p 2 v211v m+ 111 2

4

+ 2p(qm , div vm+1) + 2b(v m - vm+1, u, vm +1) + 2b(v m+ 1, u, vm + 1 ) ~

(because of (3.1 02) and (1.39)

1 ~ _ IIv m+ 1- vm 11 2 + 4p2 v 211v m+ 111 2

4

+

Bifurcation theory and non-uniqueness results

Ch.n, §4

.

2c

+ 2p(qm , div v m+ 1 ) + -

IIfI v ’ IIv m+ 1

V

2c

+- Ilfllv’ IIvm + 1 11 2 V

1

-

vm II IIv m+ 1 II

IIv m+ 1-vm Il2

:E;;; -

4c 2

223

2

2c

+ (4 p 2 v2 + - 2 Ilfl ~’ + - IIfllv ’ ) IIvm+ 1112 v V + 2p(qm , div v m+ 1 ) .

(3.130)

We take q = 2qm+ 1 in (3.123):

alqm + 112

_

alqm 12

= 2p(qm + 1,

div vm+ 1)

=_ 2p(qm, div vm+ 1 ) :E;;; -

2p(qm + 1

-

2p(qm , div vm+ 1 ) + 2plqm + 1

-

qm, div vm + 1 ) qm I IIv m+ 111

a 2p 2 n _Iqm+l - qm 12 + - - IIvm + 1 1 2 2 a - 2p(qm , div vm+ 1 )

:E;;;

(3.131)

(see (5.13), (5.25), (5.26), Chapter I). We add the last inequality (3.130) to the last inequality (3.131) and obtain 1 alqm+11 2 -alq ml2 +_lqm+1 _q ml2 + IIvm+ 1 112 2

1 m + 1 _v m ll 2 +2p ( v--lIfllv’ 2c -lIvmll2 +-llv 2 V 2

2 - 4c - 2 Ilfllv’ - 2pv 2

v

2P) IIvm +111 2 :E;;; O. a

- -

(3.132)

The conditions (3.126)-(3.127) ensure that the coefficient of m IIv + 111 2 in (3.132) is strictly positive; this inequality is then similar to the inequality (5.27) in the proof of Theorem 1.5.2; we finish the proof as in that theorem. 0 § 4. Bifurcation Theory and Non-Uniqueness Results

The uniqueness of solution of the Stationary Navier-Stokes equa› tions has only been proved under the assumptions that v is sufficiently

224

Steady-state Navier-Stokes equations

Ch.U, §4

large, or that the given forces and boundary values of the velocity are sufficiently small. It is expected that otherwise the solution is not unique, and this has been proved by V. I. Iudovich [1], [2], P. Rabino› witz [2] and W. Velte [I], [2]. The first two papers deal with the Benard problem (Navier-Stokes and heat conduction equations) and the third one shows the non-uniqueness of solutions of the Taylor problem. In this section we will establish the non-uniqueness of solutions of the Taylor problem - following W. Velte [2]. Subsection 4.1 contains the description of the problem and preliminary results. Subsection 4.2 recalls the main results of the topological degree theory that we need and then Subsection 4.3 gives the proof of the non-uniqueness theorem.

4.1. The Taylor Problem. Preliminary Results. 4.1.1. The Equations

The Taylor problem is the study of the flow of a viscous incompress› ible liquid in a doman of ~3 bounded by two infinite cylinders of radius r1 and r2 (r2 > r1 > 0), having the same vertical axis. The inner cylinder is rotating with an angular velocity e, while the other is at rest. Since we are looking for axi-symmetrical solutions, we will use cylindrical coordinates in ~3, say r, e, z, where the Oz axis is the axis of the cylinders. The fluid thus fills the domain 0:

r1
-oo
U’,

(4.1)

We denote by V’, W, the components of the velocity vector in the cylindrical coordinates, and p denotes the pressure. Then the non› dimensional form of the equations of motion is as follows:

U’) + u_ -aU’ + w_ -aU’ -

1 (_

- A

Au - - 2 r

1 (

- A 1

3r

u -

ar

r

_ aw

- 3w

w3z

3p

--Aw+u-+w-+-=O A 3z az az

a

_

a

ar (ru) + az phenomena.

r

v) + _aV’ + _ 3V’ + -

_

Av - -2 _

3z

1 -2

- v

_

(rw)

=0

1 __

r

u v

+ -ap = 0

ar

=0 (4 2)

.

(4.3)

Ch.I1, §4

where

225

Bifurcation theory and non-uniqueness results

a2

a

I

a2

A=-+--+› ar 2 r ar az 2

(4.4)

"A=Re=cxrrv-1

(4.5)

and A = Re is the Reynolds numbers The boundary conditions on the lateral surfaces of the cylinders are

u = w= 0, v = I,

for r

= rl

u=v=w=Oforr=r2’

(4.6)

A very simple solution of (4.2), (4.3), (4.6) is known, which we denote by Uo , Vo , Wo ,Po: Uo

= Wo = 0, Vo (r) =

Po (r)

= v5

r 22

-

I

r 12

(

r~r

- r )

log r + const.

(4.7)

We can then look for the solutions to (4.2), (4.3), (4.6) of the form u=uo+u,v=vo+v,w=wo

+w,’ji=Po+P,

and we obtain for u, v, w, P, the following equations:

_!..

A

(AU -

!!.-

r2

)+

v) +

I ( - Av - I - -Aw+u

A

+w

au _ !.. v2 _ ~ az r r

av + w -av + -I

u-

ar az aw aw ap - + w - + -=0 ar az az

r2

A

au ar

r

v Vo +

ap = 0 ar

I , uv + -(va + rv o) u

r

(4.8)

=0

dv dr’

where "o = -o I

a a-

a az

- (ru) + -

(rw)

= 0,

(4.9)

and the following boundary conditions deduced from (4.6): u=v=w=Oforr=rl

andr=r2’

(4.10)

We observe that the boundary conditions (4.10) are not sufficient to obtain a well set problem (as Condition (4.6) is not sufficient in the case of Equations (4.2), (4.3)): some boundary conditions at z = –oo should

226

Steady-state Navier-Stokes equations

Ch. II, §4

be added. We can either look for solutions u, v, w, which vanish at

z = –oo, or for solutions which are independent of z, or for solutions

which are periodic in z, The first possibility corresponds exactly to the type of problem studied in Section 1, the second possibility also, as this amounts to looking for bidimensional solutions. The third type of prob› lem (z-periodic solutions), is not exactly one of the problems studied in Section 1 but is very close to them, and this is actually the type of solution experimentally observed. We will seek the solutions 0[(4.8), (4.9), (4.1 0) which are periodic in z, with period L. We remember that

u = v = w = p = 0, is a trivial solution, and we want to show, in some cases, the existence o[ a non-trtvial solution. 4.1.2. The stream [unction Because of (4.9), there exists a function [= fer, z), such that

o

OZ

(rf)

= ru,

0

or (rf)

=-

(4.11 )

rw.

The boundary condition (4.10) ensures that [is a single valued func› tion. It is interesting to write Equations (4.8) using only the dependent variablesf, v. To do this we differentiate the first equation in (4.8) with respect to z, the third one with respect to r, and then subtract the resultant equations. The pressure disappears; observing that -ou - -ow = If, -a (Aw) OZ or ar

=l

(ow) or

where

1 a2 1 a a2 1 £=A--=-+--+-_:....2 2 r

OZ2

r or

or

r2

’

we obtain after expanding (4.12) with

227

Bifurcation theory and non-uniqueness results

Ch.II, §4

M(f, V) = a(r)=2v r

Q

=

~ar (aaZf £f)+ ~aZ [~(~r

2

2 2 r2-r1

-1).

(r~ r

ar

(rf)\ £fJ+

~

~ ~az (V2),

r

The second equation in (4.8) can be written as follows:

£V+A(biz +N(f,V))=O with

N(f, v) =b

-1 -a r ar

af 1 a (rv) - + - - (rf) 3z r 3r

= - ( v-o + VQ’ ) = r

(4.13)

2 r~ - rI

.

av az

-

The boundary conditions (4.10) become:

af = v = 0 for r = r1 ar

f= -

and r

= r2 .

(4.14)

The problem is thus reduced to finding the non-trivial solutions j, v, of (4.12), (4.13), (4.l4) which are periodic in z with period L.

4.1.3. The Associated Functional Equation Since the functions f and v are z-periodic with period L, it is suffi› cient to consider the restriction of these functions to the open set @L

=

1(r, z)Ir < r < r" - ~ < z < ~ i• 1

We will denote by H"((!); L) the space of functions characterized as follows: if they are prolongated to the whole set (!), (!)

= {(r,

z)lr1

<:r
tR}

as periodic functions of period L, the prolongated function belongs to L) is exactly the sub›

H" of any bounded subset of(!}(l). Actually H"((!); space of functions v of H"((!}L) such that

a/v

L

aiv 3z1

L

- . (r, - ) = - . (r, -),; = 0, ... , k - 1.

3Z1

2

2

(I) Such a function will never belong to

iI’«())

except if it is the zero function.

228

Steady-state Navier-Stokes equations

Ch. II, §4

Let us now introduce the spaces W = Wl X W2 and V= Vl X V2 :

= {[EH2«(!);L)I[=:~

Wl

=Oatr=rl andr=r 2

!

W2 = {v EH1 «(!); L)lv = 0 at r = rl’ r = r2}’ Vl =H 3«(!); L) n Wl , V2 = W2 . It is clear that Wl , W2 , Vl , V2 , equipped with the norms induced respect› ively by H 2«(!)L)’ n, «(!)L)’ H3«(!)d,H2«(!)f) are Hilbert spaces. We equip V with the product norm induced by H «(!)L) X H2«(!)L)’ but we shall equip W with another scalar product:

«
= «(fl’

h»w I + «vl’ V2»W2

«(iI.

=

lil

[,»w,

f

f

’

£I, ’ dr dz

(9£

«vl’ v2»W

2

=

(av l aV2 + aVl

(9£

az az

2 ar aVar )

r dr dz

(4.15)

where
J

(9L

sr . g r dr dz =-

Thus if [E Wl and £[= 0, then

J

(9L

£[

Ir dr dz = 0

J (a[ar ar

ag + a[ a

(9L

g)

az az

r dr dz .

(4.16)

Ch.n, §4

229

Bifurcation theory and non-uniqueness results

and f = 0 by (4.16). This shows that IIfllWI is a norm on WI’ This norm is now equivalent to the norm l£flL 2 « (!I ) and this last norm is of the regularity equivalent to the norm of H2 (mL ) by applica~ion theorems to the second order elliptic operator -£ (see for example Agmon-Douglis-Niremberg [11). The operator T The definition of the operator T will be given after the next two lemmas.

Lemma 4.2. For g given in L2(mL (resp. v in WI) such that Lu

),

there exists a unique u in W2

=g

(4.17)

(resp. £2 V

=g

(4.18)

Moreover u E H2 (mL ) and v E H4 (mL ) .

Proof. It is easy to see with (4.15) that these problems are respectively equivalent to the following ones: - To find u in W2 such that

-

J(

ou ou 1 ou ou 1 ) - + - - rdrdz or or oz oz

-

(l)L

- To find v in WI such that

f

.cv,.cv,’rdrdz=

J

gVl

rdrdz,Vv, EW,.

(!IL(fJL

The existence and uniqueness of solutions of these variational prob› lems follow from Lemma 4.1 and the Projection Theorem (Theorem 1.2.2). Then, it follows from the regularity theorems for elliptic opera› tors (£ and £2 here), that u E H2 (m L ) and v E H4 (m L)’

230

Steady-state Navier-Stokes equations

Ch. II, §4

Lemma 4.3. For {f, v}given in V, the functions

av +M(f, v),

a -

az

af

b-

az

+N(f, v),

belong to L2 (lPL ) . Proof. We have only to prove that M(f, v) and N(f, v) belong to L2 (lPL ) . This can be shown using the Sobolev imbedding Theorem which gives in particular (see Subsection 1.1): 1f2(lPL ) C ~o

(~),

HI (lPL ) C L4(lPL )

.

Now the first terms in M(f, v) are

a2f at a az . Lf - az ar (£f);

- ar

if {f, v} belongs to V, then fbelongs to H3 (lPL ) , a2 flar az and £fbelong to HI (lPL ) and thus to L 4 (lPL ) and their product is in L2 (lPL ) ; af/az is in 1f2(lPL ) and thus in~O(~), alar (£f) is in L2(lPL ) , and the product of these terms is in L2(lPL ). The proof is similar for the other terms in M and N. D We can now introduce the operator T. According to Lemmas 4.2 and 4.3, for {f, v} given in V there exists a unique pair {f’, v’} belonging to Vand also to H4 (lPL ) X 1f2 (lPL ), such that £ 2f ’ =£v

(av a ai + M(f, v) )

,= - (ab azf + N(f, v) )

(4.19)

{f’, v’} E W(or V).

Defmition 4.1. We denote by T the mapping from V into itself defined by {f, v} ~ {f’, v’}.

The relation of this operator with our problem is the following: Proposition 4.1. The two following problems are equivalent:

231

Bifurcation theory and non-uniqueness results

Ch.II, §4

- To find {f, v}in V which satisfies (4.12), (4.13), (4.14). - To find ¢ = {f, v}in V such that

(4.20)

¢="AT¢.

0

The proof is obvious.

Henceforth we will study the problem (4.12)-(4.14) in its functional form (4.20). The next subsection is devoted to the study of the opera› tor T. 0 4.1.4. Properties of the operator T.

Lemma 4.4. T is a compact operator in V. Proof. Let ¢~

¢n

= ifn , vn } be a bounded sequence

in V and let The proof of Lemma 4.3 shows more precisely

v~ }= T ¢n’

= if~,

that

a

aV n

az

+M(fn’

vn ) , b

afn

az +N(fn’ vn ) ,

are bounded sequences in L 2 «()L)’ Thus £2 f~ and £v~ are bounded se› quences in L2(~L),f~ and v~ are bounded sequences in H4((Pr) and HZ (ll:n, and they are thus relatively compact sequences respectively in H3 «(9L) (or VI) and HI «(9L) (or V2 ) (see Theorem 1.1). 0 Let us consider the linear operator B defined as follows: ¢

= {f,

v} -+¢"

sr = _ a av

= if",

v"}=B ¢,

az

(4.21 )

" af £v = - -

az

if", v"}E W.

For ¢ given in W, there exists a unique pair if", v" }in W which satis› fies (4.21) (se Lemma 4.2). Thus B is a linear continuous mapping from Winto W n {H4 «(9L) X HZ((l)}. Using again Theorem 1.1, we obtain Lemma 4.5. B is a linear compact operator in Wand in V. The relation between the operators T and B is the following one.

Steady-state Navier-Stokes equations

232

Ch.n, §4

Lemma 4.6. The operator B is the Frechet differential of T at point 0. Proof. Let us write

Rt/> = t/>* = Tt/> - At/> = t/>’ - t/>", t/> E V . We have f}f* £v*

= -M(f, v)

=-

(4.22)

N(f, v)

Since TO = 0, we must prove that IW"IIv -+ 0 as 11.1.11

’

IIt/>IIv

’#’

v

-+ 0 .

(4.23)

With (4.22) and the methods of Lemma 4.3 one easily sees that IM(f, V)IL 2( (!JL )

~collt/>II}

,

IN(f, V)IL 2((QL ) ~ cl IIt/>II} ,

(Ci

= constants).

Then, by Lemma 4.2,

Ilf*IIH 2((!J L ) ~

c21M(f, V)IL 2( (!J L )

~ c3 IIt/>II} ,

IIv*IIH 2 ((!J L ) ~

c4 1N(f, V)IL 2((!J L )

~

Cs

IIt/>II} .

In particular

IIt/>*IIv ~c611t/>II} and (4.23) follows.

0

,

4.1.5. A uniqueness result. Before starting the proof of the non-uniqueness of solutions, we es› tablish a simple uniqueness result (for A "small"), which is exactly the adaptation of Theorem 1.6. to the Taylor problem. Proposition 4.2. If A is sufficiently small O~A
r2)

(1)

(4.24)

the problem (4.12)-(4.14) (or (4.20)) possesses no solution in V other than the trivial one. (l)C(’1. ’2) is a constant depending on’1 and ’2’ made partly explicit in the proof of Proposition 4.2.

Bifurcation theory and non-uniqueness results

Ch. II, §4

233

Proof. Let if> = {f, v}be a solution of (4.12)-(4.14). We multiply (4.12) by t, (4.13) by v, and integrate these equations in (9L with respect to the measure r dr dz. We have

f

[M(f, v)’f - N(f, v)•vJ r dr dz

=0 .

If)L

Using then (4.16) we get

Integrating by parts, we see that the right-hand side of this equation is equal to (a

+ b)

at r dr dz

v -

az

.

This is bounded by

Ar2 Suple + bl Iv IL 2(If)L )

l.c
J

If)L

since

and

2

t

laa I

[1£fI +

z

L2(If)L)

<;;

1::1 + 1::1’] r 2

dr dz ,

Ch.n, §4

Steady-state Novier-Stokes equations

234

(Poincare Inequality), 2 a)"1 -a l z L2(t!!L)

~ const

(see Lemma 4.1). Finally,

[1-Ac(’,.,,)]•

2

1.c)"’L2(t!! ) L

J

t!!L

+1::I’}dr

~£fI’+I: 12

and j> v = 0 when (4.24) is satisfied.

dz <;0

0

Remark 4.1. With a slight modification of the preceding proof, one can show the uniqueness of the solution for

A<X, where A is the smallest eigenvalue of the operator (B + B*)/2 which is compact and self-adjoint in W.

Remark 4.2. Our goal is to show, in some cases, the non-uniqueness of solution for sufficiently large A. 4.2. A spectral property 0)" B. 4.2.1. Fourier Series Expansions If! belongs to L2 (~L ) then)" possesses a Fourier series expansion

L:

)"=

n=O

lfn(r) cos(n

0

z) + 1,,(r) sin(n

0

z)),

(4.25)

(0 = 27r/L), where j], and1" are L2 on the interval [r1,’2]’ Moreover the sum

I

~o..

12

(1f.1 i,<".,,) + iJ.1I,<".,,) ) 1 /

is finite and defines on L 2 (~L)’ a norm equivalent to the usual norm. The functions in Jfc (~; L) (k ;> 1) possess alike a Fourier series expansion of type (4.25), where)"n’ 1" are in Hk(’l’ r2)’ The sum

Ch. II, §4

Bifurcation theory and non-uniqueness results

i:

~

n=O

1

n 2j

j=O

(If. /2

n Hk-i(rl,r2)

+ 1112

)

n Hk-i(rl,r2)

1

/

235

2

(4.26)

is finite and defines on Hkem; L) a norm equivalent to the norm induced by Hk(mL ) . We shall consider the subspace Vof V containing all the pairs {f, v} such that fer, -z)

=-

vCr, -z)

fer, z)

= vCr,

z) .

These functionsf and v admit Fourier series expansion of type

f=

v=

~

f n sin(n a z)

z

v n cos(n a z) .

n=O

n=O

(4.27)

It is easy to particularise the preceding remarks to such functions "(V, and thus V, is a closed subspace of H3 (m; L) X HI cm; L Now, if ¢ = {f, v} E V, then ¢" = B¢ = if", r"} also belongs to V

».

and the Fourier series off" and v" are given by solving the following one-dimensional boundary value problems

(4.28)

(1- (n a)2)2 f;(r)=an avn(r) - (1- (n a)2) v~(r)

= b n a fnCr)

d2 1 d 1 j(=-+---dr 2 r dr r 2’

with the boundary conditions

df." f."n (r) = ~dr (r) = v n (r) = 0 at r = rl and r = r2

.

(4.29)

We will denote by B n the linear mapping {in.

Vn

} -.

ir;

v~},

which is continuous from L2 (rl’ r2) X L2(rl’ r2) into H4(rl’ r2) X W(rl’ r2)’

Steady-state

236

Navier-Stokes

equations

Ch.Il, §4

4.2.2. Properties of the En’ We note the following result which can be found for example in S. Karlin [1] or K. Kirchgassner [ 1] . Lemma 4.7. Let d2 d M = cx.(r) dr 2 + (j(r) dr + ’Y(r) where ex., (j, ’Y, are four times continuously differentiable on [-1, and ex > 0 and ’Y < O. Then

+ 1] ,

The Green’s function G(r, s) of M under the boundary condi› tions G(–I, s) = 0, is negative for r, s E (-1, +1) (i)

(ii) The Green’s function H(r, s) of M2 under the boundary condi› tions H(–I, s) = (3H)/3z (–I, s) = 0, is positive for r..s E (-1, + 1).

We infer from this the Lemma 4.8. The Green’s functions H(k; r, s) and G (k; r, s) of (Jt- k 2)2 and - GJt- k 2) on (r 1, r2), under the boundary conditions (4.29) are positive on (r1’ r2) X (r 1, r2)’

With these kernels, we can convert (4.28), (4.29), into integral equations f; (r)

J’2

=n a

G(n a; r, s) a(s) vn (s) ds,

(4.30)

H(na;r,s)bfn(s)ds.

(4.31)

’1

’2

J

v~(r)=na

’1

An eigenvector of En is a pair of functions lfn, vn }, such that

En lfn, vn}

= Xlfn, vn}

for some X E lR. Therefore the relations (4.28) (4.29) hold with = Xv n . These are equivalent to the following ones deduced from (4.30), (4.31):

t; = Xfn , v~

f n (r)

= An

’2

a

J

’1

G(n a; r, s) a(s)

Vn (s)

ds ,

(4.32)

Bifurcation theory and non-uniqueness results

Ch. II, §4 Vn (r)

f

237

’2

= "An

a

H(n a; r, s)b f n (s) ds .

(4.33)

’1

Eliminating Vn ’ we also get,

where Jl

K(n a; r, s) f n (s) ds ,

(4.34)

J

(4.35)

= A2 and

Kik;r, s)

= k2

’2

G(k; r, t)H(k; t, s) a(t)b dt .

’1

We shall now give some properties of the eigenvalues of the operators En. They are based on the following result whose proof can be found for example in Witting [I ] . (1) Lemma 4.9. Let K(r, s) denote a real continuous function defined on the square [rl’ r2] X [rl’ r2]’ which is strictly positive on the interior of this square. The eigenvalue problem

f(r)

’2

J

=A

K(r, s) f(s) ds, rl

< r < r2 ’

(4.36)

’1

possesses a solution Al > 0 which corresponds to an eigenfunction f etO([rl’ r2]),f(r) > Ofo rr 1 AI’ (1)This is a particular case of a general result of Krein-Rutman [1) concerning linear compact operators leaving invariant a cone of a Banach space. These results are infinite dimensional extensions of the Perron-Frobenius theorem for positive matrices, well-known in linear algebra (see for instance R. S. Varga [1]).

Ch. II, §4

Steady-state Navier-Stokes equations

238

Lemma 4.10. The operator En possesses an eigenvalue X~ > 0, which corresponds to an eigenvector {fJ, v~ } with fJ (r) > 0, v~ (r) > 0 for rl X~. Proof. Due to (4.34), {fn. vn } is an eigenvector of Bn with eigenvalue Xn , if and only if, f n is an eigensolution of (4.34) with #1 = X~. Lemma 4.9 is applicable to the equation (4.34) and the present Lemma is proved, except for the positiveness of v~. The positiveness of v~ is a consequence of Lemma 4.8, the positiveness of f~ and the relation (4.33) written with vn =v~,fn =f~. 0 Another spectral property of B n will be useful. Lemma 4.11. Let Xn be an eigenvalue of En’ Then 2 2 2

IX I> Xl ~ n

n

n

+ bl

max]s

0

Proof. Lemma 4.10 gives IXn I > X~. in (4.37). We have (Jt-

(no)2)2 f~(r)

-(Jt-

(no))

v~

= X~ (r) = X~

(4.37)

.

Let us prove the second inequality

(4.38)

a no v~(r) b no f~ (r) .

We multiply the first relation (4.38) by r f~(r), the second by r v~(r), then we integrate and integrate by parts. We obtain:

J(j~,

v~) = X~

where J(j, v) =

no (

’z

J

(a + b)f~

’1

(4.39)

dr,

’1

r [(I’n’

+ (no)4J2 +

v~r

+ 2(na)’ (

(~

)’

+

T~

(dV)2 1 ] dr + r 2 v2 + (no)2 v2

f’ ) + r dr.

Ch.n, §4

Bifurcation theory and non-uniqueness results

239

The right-hand side of (4.39) is bounded by r2

A~ n a maxls

+ bl

J

If~

Ir dr

v~

I Jr

rl

la + bl no max -2-

E;;; A~

2

no

If~

2r

1

1 dr + no

Jr

rl

2

Iv11 2 , dr

rl

I

la+bl

~

E;;; - - max - - 1(f1, vI). (no)2 2 n n

We can divide by l(f~,

v~)

which is non-zero, and (4.37) follows.

0

4.2.3. Spectral properties of B. We first observe that if If, v} is an eigenvector of B with eigenvalue A, then the ifn , v n } corresponding to the expansion (4.27) of If, v} are respectively eigenvectors of the operators B n , with eigenvalue A. We will conversely deduce from Section 4.2.2 a spectral property of B. We consider all the A~, n » 1, given by Lemma 4.10. Due to (4.37), Aln -+ 00 as n -+ 00, and therefore Inf A~

n;;.l

is fmite and strictly positive. We denote by m the largest integer n such that A~ = inf A~. It may happen that A~ = A~ for some other values p;;’l

of n, n < m. We would like to avoid this situation and actually we have: Lemma 4.12. One can choose the period L, so that Al In this case Al

=

> A~,

Vn

> 1.

Inf A~ is denoted AI; Al is a simple eigenvalue of B

n> 1

in iT and any other eigenvalue A of B satisfies IAI > AI. The eigenvector ¢ 1 = if1, vI} corresponding to A1 admits a Fourier expansion of type (4.27) with fJ = v~ = 0, V n > 1.

Proof. Let L be arbitrarily chosen and let a = 21T ILand m be defined as We set 0* = rna, above (m = the largest n such that A~ = Inf A~). n>l

L = LIm. For the corresponding operator B, Al is an eigenvalue of B l’ and Al < A~, V n > 1. The other properties stated in the Lemma

240

Steady-state Navier-Stokel equation«

are now obvious.

Ch. II, §4

0

Until the end of this Section we assume that L is chosen so that Lemma 4.12 holds.

Our last result concerns the degree of Al as an eigenvalue of B. The degree of Al is the dimension of Ker(I - Al B)p , which is independent of p, for large p. Lemma 4.13. Under the conditions of Lemma 4.12, Al is an eigenvalue of B of degree I. Proof. We will show that if (I - A1B)Pif> = 0 for p ~ 2, then (I - AlB) = 0, so that Ker(I - Al B)P is equal for each p to Ker(I - Al B), and its dimension is one, because of Lemma 4.12. We proceed by induction on p and actually we just have to show that (I - Al B)2 if> = 0 implies (I - Al B)if> = O. Let us consider some function if>0 such that (I - A1B)2if>0

= O.

We argue by contradiction and assume that (I - Al B)if>0 is not equal to O. This vector is then equal, within a multiplicative constant, to the previous eigenfunction if>l:

= (I -

if>l

Al

ew.

if>l

= Al Bif>l

(4.40)

We have,

= A1B(if>0 + if> 1),

if>0

which amounts to saying that

to = Al a aza

£2

£vo

a

= Alb az

We recall that fl (r, z)

(vo + vi

(fO +

),

r»

(4.41)

=fl (r) sin(az), vi (r, z) = vI (r) cos(az).

Let us consider the Fourier series of

to and vO ;

Ch. II, §4

Bifurcation theory and non-uniqueness results

2: 12 sin(naz),

n=l

241

00

vO

=

2:

n=l

v~ cos(naz).

The relations (4.41) imply (A’- a 2 ) 2

-i.r-

!? = Al a a (v? = Al

a 2 )v?

ba

+ vi),

ifl + 11),

(4.42)

and for n ~ 2: C/(- a 2 ) 2 ~

= Al a a n v~

-(1- a 2 ) v~

= Al

ban ~.

Since Al is not an eigenvalue of B n for n ~ 2, we see that

~ = v~ = 0 for n ~ 2. We now convert (4.42) into integral equations, as in (4.30), (4.31). Since {f"f, vi} is an eigenvector of B 1 , we obtain

v~ (r)

= Al

J(2

Gto; r, s) a(s)a v~ (s) ds

+ 11 (r),

H(a; r, s) b a!?, (s) ds +

vi (r).

rl

By elimination of v?, and using the kernel K introduced in (4.35), we get !?(r) -

A~

r2

(

i,

K(a; r, s)!?(s)

ds

= 2/1 (r).

(4.43)

The equation (4.43) satisfies the Fredholm alternative. Thus/l is orthogonal to the eigenfunction gl of the adjoint equation: gl (r) -

By Lemma

A~

f

J

r2

K(a; s, r) gl (s) ds

=o.

rl

4.9,/1 andg 1 are positive on (r1’

r2) and this contradicts

Steady-state Navier-Stokea equationa

242

Ch.II,§4

the orthogonality condition

C fl 2

Jr.

(s) gl (s) ds

= 0.

Thus (I - Al B)4Jo = 0, and the proof is complete.

0

4.3. Elements of Topological Degree Theory We recall a few definitions and properties of topological degree theory. For the proofs and further results, the reader is referred to the basic work of J. Leray and J. Schauder [1], or M. A. Krasnoselskii [I], L. Nirenberg [I], P. Rabinowitz [4]. 4.3.1. The topological degree. Let T be a compact operator in a normed space V, and let S = I - T (I = the identity in V). We denote by w, wi’ bounded domains of V; wand aw denote the closure and the boundary of w. If w is a bounded domain of V, if v E V and

v fF. S(aw), one can define an integer d(S, w, v) which is called the topological degree of S, in t», at the point v. The main properties of the degree are the following ones:

= wI

U w2’ and wI n w2 :;: v f$.S(aW2)’ then v f$.S(aw) and

(i)

If w

d(S, w, v)

= d(S,

(ii) If d(S, w, v) the equation (I - T) (u)

rp, if v f$. S(awl ), and

wI’ v) + d(S, w2’ v).

* 0, then v

E S(w), which amounts to saying that

=v

has at least one solution in w. (iii) d(S, w, v) remains constant if S, t», v, varies continuously, in such a way that v never belongs to S(aw). (1) (1) A continuous variation of Sis defined as follows:

S = S(’A.) = I - T(’A.), ’A. E tR (or any topological space), and ’A. ..... T(’A.) I/> is a mapping uni› formly continuous with respect to 1/>(1/> E V).

243

Bifurcation theory and non-uniqueness results

Ch. II, §4

4.3.2. The index. Let Uo be a point of V, v = Suo’ and let us assume that the equation Su = v admits only the solution uo, in some neighbourhood of uo’ In this case, one can define for e small enough, the degree deS, we (uo), v), where we (uo) is the open ball of radius f centred at uo’ According to the property (iii) of the degree, this number is inde› l

pendent of

f,

as

~

O.

We define then, the index of Sat Uo as this degree, for small: i(S, uo)

= deS,

we(uo), v),

f

f

sufficiently

< fO’

Some fundamental properties of this index are listed below: (i) If the equation Su = v posseses a finite number of solutions uk in a bounded domain w, and has no solutions on aw, then

.us, w, v)= L:k

;(S, uk)’

(ii) The index of the identity (T = 0) at any point Uo is one: i(l, uo) = 1.

(iii) Let us assume that T admits at the point uo, a Frechet differ› ential A. Then A is compact like T If 1 - A is one to one (i.e. 1 is not an eigenvalue of A), then Uo is an isolated solution of the equa› tion Su = Suo’ and one can define the index i(S, u 0 ). One has i(S, uo)

= it] -

A, 0)

= i(1 -

A).

(iv) If A is a linear compact operator in Vand if 1 - A is one to one, the index of 1 - A is – 1.(1) Similarly the index of 1 - M is defined on any interval A’ < A < A" containing no eigenvalue of A; the index is constant on such intervals and is equal to –1. In particular i(l- M) = 1 on the interval (0, AI)’ where Al is the smallest positive eigenvalue of A. When A crosses a spectral value Ai of A, the index i(1 - M) is multi› plied by (_I)m where m is the degree of Ai’ i.e. the dimension of Ker(1 - AiA)k which is independent of k, when k is sufficiently large. (1) One can define the index of I - A if and only if I - A is one to one; when it is defined, the index is the same at every point u u ’

244

Steady-state Navter-Stokes equations

Ch. II, §4

4.4 The non-uniqueness theorem.

Our purpose is to prove the following result. Theorem 4.1. For A sufficiently large, and for suitable values of L, the problem (4.2), (4.3), (4.6) possesses z-periodic solutions of period L which are different from the trivial solution (4.7). Proof. We will prove that the equation (4.20) has a non-trivial solu› tion in V, when A is sufficiently large. According to Lemma 4.12, we can choose L so that Al is a simple eigenvalue of B in V: these are the values of L mentioned in Theorem 4.1. It is known from Proposition 4.2 that (4.20) possesses only the trivial solution for A sufficiently small (A <; c(rl’ r2)’ see (4.24». The theorem is already proved if the equation (4.20) has a non trivial solution for some A E [0, AI]’ Therefore we will assume from now on that ¢

= 0 is the only solution of ¢ = A T ¢ for any ¢ E

[0, AI] (4.44)

With this assumption, the next lemmas, using the degree theory, will show the existence of non zero ¢ of V, satisfying ¢ = A T ¢, with A > Al . Lemma 4.14. Let w be some open ball of V centered at O. There exists some 6 > 0 such that ¢ :II: A T ¢ has no solution on the boundary aw of t», for each A in the interval [AI’ Al + 6]. Proof. We argue by contradiction. If this statement is false, there exists a sequence of An decreasing to AI, and a sequence of Uk belonging to aw. such that

Since the sequence un is bounded, the sequence T un is relatively compact (by Lemma 4.4), and there exists a subsequence T uni con› verging to some limit v in V. Then u ni = Ani T u ni converges to Al v. Since T is continuous, we must have

Al v = Al T(AI v). Thus Al V is a solution of ¢ = Al T¢, and because of (4.44), Al v = 0, v = 0. This contradicts the fact that I!A I v II v is equal to the radius of the ball w (llu n II = radius of w, V n).

Ch. II, §4

Bifurcation theory and non-uniqueness results

245

Lemma 4.15. Under the assumption (4.44), if wand 8 are as in Lemma 4.14, the equation cP = A T cP has no solution on aw, for any A E [0, Al + 8). Obvious Corollary of Lemma 4.14. This lemma allows us to define the degree d(l - AT, w, 0) for A E [0, x, + 8). Lemma 4.16. With 8 and w as before, d(I - AT, w, 0)

= 1,

for A E [0, Al + 8).

Proof. It follows from the property (iii) of the degree that d(l - AT, w, 0) = di], w, 0) = i(l) and this index is equal to one (the index of the identity). Lemma 4.17. Under the assumption (4.44), there exists for any A E (AI’ Al + 8) at least one non-trivial solution of cP = A T cp. Proof. According to Lemma 4.13, and the properties (iv) of the index, i(I - AB) is equal to 1 on [0, AI) and is equal to -Ion (A. 1 , Al + 8). According to the property (iii) of the index, i(l - AT, 0) is + 1 for A E (0, AI) and -1 for A E (AI’ Al + 8).’ If A E (AI’ Al + 8) and if zero is the only solution of cp = A. T cp in w, we should have d(I - AT, w, 0)

= i(l -

AT, 0)

according to the property (i) of the index. But we proved that d(l - AT, w, 0)

= +1,

i(l - AT, 0)

= -1,

A E (AI’ Al + 8).

Thus the equation cp = A T cp has a non-trivial solution for any A E (A.1’ Al + 8). The proof of Theorem 4.1 is complete. Remark 4.3. The condition "A sufficiently large", amounts to saying that the angular velocity a is large or that the viscosity v is small (for fixed r1’ ’2)’ Remark 4.4. Under condition (4.44), there exists for each A E (AI’ Al + 6) a non-trivial solution Cp", of (4.20). One can prove that CPA -+ 0 in V, as A decreases to AI’ This is the bifurcation

246

Steady-state Navier-Stokes equations

Ch.II, §4

In case of the Benard problem the situation is very similar, but it can be proved that there only exists the trivial solution for A E [0, AI]’ Thus the assumption (4.44) is unnecessary, and one does prove the occurrence of a bifurcation (see V. I. Iudovich [2] , Rabinowitz [1], Velte [1] ). A study of the Taylor problem by analytical methods is developed in Rabinowitz [5]. 0 Acknowledgment. The author gratefully acknowledges useful remarks of P. H. Rabinowitz on Section 4.

CHAPTER III THE EVOLUTION NAVIER-STOKES EQUATION Introduction This final chapter deals with the full Navier-Stokes Equations; i.e., the evolution nonlinear case. First we describe a few basic results con› cerning the existence and uniqueness of solutions, and then we study the approximation of these equations by several methods. In Section I we briefly examine the linear evolution equations (evolution Stokes equations). This section contains some technical lemmas appropriate for the study of evolution equations. Section 2 gives compactness theorems which will enable us to obtain strong con› vergence results in the evolution case, and to pass to the limit in the non› linear terms. Section 3 contains the variational formulation of the problem (weak or turbulent solutions, according to J. Leray [1], [2], [3]; E. Hopf [2]) and the main results of existence and uniqueness of solution (the dimension of the space is n = 2 or 3); the existence is based on the construction of an approximate solution by the Galerkin method. In Section 4 further existence and uniqueness results are pre› sented; here existence is obtained by semi-discretization in time, and is valid for any dimension of the space. In the final section we study the approximation of the evolution Navier-Stokes equations, in the two- and three-dimensional cases. Several schemes are considered corresponding to a classical discretiza› tion in the time variable (implicit, Crank-Nicholson, explicit) asso› ciated with any of the discretizations in the space variables introduced in Chapter I (finite differences, finite elements). We conclude with a study of the nonlinear stability of these schemes, establishing sufficient conditions for stability and proving the convergence of all these schemes when they are stable.

§ 1. The Linear Case In this section we develop some results of existence, uniqueness, and regularity of the solutions of the linearized Navier-Stokes equations. After introducing some notation useful in the linear as well as in the 247

248

The evolution Navier-Stokes equations

Cli. III, § 1

nonlinear case (Section 1.1), we give the classical and variational formu› lations of the problem and the statement of the main existence and uniqueness result (Section 1.2); the proofs of the existence and of the uniqueness are then given in Sections 1.3 and 1.4.

1.1. Notations Let 0 be an open Lipschitz set in eRn ; for simplicity we suppose 0 bounded, and we refer to the remarks in Section 1.5 for the unbounded case. We recall the definition of the spaces "Y, V; H, used in the pre› vious chapters and which will be the basic spaces in this chapter too: "1/= {u E ~(O),

div u = O}

V = the closure of fin Hfi (0),

H

= the closure

of fin L 2 (0).

(1.1) (1.2) (1.3)

The space H is equipped with the scalar product (. , .) induced by L 2(0); the space V is a Hilbert space with the scalar product n

«u, v)

= 2: i= 1

(DiU, Div),

(1.4)

since 0 is bounded. The space V. is contained in H, is dense in H, and the injection is con› tinuous. Let H’ and V’ denote the dual spaces of H and V, and let i denote the injection mapping from V into H. The adjoint operator t is linear continuous from H’ into V’, and is one to one since i( V) = V is dense in Hand i’(H’) is dense in V’ since i is one to one; therefore H’ can be identified with a dense subspace of V’. Moreover, by the Riesz representation theorem, we can identify H and H', and we arrive at the inclusions VCH=H'CV',

(1.5)

where each space is dense in the following one and the injections are continuous. As a consequence of the previous identifications, the scalar product in H of f E Hand u E V is the same as the scalar product of f and u in the duality between V’ and V: (f, u) = if, u), V fEH, VuE V.

For each u in V, the form

(1.6)

Thelil1ear case

Ch. III, § 1

249

v E V-+ ((U, V)) E lR

(1.7)

is linear and continuous on V; therefore, there exists an element of V’ which we denote by Au such that

= «U, v)), V v E V.

(1.8)

It is easy to see that the mapping u -+ Au is linear and continuous, and, by Theorem 1.2.2 and Remark 1.2.2, is an isomorphism from V onto V’.

If n is unbounded, the space V is equipped with the scalar product [u, v]

= «u,

v)) + (u, v);

(1.9)

the inclusions (1.5) hold. The operator A is linear continuous from V into V’ but it not in general an isomorphism; for every e > 0, A + el is an isomorphism from Vonto V’. Let a, b be two extended real numbers, -00 ~ a < b ~ 00, and let X be a Banach space. For given ex, I ~ ex < +00, LI:I.(a, b; X) denotes the space of LI:I.-integrable functions from [a, b] into X, which is a Banach space with the norm

L I b

11!tt) 11’jC cit

11/1:1.

(1.10)

The space L (a, b; X) is the space of essentially bounded functions from [a, b] into X, and is equipped with the Banach norm GO

Ess Sup "f(t) [a, bl

"x'

(1.11)

The space ~([a, b] ; X) is the space of continuous functions from [a, b] into X and if -00 < a < b < 00 is equipped with the Banach norm Sup

tE

[a. bl

"f(t)

"x'

(1.12)

°

Most often the interval [a. b] will be the interval [0, T], T> fixed; when no confusion can arise, we will use the following more con› densed notations, LI:I.(X) = LI:I.(O, T; X), I ~(X)

=<6’([0, T] ; X).

~

ex ~ +00

(1.13) (1.14)

The remainder of this Section 1.1 is devoted to the proof of the following technical lemma concerning the derivatives of functions with

Theevolution Navier-Stokes equations

250

Ch. III, §l

values in a Banach space: Lemma 1.1. Let X be a given Banach space with dual X’ and let U and g be two functions belonging to L 1 (a, b; X). Then, the following three conditions are equivalent (i) u is a. e. equal to a primitive function ofg, u(t)

(ii)

=t +

1:

g(s)ds,

t EX, a.e. t E

r

( 1.15)

[a, b]

For each test function ¢ E ,!»«a, b)),

f

J

bu(t)4J'

(Odt

a

=-

a

g(t)~(t)dt

(~’

=:: );

(1.16)

(iii) For each 11 E X’,

d dt (u, 11) = (g, 11),

(1.17)

in the scalar distribution sense, on (a, b). If (i) - (iii) are satisfied u, in particular, is a. e. equal to a con tinuous function from [a, b] into X Proof. We suppose for simplicity that the interval [a, b] is the interval [0, T]. A 1egitamate integration by parts shows that (i) implies (ii) and (iii); it remains to check that the property (iii) implies the property (ii) and that (ii) implies (i). If (iii) is satisfied and ¢ E 9}«O, T)), then by definition, T

or

J
<[

u(t)~’(t)dt+

~’(t

)dt

=-

J (g(t),~) T

~(t)dt

(1.18)

0

1: g(t)~(t)dt,~)=O,

V~EX’,

so that (1.16) is satisfied. Let us now prove that (ii) implies (i). We can reduce the general case to the case g = 0. To see this, we set v = U - Uo with

Ch. Ill, § 1

uo(t) =

Thelinear case

L

251

(1.19)

g(s)ds;

it is clear that "0 is an absolutely continuous function and that (1.16) holds with" replaced by "0 and

"0 = g; hence

J

T

0 .(t)
~ E~(O,

= 0, V

(1.20)

T)).

The proof of (i) will be achieved if we show that (1.20) implies that

v is a constant element of X. Let

rJ>0

be some function

J: ~o(t)dt

Any function

rJ>

in~«O,

T»,

such that

= I.

in.@«O,

T»

can be written as

J T

~ =1-4>0 + «, A = 0 ~(t)dt,

r

'iJ E ~«O,

T));

(1.21)

indeed since

(if>(t) -

o

A~o(t) dt

= 0,

the primitive function of rJ> - ’ArJ>0 vanishing at 0, belongs to ’@«O, T», and I/J is precisely this primitive function. According to (1.20) and (1.21),

J

T 0 (>
where

J

~=

~)
T

0 ·(s)
= 0, V ~

E~«O,

T))

(1.21a)

Theevolution Navier-Stokes equations

252

Ch.III, §1

To achieve the proof, it remains to show that (1.21 a) implies that

=~

vet)

a.e.,

i.e., that a function w ELI (X) such that

f

T

0 w(t)lf>(t)dt

= 0, V

~

(1.22)

E.’»((O, 1),

°

is zero almost everywhere.

This well-known result is proved by regular› ization: if is the function equal to won [0, T] and to outside this interval, and if Pe is an even regularizing function, then for small enough, Pe * ~ belongs to 1»«0, n), V ~ E!iP«O, n), and

w

+ ..

= _..

J

°

(Pe

w)

(t)~(t)dt

= 0.

°

Hence, for any 71 > fixed, Pe * Wis equal to on the interval [71, T-71] , for small enough; as e -+ 0, Pe Wconverges-to win L 1 (-00, +co; X). Thus w is zero on [71, T - 71]; since 71 > is arbitrarily small, w is zero on the whole interval [0, T]. 0

°

1.2. The Existence and Uniqueness Theorem

°

Let n be a lipschitz open bounded set in &in and let T> be fixed. We denote by Q the cylinder n X (0, n. The linearized Navier-Stokes equations are the evolution equations corresponding to the Stokes problem: To find a vector function

u : n X [0,

T] -+

lR n

and a scalar function p :

n X [0, T]

-+

lR,

respectively equal to the velocity of the fluid and to its pressure, such that

Ch.IIl, §1

Thelinear case

auat - VlYl + grad p =I in Q = n X (0, T), div u

= a in Q,

253

(1.23) (1.24)

= a on an X [0, T] , u(x, 0) = uo(x), in n, u

(1.25) (1.26)

where the vector functions I and Uo are given, I defmed on n X [0, T] , Uo defined on n; the equations (1.25) and (1.26) give respectively the boundary and initial conditions. Let us suppose that u and p are classical solutions of (1.23)-( 1.26), say u EfG2(Q), P E ~1 (Q). If v denotes any element of ~ it is easily seen that

(

~"

) + I’«u

))

=if, v),

(1.27)

By continuity, the equality (1.27) holds also for each v E V; we observe also that

) at ’ v ( au

d

= dt

(u. v).

This leads to the following weak formulation of the problem (1.23)› (1.26): For I and Uo given,

IE £2(0, T; V’)

(1.28)

uOEH,

(1.29)

to find u, satisfying

u E £2(0, T; V)

(1.30)

and d dt (u. v) + v«u. v» u(O)

= uo'

= (f, v>,

Vv E V.

(1.31) (1.32)

If u belongs to £2(0, T; V) the condition (1.32) does not make sense in general; its meaning will be explained after the following two remarks: (i) The spaces in (1.28), (1.29), and (1.30) are the spaces for which existence and uniqueness will be proved; it is clear at least that a smooth

Theevolution Navier-Stokes equations

254

Ch. III, § 1

solution u of (1.23)-(1.26) satisfies (1.30). (ii) We cannot check now that a solution of (1.30)-(1.32) is a solution, in some weak sense, of (1.23)-(1.26); hence we postpone the investigation of this point until Section 1.5. By (1.6) and (1.8), we can write (1.30) as d dt tu, v) = (f - vAu, v), Vv E V.

(1.33)

Since A is linear and continuous from V into V’ and u E L 2 (V), the function Au belongs to L 2(V '); hencef - vAu E L2(V') and (1.33) and Lemma 1.1 show that u' E L 2(0, T; V’)

(1.34)

and that u as a.e. equal to an absolutely continuous function from [0, T] into V'. Any function satisfying (1.30) and (1.31) is, after modi› fication on a set of measure zero, a continuous function from [0, T] into V’, and therefore the condition (1.32) makes sense. Let us suppose again that j' is given in L2(V ') as in (1.28). Ifu satisfies (1.30) and (1.31) then, as observed before, u satisfies (1.34) and (1.33). According to Lemma 1.1 the equality (1.33) is itself equivalent to u ’ + vAu

=f.

(1.35)

Conversely if u satisfies (1.30), (1.34), and (1.35), then u clearly satisfies (1.31), V v E V. An alternative formulation of the weak problem is the following: Given f and Uo satisfying (1.28)-(1.29), to find u satisfying u E L 2(0, T; V), u’ E L2(0, T; V'), (1.36) u’ + vAu = f, on (0, T),

(1.37)

u(O) = uo'

(1.38)

Any solution of (1.36)-(1.38) is a solution of (1.30)-(1.32) and conversely. Concerning the existence and uniqueness of solution of these problems, we will prove the following result.

Theorem 1.1. For given f and Uo which satisfy (1.28) and (1.29), there exists a unique function u which satisfies (1.36)-(1.38). Moreover

255

The linear case

Ch. III, § 1

(1.39)

T] ; H).

U E~([O,

The proof of the existence is given in Section 1.3, that of the unique› ness and of (1.39) are in Section 1.4.

1.3. Proof of the Existence in Theorem 1.1 We use the Faedo-Galerkin method. Since V is separable there exists a sequence of linearly independent elements, WI, ... , W m , ... , which is total in V. For each m we define an approximate solution um of 0.37) or (1. 31) as follows: m

um

=

2:

(1.40)

Kim (t)Wi’

i= 1

and (U:n, Wj) + U m (0)

v«u m Wj»

= (f,

Wj), j

= 1, ... , m,

= UOm ’

(1.41) (1.42)

where "Om is, for example, the orthogonal projection in H of Uo on the space spa nne db y wI’ ... , wm • (l) The functions Kim’ 1 ~ i ~ m, are scalar functions defined on [0, T], and 0.41) is a linear differential system for these functions; indeed we have m

m

2:

2:

i= 1

i= 1

= <{(t),

Wj),j

= 1, ... , m;

(1.43)

since the elements WI’ ... , W m ’ are linearly independent, it is well› known that the matrix with elements (Wi’ Wj) 0 ~ i, j ~ m) is non› singular; hence by inverting this matrix we reduce (1.43) to a linear system with constant coefficients m

Kim (t) + J=~ 1

m

2:

j= 1

1 «t
in the norm of H. as m ...... 00.

(1.44)

The evolution Navier-Stokes equations

256

Ch. III, § 1

where 04/, (3i/ E cR The condition (1.42) is equivalent to m equations Kim (0)

= the i th component of uO m

(l.45)

The linear differential system (1.44) together with the initial condi› tions (l.45) defines uniquely the Kim on the whole interval [0, T] Since the scalar functions t -+
um

E L 2(0, T;

n, u~

E L 2(0, T;

n.

(1.46)

We will obtain a priori estimates independent of m for the functions

um and then pass to the limit. A Priori Estimates. j

We multiply equation (1.41) by K/m (t) and add these equations for We get

= 1, ..., m.

(t), u m (r)

(u~

+ vllu m (t)1I2 =
But, because of (1.46), ,

_ d

2(u m (t), U m (t» - dt

IU m (t)1

2

,

and this gives d

-Iu m (t)1 2 + 2vllu m (t)112 dt

= 2
(1.47)

The right-hand side of (1.47) is majorized by 2I1f(t)lI v ’ Ilum (t)1I ~ vllu m (t)!12

1

+-

v

IIf(t)II~,.

Therefore d

1

-Iu m (t)1 2 + vllu m (t)1I2 ~ - IIf(t)II~ dt v

Integrating (1.48) from

°

•.

to s, s > 0, we obtain in particular

(1.48)

Thelinear case

Ch. III, § 1

1 + -;

257

JT0 IIf(t) 11~,dt.

(1.49)

2 ~ JT I f(t)I ~’dt. V

(1.50)

Hence:

SE

Sup IU m (s)1 [0, T]

2~

lu ol +

o

The right-hand side of (1.50) is finite and independent of m; there› fore The sequence u m remains in a bounded set of L""(O, T; H).

We then integrate (l.48) from

f

(1.51 )

°

to T and get

T

Iu.. (1')1 2 + v

0

Ilu.. (t) 112 dt ..

(1.52)

This shows that The sequence u m remains in a bounded set of L2(0, T; V). Passage to the Limit. The a priori estimate (1.51) shows the existence of an element L"" (0, T; H) and a subsequence m’ ~ 00, such that

(1.53)

U

in

urn' converges to u, for the weak-star topology of

(1.54)

L""(O, T; H);

(1.54) means that for each

vEL 1 (0,

T; H),

258

The evolutionNavier-Stokel equations

J

T

0 (um,(f) - U(f), v(t))df -+

0, m’ -+~.

Ch. III, § 1

(1.55)

By (1.53) the subsequence urn' belongs to a bounded set of L 2 (0, T; V); therefore another passage to a subsequence shows the existence of some U. in L 2(0, T; V) and some subsequence (still denoted urn' ) such that urn' converges to U., for the weak topology of

L 2(0, T; V).

(1.56)

The convergence (1.56) means

~

T 0 (um,(f) - U. (f), v(f)>df -+

In particular, by (1.6),

J

T

0, V v E £2(0, T; V’).

J

T

0 (u m' (f), v(f»df -+

0 (u" (f), v(t))df,

(1.57)

for each v in L2(0, T; H). Comparing with (1.55) we see that

J T

0 (u(t) - u;

ro. v(t))df = 0,

(1.58)

for each v in L 2 (0, T; H); hence U = U. E L 2(0, T; V) n L (0, T; H). GO

(1.59)

In order to pass to the limit in equations (1.41) and (1.42), let us con› sider scalar functions v continuously differentiable on [0, T] and such that (1.60) "’(T) = For such a function v we multiply (1.41) by ’" (t), integrate with respect to t and integrate by parts: .

°

L T

T

(U;" (f), Wj)1/!(f)dt = -

So

(um (f)1/!'(f), wj)dt

- (urn (0), Wj)"’(O).

The linear case

Ch. III, § 1

Hence we find,

-f

T

259

f

T

0 (u m (I), 1/i(t)wj)dl +v

=(uo m , wj )1/1(0) +

J

0 «u m (t), 1/I(t)wj»dl

T

0 dl.

(1.61)

The passage to the limit for m = m’ ~ 00 in the integrals on the left› hand side is easy using (1.54), (1.57), and (1.59); we observe also that UOm ~

uo,

(1.62)

in H, strongly.

Hence we find in the limit T

T

- So (u(t), 1/I’(t)w,)dl + v So «u(I), 1/I(I)wj»)d1 T

=(uo, Wj)1/I(O) + SO (f(I), Wj>1/I(I)dI.

(1.63)

The equality (1.63) which holds for eachj, allows us to write by a linearity argument: T

f

T

- So (u(I), v)1/I'(t)dl + v

0 «u(t), v))1/I(I)d1

f

T

= (uo, v)1/I(O) + 0 (f(t), v>1/I(I)dI,

(1.64)

for each v which is a finite linear combination of the w;’s. Since each term of (1.64) depends linearly and continuously on v, for the norm of V, the equality (1.64) is still valid, by continuity, for each v in V. Now, writing in particular (1.64) with VJ = ~ E9}«O, T», we find the following equality which is valid in the distribution sense on (0, T): d dt (u, v) + lI«U, v»

= (f,

v), V v E V;

(1.65)

The evolutionNavier-Stokel equations

260

Ch. III, § 1

which is exactly (1.31). As proved before the statement of Theorem 1.1, this equality and (1.59) imply that u' belongs to L 2(0, T; V’) and

u' + vAu

=f.

(1.66)

Finally, it remains to check that u(O) = Uo (the continuity of u is proved in Section 1.4). For this, we multiply (1.65) by 1/1(1), (the same 1/1 as before), integrate with respect to t, and integrate by parts:

~ We get

T

d

dt (u(t), v)1/I(t)dt

o·

=- JT (u(t), v)1/I'(t)dt 0

+ (u(O),

-J

T

v)1/I(0).

J T

0 (u(I), _l\!/(I)dl

+ v 0 «u(I), v))>/I(ljdl

J T

= (u(O), _)>/1(0) + 0 (/(1), v)>/I(t)dl.

(1.67)

By comparison with (1.64), we see that (uo - u(O), v)1/I(0)

= 0,

for each v E V, and for each function 1/1 of the type considered. We can choose 1/1 such that 1/1(0) =1= 0, and therefore (u(O)-uo, v)=O, VvE V.

This equality implies that u(O)

= Uo

and achieves the proof of the existence.

0

1.4. Proof of the Continuity and Uniqueness This proof is based on the following lemma which is a particular case of a general theorem of interpolation of Lions-Magenes [I] :

Lemma 1.2. Let V; H, V’ be three Hilbert spaces, each space included

Thelinear case

Ch. Ill, § 1

261

in the following one as in (l.5), V’ being the dual of V If a function u belongs to L 2(0, T; V) and its derivative u’ belongs to L2(0, T; V’), then u is almost everywhere equal to a function continuous from [0, T] into H and we have the following equality, which holds in the scalar distribution sense on (0, D: d

dt lul 2 = 2(u’, u).

(1.68)

The equality (1.68) is meaningful since the functions t -+ lu(t)1 2 , t -+ (u’(t), u(t»

are both integrable on [0, T]. An alternate elementary proof of the lemma is given below. If we assume this lemma, (1.39) becomes obvious and it only remains to check the uniqueness. Let us assume that u and v are two solutions of (1.36)-(1.38) and let w = u - v. Then w belongs to the same spaces as u and v, and w’ + vAw = 0, w(O) = 0.

(1.69)

Taking the scalar product of the first equality (1.69) with w(t), we find

=

(w’ (t), w(t» + v II w(t) 11 2

°

a.e.

Using then (1.68) with u replaced by w, we obtain d -lw(t)1 2 + 2vllw(t) 11 2 dt

=

°

Iw(t)1 2 '" Iw(0)1 2 = 0, t E [0, T] . and hence .,(t) = v(t) for each t,

0

Proof of Lemma 1.2. The elementary proof of Lemma 1.2 which was announced before, is now given in the two following lemmas. Lemma 1.3. Under the assumptions of Lemma 1.2, the equality (1.68) is satisfied. Proof. By regularizing the function U, from cR into V, which is equal to u on [0, T] and to 0 outside this interval, we easily obtain a sequence of functions urn such that V m, urn is infmitely differentiable from [0, T] onto V, (1.70)

The evolutionNavier-Stokes equations

262

as m ~OO,

Ch. III, § 1

Llo c ( ] 0, T[; V),

Um

~U in

u~

~u’inLfoc(]O,T[;

V').

(1.71)

Because of (1.6) and (1.70), the equality (1.68) for um is obvious: -

d

dt As m ~

IU m (r)]

2 -_ 2(u’m (t), U m (t)) _ - 2(u’m (t), u m (t) ) ,V m.

(1.72)

it follows from (1.71) that lu ml 2 ~lul2 in Lloc(] 0, T[) (u~, u m ) ~ (u', u) in Lloc(] 0, T[) 00,

These convergences also hold in the distribution sense; therefore we are allowed to pass to the limit in (1.72) in the distribution sense; in the limit we find precisely (1.68). Since the function t ~ (u'(t), u(t)

is integrable on [0, T] , the equality (1.68) shows us that the function u of Lemma 1.3 satisfies (1. 73) u E L" (0, T; H). In the particular case of the function u satisfying (1.36)-(1.38), this was proved directly in Section 1.3. According to Lemma 1.1., u is continuous from [0, T] into V'. Therefore, with this and (1.73), the following Lemma 1.4 shows us that u is weakly continuous from [0, T] into H, i.e., V v EH, the function t .... (u(t), v) is continuous.

(1.74)

Admitting temporarily this point we can achieve the proof of Lemma 1.2. We must prove that for each to E [0, T], lu(t)-u(to)1 2 ~O,

ast~to’

(1.75)

Expanding this term, we find lu(t)1 2

+ lu(to)1 2

- 2(u(t), u(to))'

When t ~ to, lu(t)1 2 ~ lu(to)1 2 since by (1.68), lu(t)1 2

= lu(to)1

2

+

2J

t

(u'(s), u(s)>ds; to

The linear case

Ch. III, § 1

263

and because of (1.74) (u(t), u(to)) -+ lu(to)/2,

so (1.75) is proved. The proof of Lemma 1.2 is achieved as soon as we prove the next lemma. This lemma is stated in a slightly more general form.

Lemma 1.4. Let X and Y be two Banach spaces, such that

XeY

(1.76)

with a continuous injection. If a function tP belongs to L"" (0, T; X) and is weakly continuous with values in Y, then tP is weakly continuous with values in X. Proof. If we replace Y by the closure of X in Y, we may suppose that X is dense in Y. Hence the dense continuous imbedding of X into Y gives by duality a dense continuous imbedding of Y’ (dual of Y), into X’ (dual of X):

y'ex'.

(1.77)

By assumption, for each 11 E y’, (tP(t), 11)-+ (tP(to), Tl), ast-+to,Vto,

0.78)

and we must prove that (1.78) is also true for each 11 EX’. We first prove that tP(t) E X for each t and that IItP(t)lI x ~ IItPIIL""co,T;X)' VtE[O,T].

(1.79)

°

Indeed, by regularizing the function 1> equal to tP on [0, T] and to outside this interval, we find a sequence of smooth functions tP m from [0, T] into X such that IItPm (t)lI x ~ IItPll

ee

L

(X)

,V m, V t E [0, T]

and (tP m (t), 11) -+ (tP(t), 11), m -+ 00, V 11 E y’.

Since l(tP m (t), 11)1 ~ IltP II

L

co

(X)

1I11l1x', V m, V t,

we obtain in the limit l(tP(t), 11)1 ~ IItP ll

L

ee

(X)

1I11 l1x" V t E [0, T], V11 E y’.

This inequality shows that tP(t) EX and that (1.79) holds.

264

Theevolution Navier-Stoke« equations

Ch. III, § 1

Finally let us prove (1.78) for 'TI in X’. Since y’is dense in X’, there exists, for each > 0, some 'TIe E y’ such that

1I'T1 - 'TIe"x' We then write

=
-+

'TIe> +
-
'TIe>

lI ~ 2 II¢ ilL" (X) ;t I<¢(t) - cP(t 0)’ 'TIe>' to, since 'TIe E y’, the continuity assumption implies that


'TIe> -+

°

and hence lim l I ~ 2e II cP IlL "(X)·

°

Since e > is arbitrarily small, the preceding upper limit is zero, and (1.78) is proved. 0 1.5. Miscellaneous Remarks We give in this section some remarks and complements to Theorem 1.1. An Extension 01 Theorem 1.1. Theorem 1.1 is a particular case of an abstract theorem, involving abstract spaces V and H, and an abstract operator A; see Lions› Magenes [I] . If instead of (1.28) we assume that (1.80) 1=/1 +/2,11 EL 2(0,T;V’), h E L 1(0, T; H), then all the conclusions of Theorem 1.1 are true with only one modifi› cation: U’ E L2(0, T; V’) + L 1(0, T; H). (1.81) In the proof of the existence, we write after (1.47): d dt IU m (t)1 2 + 2vllu m (t)1I 2 ~ 211t1 (t)lI v' lIum (t)1I

+ 2Ih(t)llum (t)1 ~ vllu m (t)1I2 I +-

V

IIfl (t) II}. + Ih(t)1

{l + Iu m (t)12 }.

(1.82)

Thelinear case

Ch. III, § 1

Hence, in particular d I - {I + Iurn (t)1 2 } :E;;; - IItl (t)II~, dt v

265

+ Ih(t)/ {I + Iurn (t)1 2 }.

(1.83)

Multiplying this by

we obtain

d~

I

exp ( - (

.;;

~

If,(a)lda)- (I + IU m (tll')

nfl (t)II}. exp ( - (

I

If,(a)Ida) .

°

Integrating this inequality from to s, s > 0, we obtain a majoration similar to (1.50) which implies (1.51). Then integrating (1.82) from to Twe obtain (1.53). The proof of the existence is then conducted exactly as in Section 1.3. Concerning the derivative u’, we have u’

= -vAu + t 1 + h

E L 2(0, T; V’)

°

+ L 1 (0, T; H).

(1.84)

It is easy to see that Lemma 1.2 is also valid if

u E L 2(0, T; V) n ir (0, T; H), u’ E L2(0, T; V’) + Ll (0, T; H).

(1.85)

Noting this, we can prove the uniqueness and the continuity of u, u E
The Case n Unbounded. For the evolution problem, when n is unbounded, the introduction of the space Y considered in the stationary unbounded case (Chapter I, Section 2.3) is no longer necessary. All the previous results hold if n is unbounded and V is equipped with the norm (1.9). Let us assume, in the most general case, that t satisfies (1.80). We have exactly the same results as in Theorem 1.1, if t satisfies (1.28), and the same results as in

The evolution Navier-Stokea equations

266

Ch. III, § 1

Remark 1.1 iff satisfies (1.80). The only difference is that we must replace (1.82) by

.!

dt

lum( t) 12 + 2vllum(t)lIl1um(t)1I 2 21!2(t)llu m(t)1 1 ~ vllu m( t) 1I 2 + v IUm(t) 12 + -lIt1 (t)II~1

~ 21Itl(t)llv’~um(t)]+

V

Hence

( 1.86)

+ Ih(t)!O + lum (t ) \2) d dt {l + IUm(t)12 } ~ (Ih(t)1 + v) {l + I~m (t)12

}

+~ I f1(t)I ~,.

(1.87) v This inequality is then treated exactly as(1.83), to obtain (1.51).. After that, integrating (1.86) from 0 to T, we obtain

[

lu m (t)l'dt <; Const.

This estimation, together with (1.51), gives (1.53). The proofs of the existence, the uniqueness, and the continuity are then exactly the same as before. [J . , Interpretation of the Variational Problem We wish to make precise in what sense the function u defined by Theorem 1.1 is a solution of the initial problem (1.23)-(1.26).

Proposition 1.1. Under the assumptions of Theorem 1.1, there exists a distribution p on Q = n X (0, n, such that the function u defined by Theorem 1.1 and p satisfy (1.23) in the distribution sense in Q; (1.24) is satisfied in the distribution sense too and (1.26) is satisfied in the sense u(t) ~ Uo in L 2 (n ), as t ~ O. (1.88) Proof. The equality (1.24) is an easy consequence of u E L 2 (0, T; V); (1.26) and (1.88) follow also immediately from Theorem 1.1; (1.25) is satisfied in a sense which depends on the trace theorems available for n since u is in L2(0, T; nA(n». To introduce the pressure, let us set U(t)

J

=

t

0 u(s)cls, F(t) =

J t

0 f(s)ds.

(1.89)

267

The linear case

Ch. III, § 1

It is clear that, at least, U E’6’([O, T]; V), F E’6’([O, T] ; V').

Integrating (1.31), we see that (u(t) - uo. v)

+ v«U(t), v»

= (F(t), v), V t E

[0, T], (1.90)

VvEV,

or (u(t) - Uo - vAU(t) - F(t), v)

VtE [0,

n. VvE v.

= 0,

By an application of Proposition I. 1.1 and 1.1.2, we find, for each

t E [0, T], the existence of some function P(t), P(t) E L2(n),

such that u(t) - Uo - vAU(t) + grad P(t)

= F(t).

(1.91)

We infer from Remark 1.1.4 (ii) that the gradient operator is an isomorphism from L 2(n)/6? into a-I (n). Observing that gradP=F+ vAU - u

+ uo,

we conclude that grad P belongs to CC ([0, T] ; a-I (n» as does the right-hand side of (1. 92); therefore P E’6’([O, T]; L2(.o».

(1.93)

This enables us to differentiate (1.91) in the t variable, in the distri› bution sense in Q = n X (0, T); setting p=

ap

at ’

(1.94)

we obtain precisely (1.23). We do ’10t have in general any information on p better than (1.93)› (1.94). In the next proposition, we will get more regularity on p after assuming more regularity on the data f and uo. 0

Some Results of Regularity Assuming that the data n. f. uo, are sufficiently smooth, we can obtain as much regularity as desired for u and p. We will only prove a simple result of this type: Proposition 1.2. Let us assume that

fEL 2(0,

T; H)

n is of class~2.

that

The evolution Navier-Stoke. equationr

268

Ch. III, § I

and

(1.96)

Uo E V

Then

(1.97)

E £2(0, T; n 2 (n )),

U

U/ E

£2 (0, T; H), i.e.,

(1.98)

u’ E £2 (Q),

(1.99)

p E £2(0, T; HI (n)).

Proof. The first point is to obtain (1.98); this is proved by getting another a priori estimate for the approximate solution u m constructed by the Galerkin method. Using the notation of Section 1.3, we multiply (1041) by g’im (f), and add these equalities for j = 1, ..., m; this gives (t)12 +'v((u m (t), U~ (f))) = (f(t), U~ (f))

Iu~

or (t)1 2

21u~

+v -

d

df

lIum (t) 11 2

= 2(f(t), U~

(t)).

(1.100)

We then integrate (1.100) from 0 to T, and use the Schwarz inequality; we obtain

J

T

2

0

lu;' (t)I'dt + vllu.. (D II' =v Iluo.. II'

J T

+2

0 (f(t),

u;. (t))dt'; vlluo.. II'

L L J T

+

J

T

0

T

1!(t)I'dt +

lu;' (t)I'dt,

T

lu;' (t)I'dt.; vnuo.. II' +

0

If(t)I'dt.

(1.101)

The basis Wi used for the Galerkin method may be chosen so that Wi E V for each j and we can take

Compactness theorems

Ch. III, § 2 UOm

by

= the projection in V ofuo

WI"’"

269

on the space spanned

wm .

Therefore UOm ~ Uo

in V strongly as

m ~

(1.102)

00,

and (1.103)

lIuOm II ~ lIuo II. With these choices of the wjs and of uOm’ (1.101) shows that u~

belongs to a bounded set of L 2 (0, T; H),

(1.104)

and (1.98) is proved. We then come back to the equalities (1.23), (1.24), (1.25) and we apply the regularity theorem of the stationary case (Theorem 1.2.4): For almost every fin [0, T] ,

+ grad pet) =! -

-v~u(t) div u(t) u(f)

u' E L

2(0,

= 0 in il

T; L

2(il))

}

(1.105)

= 0 on ail

so that u(t) belongs to H 2(il) and pet) belongs to HI (il). Moreover, since the mapping !(t) - u'(t) ~ {u(f), pet)}

is linear continuous from L 2(il ) into H 2(il) x HI(il) , due to 1.(2.40), and since

!- u’EL2(0, T; L2 (il ) ), it is clear that (1.97) and (1.99) are satisfied. §2, Compactness Theorems

The compactness theorems presented in Chapter II are not sufficient for the nonlinear evolution problems. Our goal now is to prove some compactness theorems which are appropriate for the nonlinear prob› lems of the remainder of this chapter. After a preliminary result given in Section 2.1, we prove in Section 2.2 a compactness theorem in the frame of Banach spaces. In Section 2.3 we prove two other compactness theorems in the frame of Hilbert

270

The evolution Navier-Stokes equations

Ch. III, §2

spaces; one of them involves fractional derivatives in time of the functions. Some related discrete forms of these theorems will be studied later on. 0 2.1. A Preliminary Result The proofs of the compactness theorems of the next two sections will be based on the following lemma. Lemma 2.1. Let Xo, X, and Xl be three Banacb spaces such that (2.1 )

Xo CXCXl,

the injection of X into Xl being continuous, and: the injection of X 0 into X is compact.

(2.2)

Then for each 11 > 0, there exists some constant c Tl depending on 11 (and on the spaces X o, X, Xl) such that:

IIv IIx

:eo;; 11llv IIxo’ + c Tl IIv II x 1,

Vv E X o.

(2.3)

Proof. The proof is by contradiction. Saying that (2.3) is not true amounts to saying that there exists some 11 > 0 such that for each c in lR,

IIv IIx

~

11 llv II xo + cIIv IIx 1,

for at least one v. Taking c = m, we obtain a sequence of elements V m satisfying

IIvm II x

~

11 IIv m II x 0 + mIIv m II x l , V m.

We consider then the normalized sequence Wm

= II Vm IIXo

'

which satisfies

IIwm IIx

~

11 + m Ilwm IIx 1, V m.

Since IIw m II xo = I, the sequence that

IIw m IIx 1 -+ 0, as m -+ 00.

Wm

(2.4)

is bounded in X and (2.4) shows (2.5)

Compactness theorems

Ch. III, § 2

271

In addition, by (2.2), the sequence wm is relatively compact in X; hence we can extract from W m a subsequence w~ strongly convergent in X. From (2.5) the limit of w~ must be 0, but this contradicts (2.4) as:

IIx ~

IIw~

f/

> 0,V IJ..

2.2. A Compactness Theorem in Banach Spaces

Let Xo, X, Xl, be three Banach spaces such that (2.6)

Xo CXCXI,

where the injections are continuous and: Xi

is reflexive,

i

the injection Xo

= 0, -+

I,

(2.7)

X is compact.

(2.8)

Let T> 0 be a fixed finite number, and let ao, al , be two finite numbers such that ai > I, i = 0, I. We consider the space :v=,~(0,

j’ =

1.

T; ao, al; X o, Xl) E L" (0, T; X 0), .’ = :; E La, (0, T,' Xl)

!

(2.9) (2.10)

The space Y is provided with the norm

IIvlLw = IIvllL ~

(O,T;Xo)

+ IIv'lIL

Ci

1(O,T;X 1)

'

(2.11)

which makes it a Banach space. It is evident that :VC LO
in :v weakly, as IJ.

-+

00,

(2.12)

The evolution Navie,..Stokes eqU/ltions

272

Ch. III, §2

which means Ujl ..:,.u

u~

in

..:,.u'

Lao (0,

T; Xo) weakly

in Lal(O, T; Xf}weak1y.

(2.13)

It suffices to prove that vjl

=ujl

- U

converges to 0 in Lao(O, T; X) strongly.

(2.14)

(ii) The theorem will be proved if we show that vjl ..:,.

0 in LaO(O, T; Xf} strongly.

(2.15)

In fact, due to Lemma 2.1, we have IlvjlIlLaO(O, T; X)

and since the sequence IIvjlIlLaO(O, T; X)

~

71 l1 v jl "L a o(o, T; Xo) + C,., IIvjlIILao(o, T; Xl)

vjl

is bounded in I!!I:

~ C71 + C,., IIvjlIlLaO(O, T; Xl>'

(2.16)

Passing to the limit in (2.16) we see by (2.15) that

!~oo

~C71;

IIvjlIlLaO(O,T;X)

(2.17)

since 71 > 0 is arbitrarily small in Lemma 2.1, this upper limit is 0 and thus (2.14) is proved. (iii) To prove (2.15) we observe that (2.18)

T] ;Xf},

~C
with a continuous injection; the inclusion (2.18) results from Lemma 1.1, and the continuity of the injection is very easy to check. We infer from this, the majoration IIvit)lIxI ~ c.

V t E [0, T], V IJ..

(2.19)

According to Lebesgue’s Theorem, (2.15) is now proved if we show that, for almost every t in [0, T] , vjl (t) ..:,.

0 in Xl strongly, as IJ. ..:,. 00.

We shall prove (2.20) for t other t, We write vv(O)

=v; (I) -

= 0; the proof would be similar

J v~ t

0

(T)cIT

(2.20) for any

Compactness theorems

Ch. III, §2

273

and by integration 1

v (0) =S

II.

Hence

(2.21 )

with (2.22) For a given

> 0, we choose s so that

,.;f Ilv~(t)lIx, s

Ilb.llx

0

dt’;~.

Then, for this fixed s, we observe that, as J.l. -+- 00, 0Il. -+- 0 in X o weakly and thus in Xl strongly; for J.l. sufficiently large 1I0ll.lIx1 ~:2’

and (2.20), for t = 0, follows. 2.3. A Compactness Theorem Involving Fractional Derivatives

The next compactness theorem is in the frame of Hilbert spaces and is based on the notion of fractional derivatives of a function. Let us assume that X 0, X, Xl, are Hilbert spaces with Xo CXCX1,

(2.23)

the injections being continuous and the injection of X 0 into X is compact. If v is a function from lR into Xl, we denote by form

f~

(2.24)

vits Fourier trans›

+00

t(T)

=

ee

e- ,"’" v(tJdt.

(2.25)

TheevolutionNavier-StOkeB equations

274

Ch. III, § 2

The derivative in t of order ’Y of v is the inverse Fourier transform of

(2i7l'T)'Yv or

.----....... ) DjV(T) = (2i7l'T)'Y " v (T).(l

For given ’Y> 0

(2),

(2.26)

we define the space

;J{’’Y(K Xo, XI>

= {v EL2(6t; Xo), Div EL2(6t; Xl)}'

(2.27)

This is a Hilbert space for the norm.

= {lIvlli 2(~. , x 0 ) + IIITI’Y V1I12(~.

IIvll ;f( ’Y (~,XO,Xl)

X

1

)}~.

We associate with any set K C 6t, the subspace JCk of;J{"Y defined as the set of functions U in;J{"Y with support contained in K: ;J{’1(6t; Xo, Xl)

= {u E ;J{’’Y(K

Xo, XI>, support U C K}.

(2.28)

The compactness theorem may now be stated: Theorem 2.2. Let us assume that Xo, X, Xl are Hilbert spaces which satisfy (2.23) and (2.24). Then for any bounded set K and any ’Y> 0, the injection of ;J{’l (6t; Xo. Xl) into L2(6t, X) is compact.

Proof. (i) Let ’Y and K be fixed, and let Urn be a bounded sequence in ;J{’l (6t; Xo, Xl). We must show that Urn contains a subsequence strongly convergent in L2(6t; X). Since ;J{"Y(6t; X o, Xl) is a Hilbert space, the sequence UIJ contains a subsequence weakly convergent in this space to some element u. It is clear that U must also belong to ;J{’l; therefore, setting the sequence vIJ appears as a bounded sequence of;J{’1 (K Xo, Xl), which converges weakly to 0 in ;J{’’Y; this means

0 in L2(K Xo ) weakly

(2.29)

ITI’Y vIJ -+ 0 in L2(K Xl) weakly.

(2.30)

VIJ

-+

The theorem is proved if we show that uIJ converges strongly to U in

L2(6t; X), which is the same as

(l)The defmition (2.26) is consistent with the usual defmition for (2)In the applications, 0 < ’Y

" I in general.

’Y an integer.

275

Compactness theorems

ell. Ill, *2

0 in L2 (61.; X) strongly.

Vj.L ~

(2.31)

(ii) The second point of the proof is to show that (2.31) is proved if we prove that

a in L2(61.; Xl) strongly.

vj.L ~

(2.32)

Due to Lemma 2.1,

+ c l1l1vj.LII L2(iR;X1 )

IIvj.LIIL2(iR;x) ~71l1vj.lIIL2(iR;xo)

(2.33)

and since vj.l is bounded in L 2 (61.; Xo), Ilvj.L: II L2(iR ; X) ~ c71

+ cl1llvj.lIlL2(iR; Xl)'

If we assume (2.32), then letting Po

~

00

(2.34)

in (2.34), we obtain

lim IIvj.L IIL 2 (iR; X) ~ C71·

j.L-+OO

Since 71 is arbitrarily small in Lemma 2.1, this upper limit must be and (2.31) follows. (iii) Finally let us prove (2.32). According to the Parse val theorem,

I, =

r~

A,,(t)H}, dt =

r~

IIv,(T)II},

d~,

a

(2.35)

where Vj.L denotes the Fourier transform ofvj.L' We must show that Ij.L ~

a as Po ~

(2.36)

00.

For this, we write

i,

=f

1 ~j.l(T)I }l ITIE;;M

dr +

J

ITI>M

since vj.L is bounded in Jf'Y. For a given e > 0, we choose M such that

(l

+ ITI 2’Y)lIvj.l(T)II}1

276

The evolutionNavier-Stokes equations

Ch.JII, §2

Hence

and (2.36) is proved if we show that, for this fixed M,

i,

=

J

Ilv/l(T)lIkl dr -+ 0, as

[J -+

(2.37)

00.

ITI <:M

This is proved via the Lebesgue theorem. If X denotes the characteris› tic function of K, then v/lX = v/l and

~.(T)

L~ + ..

=

e- 2I, " X(t)v. (I)dt.

Thus IIV/l(T)lIx

l ""

Ilv/lIIL2(~;

IIv/l(T)lI x

l ""

Const.

Xl)

lIe-2ilTtT xIIL2(~),

(2.38)

On the other hand for each a in Xo, and each fixed

«~.

(T),

.»xo

=J~~

T,

«v. (I), e- 21", X(t).»xo dt ,

which goes to 0 as [J -+ co because of (2.29). The sequence V/l (T) con› verges to 0 weakly in Xo and therefore strongly in X and Xl . With this last remark and (2.38), the Lebesgue theorem implies (2.37). 0 Using the methods of the last theorem, we can prove another com› pactness theorem similar to Theorem 2.1. Nevertheless, this theorem is not contained in, nor itself contains, Theorem 2.2.

Theorem 2.3. Under the hypotheses (2.23) and (2.24), the injection of T; 2, 1; Xo, Xl )(1) into L 2 (0, T; X) is compact.

~(O,

(1) For the definition of this space see (2.9)-(2.10).

Ch. III, §2

277

Compactness theorems

Proof. Let U m be a bounded sequence in the space 0. Because of Lemma 1.1, each function U m is, after modification on a set of measure 0, continuous from [0, T] into Xl; more precisely the injection of
um the function defined

°

°

u u

-dtd um = 1m + U m (0)5 0 -

Um

(T)5 T ,

where 50 and 5T are the Dirac distributions at gm

= u~ = the derivative

(2.39)

°

and T, and (2.40)

of Um on [0, T].

After a Fourier transformation, (2.39) gives 2i7rrU m (T)

u

i

=1m (T) + Um (0) -

Um (1) exp(-2i7rTT),

u

(2.41 )

where m and m denote the Fourier transforms of 1m and m respectively. Since the functions gm remain bounded in L 1(0, T; Xl ), the func› tions 1m remain bounded in L I (lR; Xl ) and the functions gm are bounded in L (lR; Xl): GO

IIgm (T) lI x l

Vm,V TElR.

~Const.,

(2.42)

We have noted that the injection of:y into<6’([O, T] ; Xl) is continuous; thus IIum (0) II Xl ~ Const., IIU m (T)II x l ~ Const.,

and (2.41) shows us that ITI2

IIum (T)"~l

For a fixed ’Y

~ c,

v m. VT E lR.

< 1/2, we observe

that

~ 1 + T2 ITI 2"Y ....,co<’Y) 1 + ITI2(l-"Y) , VTElR. Therefore

(2.43)

278

Theevolution Navie,..Stokes equations

+~

.; C,

~ _ ~ 1 + IT1:C,-') + co(-y)

t

Ch. III, §3

+~

ee

Ilum (T) IIi, dr

by (2.43). Since ’Y < 1/2, the integral

is convergent; on the other hand, by the Parseval equality, we see that

C~

lIum (T)"i, dr =

ell..

(t)lIi, dt,

and these integrals are bounded. We conclude that +~

~ _~ ITI" IIUm (T)Ii, dT’; c" where

C2 depends on ’Y. It is now clear that the sequence

and the proof is achieved.

0

Um

(2.44)

is bounded in ’J(Y (R XO. Xl)

Remark 2.1. Assuming only that Xl is a Hilbert space, Xo. X being Banach spaces, it can be proved in a similar way that the injection of ;V(O, T; Q'o, 1; Xo. Xl),

into Lao (0, T; X) is compact for any finite number

aa > 1. 0

§3. Existence and Uniqueness Theorems (n <; 4) This section is concerned with existence and uniqueness theorems for weak solutions of the full Navier-Stokes equations (n <; 4). In Section

Existence and uniqueness theorems (n <; 4)

Ch. III, §3

279

3.1 we give the variational formulation of these equations, following J. Leray, and state an existence theorem for such solutions for n ~ 4. The proof of this theorem, due to J. L. Lions, is given in Section 3.2. It is based on the construction of an approximate solution by the Galerkin method; then a passage to the limit using, in particular, an a priori esti› mate on a fractional derivative in time of the approximate solution, and a compactness theorem contained in Section 2. An alternate proof based on a semi-discretization in time and valid in all dimensions is dis› cussed in Section 4. In Section 3.3 we develop the uniqueness theorem of weak solutions (n = 2). In the three-dimensional case there is a gap between the class of functions where existence is known, and the smaller classes where uniqueness is proved; an example of such a uniqueness theorem is developed in Section 3.4 (n = 3). In Section 3.5 we show in the two› dimensional case the existence of more regular solutions, assuming more regularity on the data; a similar result holds in the three-dimensional case for local solutions, that is solutions which are defined on some "small" interval of time, assuming that the data is sufficiently small. 3.1. An Existence Theorem in eRn (n

~

4)

Notations are as above, in particular that recalled at the beginning of Section 1.1; .0 is an open Lipschitz set which for simplicity, we suppose bounded; the unbounded case is discussed in Remarks 3.1 and 3.2. We recall (1) that since the dimension is less than or equal to 4, one can define on Hfi (.0), and in particular on V, a trilinear continuous form b by setting b(u, v, w)

=

.i f

I,J= 1

n

ui(DiVj)wjdx.

(3.1)

Ifu E V, then b(u, v, v)=O, VvEHfi(n).

(3.2)

For u, v in V, we denote by B(u, v) the element of V’ defined by (B(u, v), w> = b(u, v, w), V w E V,

and we set (l)c!. Section 1.1, Chapter II.

(3.3)

en, Ill,

The evolution Navier-StokefJ equationfJ

280

B(u)

= Btu, U) E V', VuE V.

§3

(3.4)

In its classical formulation, the initial boundary value problem of the full Navier-Stokes equations is the following: To find a vector function

n

u:

X [0, T]

~lRn

and a scalar function

n X [0, T]

p:

~

lR,

such that

au -

at

n

~

- vLlu + k

divu

=

°

°

i= 1

UiDiU

+ grad p

=/ in Q = n X (0, T),

(3.6)

in Q,

an X (0, T), u(x, 0) = uo(x), in n.

u

=

(3.5)

(3.7)

on

(3.8)

As before, the functions/and Uo are given, defined on

n respectively.

n X [0,

T] and

Let us assume that u and p are classical solutions of (3.5)-(3.8), say u E~2 «(2), p Ecffl «2). Obviously u E L 2 (0, T; V), and if v is an element of r; one can check easily that d dt (u. v) + v«u. v» + btu, u, v)

=
(3.9)

By continuity, equation (3.9) will hold for each v E V. This suggests the following weak formulation of the problem (3.5)› (3.9) (cf. J. Leray [1] , [2], [3]): Problem 3.1. For/and u given with

/E L 2(0, T; V’)

(3.10)

Uo EH,

(3.11 )

to find u satisfying (3.12)

and

281

Existence and uniqueness theorems (n " 4)

Ch. Ill, §3

d - (U, v)

dt

u(o)

+ V((U. V» + b(u, u, V) = (f,

V), V V E V

(3.13)

= UO’

(3.14)

If u merely belongs to L 2 (0, T; V), the condition (3.14) need not make sense. But if u belongs to L 2 (0, T; V) and satisfies (3.13), then we will show as in the linear case (using Lemma 1.1) that u is almost everywhere equal to some continuous function, so that (3.14) is mean› ingful. Before showing this, we recall that we are considering the case n ~ 4; we will modify slightly the preceding formulation in higher dimensions (see Section 4.1 ).

Lemma 3.1. We assume that the dimension of the space is n that u belongs to L 2 (0, T; V). Then the function Bu defined by (Bu(t), v) belongs to

L 1 (0,

= b(u(t), u(t), v), V V E

~

4 and

V, a.e. in t E [0, T],

T; V’).

Proof. For almost all t, Bu(t) is an element of V’, and the measur› ability of the function t E [0, T] ~ Bu(t) E V'

is easy to check. Moreover, since b is trilinear continuous on V,

IIBwl1 Vi

-c cIIw11 2 , V w E

so that T (

J0

f

T

IIBu(t)ll y' dt .;; c

and the lemma is proved.

(3.15)

V,

Ilu(t)II' dt

< -,

0

0

Now if u satisfies (3.12)-(3.13), then according to (1.6), (1.8), and the above lemma, one can write (3.13) as d - (u, v) = (I - vAu - Bu, v), Y v E V dt

Since Au belongs to L 2 (0, T; V’), as in the linear case, the function

The evolution Navie,..Sto!ces equadons

282

I

Ch. III, §3

1- vAu - Bu belongs to L 1 (0, T; V’). Lemma 1.1 implies then that u’ ELI(O, T; V’) u’

=1- vAu

- Bu,

(3.16)

and that u is almost everywhere equal to a function continuous from [0, T] into V'. This remark makes (3.14) meaningful. An alternate formulation of the problem (3.12)-(3.14) is: Problem 3.2. Given / and Uo satisfying (3.10)-(3.11), to find u satisfying u E L 2(0, T; V), u’ ELI (0, T; V’), u’ + vAu + Bu

=Ion (0, T),

(3.17)

(3.18)

u(O)=uo.

(3.19)

We showed that any solution of Problem 3.1 is a solution of Problem 3.2; the converse is very easily checked and these problems are thus equivalent. The existence of solutions of these problems is ensured by the follow› ing theorem. Theorem 3.1. The dimension is n :E;;; 4. Let there be given I and Uo which satisfy (3.10)-(3.11). Then there exists at least one function u which satisfies (3.17)-(3.19). Moreover,

u E L (0, T; H)

(3.20)

GO

and u is weakly continuous from [0, T] into H. (1)

The proof of the existence of a u satisfying (3.20) is developed in Section 3.2; the continuity result is a direct consequence of (3.20), the continuity of u in V’, and of Lemma 104. Remark 3.1. (i) Theorem 3.1 also holds if we assume that

1= 11 + h;

/1 E L 2(0, T; V’),

h

ELI (0, T; H).

For the corresponding modifications of the proof of the theorem, the reader is referred to Section 1.5. (ii) Theorem 3.1 is also valid if n is unbounded; for details, see Remark 3.2. 0 (1)i.e., V

II

EH, t -+

(u(t), II) is a continuous scalar function.

Existence and uniqueness theorems

Ch. III, §3

(n " 4)

283

3.2. Proof of Theorem 3.1 (i) We apply the Galerkin procedure as in the linear case. Since V is

separable and ’i""is dense in V, there exists a sequence wI’ ..., W m ' ••. of elements of ’i"", which is free and total in V (l). For each m we de› fine an approximate solution um of (3.13) as follows: m

um

=

and

2:

i= I

(U:n (t), Wj)

U m (0)

gim (OWi

(3.21 )

+ v«u m (t), Wj» + b(u m (t), Um (t), Wj) =
(3.22)

= uO m ’

(3.23)

where uom is the orthogonal projection in H of Uo onto the space (2) spanne d b y WI , , Wm • The equations (3.22) form a nonlinear differential system for the functions gIm’ ... , gmm : m

2:

j= I

m

(wi’ Wj)g;m (t)

2:

+v

«Wi’ Wj»gim (t)

i= I

m

+

2:

t. Q= I

=
(3.24)

Inverting the nonsingular matrix with elements (wi’ Wj), 1 ~ i, j we can write the differential equations in the usual form m

g;m (t)

+

2:

j= I

~

m,

m

Ol;jgjm (t)

+

L

j. k= I

Ol;jkgjm (t)gkm (t)

m

=

L

j= I

(3ij
(3.25)

(l )The Wj are chosen in ’Y for simplicity. With some technical modifications we could take the Wjin V.

(2)We could take for uom any element of that space such that Uom

--+ uo.

in H, as m --+ 00.

TheevolutionNavier-Stokes equations

284

cs. Ill,

§3

where Ol;j’ Ol;jk’ (3ij E tR. The condition (3.23) is equivalent to the m scalar initial conditions Kim

(0) = the i th component of uO m •

(3.26)

Thenonlinear differential system (3.25) with the initial condition (3.26) has a maximal solution defined on some interval [0, tm ]. If t m < T, then IU m (t)1 must tend to +00 as t ~ t m ; the a priori estimates we shall prove later show that this does not happen and therefore t m = T. (ii) The first a priori estimates are obtained as in the linear case. We multiply (3.22) by Kjm (t) and add these equations for j = I, ..., m. Taking (3.2) into account, we get

=
(U;" (t), u m (t)) + VIIl1 m (t) 11 2

(3.27)

Then we write d

- 1u m (t)1 2 + 2v lIum (t) 11 2

dt

= 2
~ 2I1f(t) " v' lIum (t) II I ~ v lIum (r) 11 2 + -lIf(t) II~,

v

so that d

I

2 lu +vllu m(t)1I2 ~-lIf(t)1I2v" dt m(t)1 v

-

(3.28)

Integrating (3.28) from 0 to s we obtain, in particular,

Hence Sup

sE [0, T)

IU m

(s)1 2 ~ luol2 + -I

v

JT IIf(t)II~,dt o

(3.29)

Existence and uniqueness theorems (n

Ch. III, §3

285

<; 4)

which implies that The sequence u m remains in a bounded set of L (0, T; H) (3.30) We then integrate (3.28) from a to T to get 00

Ium (T)1 2 + v

J(T lIu m(t) 11 o

2

1JT Ilf(t) 1I}.dt

dt ~ IUOm 12 +;-

0 T

~ Iuol

2

( 12 +;;1 J 0 IIf(t) I v•dt .

This estimate enables us to say that The sequence um remains in a bounded set of L 2 (0, T; V). (3.31 ) (iii) Let

um denote the function from

um on [0, T] and to

tR into V, which is equal to

a on the complement of this interval.

The Fourier transform of um is denoted by am . In addition to the previous inequalities, which are similar to the esti› mates in the linear case, we want to show that

J

+ OO

Irl2'r Itim (r)1 2 dt ~ Const., for some

’Y>

O.

(3.32)

Along with (3.31), this will imply that

um belongs

to a bounded set of’JC’Y (lR; V, H)

(3.33)

and will enable us to apply the compactness result of Theorem 2.2. In order to prove (3.32) we observe that (3.22) can be written (1) d dt (u m , Wj)

-

=
where 6 0 , 6T are Dirac distributions at

= 1, ... , m

a and T and

=f- vAu m -Bu m f m =f m on [0, T], 0 outside this interval.

(3.34)

~ Jm

(1)Compare this with the proof of Theorem 2.3.

(3.35)

286

The evolution Navier-Stokes equations

Ch. III. §3

By the Fourier transform, (3.34) gives 2i1rr(D ml• wI)

- (U m (T), wI) exp(-2i1rTr),

t:

u

=
and denoting the Fourier transforms of um and 1m respectively. We multiply (3.35) by Ktm (r) (= Fourier transform of i lm ) and add the resulting equations for j = 1, ..., m; we get: m

2i1rrlD m (r)1 2

=
+ (uo m , Dm (r))

- (u m (T), Dm (r)) exp(-2i1rTr).

(3.37)

t

Because of inequality (3.15),

~o

T

T

Ifm (I) II v•dl <;;

(1If(1) II v’ + vllu m (I) II + C lIum (I) R’)dl,

and this remains bounded according to (3.31). Therefore Sup Itfm (r)1I v' ~ Const., V m.

TE iR

Due to (3.29),

IU m (0)1 ~ const., IU m (T)I

~

const.,

and we deduce from (3.37) that IrIIDm(r)1 2 ~c2I1Dm(r)ll+c3IDm(r)1

or IrliDm (r)1 2 ~c4l1Dm

(3.38)

(r)11.

For ’Y fixed, ’Y < 1/4, we observe that Irl

2'Y

~C5(’Y)

1 + lr] 1 + IrI 1- 2'Y ,V rE~.

Thus

f 2'Y IDm(r)1 2dr~c5(’Y) J_oolrl +~

~

J +~

1 + Irl ~ 2 _ool+lrI1-2'Ylum(r)1 dr

(by (3.38))

Existence and uniqueness theorems (n 0;; 4)

Ch. III, §3

287

Because of the Parseval equality and (3.31), the last integral is bounded as m -+ 00; thus (3.32) will be proved if we show that

f

+00

IIDrn (T)II

(3.39)

-1-+---’I’:":"’T,-’-l--"""2"Y= dr ~ const.

-

00

By the Schwarz inequality and the Parseval equality we can bound these integrals by

(1 +

00

-

(l

d~

+ ITI -

2"Y 2) 1/2( )

1T

00

lIu (t)1I2 rn

dt)

1/2,

0

which is finite since 'Y < 1/4, and bounded as m -+ 00 by (3.31). The proof of (3.32) and (3.33) is achieved. (iv) The estimates (3.30) and (3.31) enable us to assert the existence of an element U E L 2(0, T; V) n Lao (0, T; H) and a sub-sequence urn' such that urn' -+ U in L 2(0, T; V) weakly, and in

L (0, T; H) weak-star, as m’ -+ 00. 00

(3.40)

Due to (3.33) and Theorem 2.2, we also have Urn'

-+ U

in L 2 (0, T; H) strongly.

(3.41)

The convergence results (3.40) and (3.41) enable us to pass to the limit. We proceed essentially as in the linear case. Let I/J be a continuously differentiable function on [0, T] with I/J(T) = 0. We multiply (3.22) by I/J(t), and then integrate by parts. This leads to the equation

-L T

f

(urn (f), ,J/(t)Wf)df + v

T 0 ((Urn (f), wf1/l(f)))df

288

Theevolution Navier-Stokes equations

Io

Ch. iII, §3

T

+

([(f), wi!/J(f)ldf.

(3.42)

Passing to the limit with the sequence m’ is easy for the linear terms; for the nonlinear term we apply the next lemma, Lemma 3.2. In the limit we find that the equation

-I:

(u(f), v!/J'(t))df

+[

+v

J:

h(u(f), u(l), v!/J(t»)cIf

«u(f), vl/J(t»)df

=(uo; v)!/J(O)

+ [ (f(t), v!/J(f)ldf,

(3.43)

holds for v = wl. w2’ ... ; by linearity this equation holds for v = any finite linear combination of the wi' and by a continuity argument (3.43) is still true for any v E V. Now writing, in particular, (3.43) with I/J = ep E91((O, D), we see that u satisfies (3.13) in the distribution sense. Finally, it remains to prove that u satisfies (3.14). For this we multi› ply (3.13) by I/J, and integrate. After integrating the first term by parts, we get

I

T

-

0 (u(f), v.p(I))dt + v

J

J T

0 «u(f), v!/J(t»)df

T

+

0 h(u( I),

u(f), v !/J(f»)cIf = (u(O), v)!/J(O) +

Ch. 1II, § 3

+

J:

Existence and uniqueness theorems (n

0;;; 4)

289

(3.44)

(((I), v1/<(I)}dl.

By comparison with (3.43),

= O. IjI with 1jI(0) = 1; thus uo. v) = 0, V v E V,

(u(O) - uo. v)IjI(O)

We can choose (u(O) -

and (3.14) follows. The proof the Theorem 3.1 will be complete once we prove the following lemma.

Lemma 3.2. Iful-l converges to u in L 2(0, T; V) weakly and L 2(0, T; H) strongly, then for any vector function w with components in ~1 (Q), btu. (I), »; (I), w(t))dl -+

[

r:

Proof. We write b(u., u.' w)dt = -

J:

J:

b(u(t), utt), W(I))dI.

b(u,., w, u.) =

These integrals converge to n

2:

t.i» 1

-faT

btu, w, u)dt

and the lemma is proved.

= IT0 btu, u, w)dt, 0

(3.45)

290

The evolution Navier-Stokes equations

Ch.III, §3

Remark 3.2. (i) The unbounded domain. When 0 is unbounded, we prove (3.30) and (3.31) as we did in Section 1.5 for the linear case. Then (3.32) and (3.33) follow as before. The main difference lies in the fact that the injection of V into H is no longer compact. Nevertheless we can extract a sub-sequence "rn’ which satisfies (3.40). Then, for any ball (!) included in 0, the injection of H l «(!)) into L 2«(!)) is compact and (3.33) shows that: L 2 ( {) ) ), V{).

urn I~ belongs to a bounded set ofjf’Y(~;Hl«(!)),

(3.46)

Theorem 2.2 then implies that "rn,l~ -+ "I~ in L2(0, T; L2(())) strongly, V{). which means

"rn’-+" in L2(0, T; L1oc(0)) strongly. In particular, for a fixed j, "rn'lo'-+ "10' in £2(0, T; L 2(O’)) strongly

where 0’ denotes the support of wI' and this suffices to pass to the limit in (3.42). (li) Energy inequality By integration of (3.27) we see that 2

f: f

IUm (t)1 + 2.

Ilum «)l 2 cb = IUOm 12

t

+2

0 (f(s), u m (.»cb,

We multiply this equality by cp(t), where cp E9 (] 0, and integrate in t:

f\

2

lum (t)1 +2.

J:

lum « ) 1I2 cb

= [ Iluom 12 + 2

I~(t)dt

f:(/(.),

TD, cp(t) ;> 0,

um «)) ds

I~(t)

dr,

Ch. III, §3

Existence and uniqueness theorems (n <; 4)

291

Using (3.40) we now pass to the lower limit in this relation and obtain

f: \IU(t)I' J: + 2"

Ilu(s) II' ds

I~(t)

dt

.; [ IIU I' + 2 J: (j(s), u(s)}ds O

for all ¢> E

~ (]

I

W) dt,

0, TD, ¢> ~ O. This amounts to saying that

lu(t)I' + 2"

J:

Burs)!' ds’; luo I' + 2

J:

(1(,,), u(s)} ds,

(3.47)

for almost every t E [0, T]. This is an energy inequality satisfied by the function u given by Theorem 3.1. We shall show later that if n = 2, the corresponding equality holds (see Theorem 3.2 and Lemma 2.1). We do not know if the corresponding equality is satisfied in general (cf. Theorem 3.9). 3.3. Regularity and Uniqueness (n

= 2)

When the dimension of the space is n = 2, the solution of (3.17)› (3.19) whose existence is ensured by Theorem 3.1 satisfies some further regularity property and is actually unique. The proof of these results is based on the following lemmas.

Lemma 3.3. If n = 2,/or any open set 0" IIvll

4

L ({l)

~21/411vlllq)

L ({l

IIgradvIIL1~:

\{l

)' VvEHA(o,).

(3.48)

Proof. It suffices to prove (3.48) for v ED (0,). For such a v, we write

and therefore v 2(x ) =s:;;; 2vl (x2)'

(3.49)

292

The evolution Navier-Stokes equations

o, III, §3

where

"(X,)=f+-I.(~,.X,)"D,.(~,.

x,)ld~I’

(3.50)

-00

Similarly v 2(x) E;;; 2v2(x 1 ),

where " (x,)

f

=

+00

(3.51)

I.(x,. h)1 !D,.(x,. ~2)1d~,

(3.52)

-00

and thus

E;;;411v1lL2 2 (iR 2 ) IID1vllL 2 (iR 2)IID2 v 11L 2 (iR 2 ) E;;; 211vIII2 (iR 2) IIgrad vllL2(iR 2).

Lemma 3.4. If n

= 2,

Ib(u, v, w)1 E;;; 2 1/21ul1/211u 1I1/2I1vlllwI1/2I1wIl1/2, V u, v, w EHa(f’n.

(3.53) If u belongs to L 2 (0, T; V)

L 2 (0,

T; V’) and

n L (0, 00

T; H), then Bu belongs to

12

IIBu "L 2 (0, T; V’) E;;; 2 / 1ul L (0, T; H) Ilu"L 2 (0 , T; V)• 00

(3.54)

Proof. By repeated application of the Schwarz and Holder inequalities we find: Ib(u, v, w)] E;;;

o~

2

1./=1

J(IUj(Djv/o)w/oldx n

Existence and uniqueness theorem, (n < 4)

Ch. III, §3

293

2

L

~ ~

;,j=1

2

(

L i,j=1

(

L j= 1

2

IIu; II

L

4(

n

)

IID;vJ.11

2 L (n)

)

IID;vj 11 2 2

1/2 (

L (n)

IIwJ.II 2 4

IIwJ. II 4 L (n) 2

L ;=1

) 1/2

L (n)

Due to (3.48),

,~

Ilu,

Ii

’(n) . .

21/2

~

1

(IIU,IIL' (n) llgrad uillL•

(n))

~ 21/ 2 lui IIu II.

With a similar inequality for w, we finally get (3.53). If u, v, W belong to V, the relation b(u, v, w)

= -b(u, w, v)

gives another estimate of b: Ib(u, v, wi ~ 2 1/ 2 lu1 1/ 2 IIu 11 1/2 Ivl 1/2 IIvill/2 IIwII, V u, v, wE V

(3.55)

In particular, Ib(u, u, v)] ~ 2 1/2 lui IIuli Ilvll, V u, v E V,

(3.56)

and hence IIBu " v ’ ~ 2 1/2 lui lIu II, VuE V.

(3.57)

If now u E L 2 (0, T; V) n L (0, T; H), then Bu(t) belongs to V’ for almost every t and the estimate 00

IIBu(t) II v’ ~ 2 1/ 2 Iu(t)1 IIu(t) II

shows that Bu belongs to

L 2(0,

T; V') and implies (3.54).

(3.58)

D

We can now state and prove the main result (cf.. J. L. Lions and G. Prodi [I]).

Theevolution Navier-Stoke, equation,

294

Ch. III, §3

Theorem 3.2. In the two-dimensional case, the solution u of Problems 3.1-3.2 given by Theorem 3.1 is unique. Moreover u is almost every› where equal to a function continuous from [0, T] into Hand u(t) -+ uo, in H, as t -+ 0.

(3.59)

Proof. (i) We first prove the result of regularity. According to (3.18) and Lemma 3.4, u'

=f -

vAu - Bu,

and since each term in the right-hand side of this equation belongs to L 2(0, T; V'), u ’ also belongs to L 2(0, T; V'); this remark improves (3.17): u’ E L 2(0, T; V'). (3.60) This improvement of (3.17) enables us to apply Lemma 1.2, which states exactly that u is almost everywhere equal to a function con› tinuous from [0, T] into H. Thus

(3.61)

u E<6’([O, T] ; If)

and (3.59) follows easily. We also recall that Lemma 1.2 asserts that for any function u in 2(0, L T; V) which satisfies (3.60), the equation below holds:

~ Iu(t)1 2 = 2(u'(t), u(t)}.

dt

(3.62)

This result will be used in the following proof of uniqueness which we will start now. (ii) Let us assume that u1 and u2 are two solutions of (3.17)› (3.19), and let u = u1 - u2' As shown before, u1' U2' and thus u, satisfy (3.60). The difference u = u1 - u2 satisfies u'

+ vAu = -Bul + BU2

u(O)

=

°

(3.63) (3.64)

We take a.e. in t the scalar product of (3.63) with u(t) in the duality between V and V’. Using (3.62), we get d

- lu(t)1 2 + 2vllu(t) 11 2 dt

= 2b(u2(t), u2(t), utt)

- 2b(u1 (t), u1 (t), u(t».

(3.65)

Existence anduniqueness theorems (n

Ch. III, §3

295

<; 4)

Because of (3.2) the right-hand side of this equality is equal to -2b(u(t), u2 (t), u(t».

With (3.53) we can majorize this expression by 1 3 2 / 21 u(t)1 lIu(t) II lIu2(t) II ~ 2v lIu(t) 11 2 + -lu(t)1 2 IIu2 (t) 11 2 . V

Putting this in (3.65) we find d 1 -d lu(t)1 2 ~ -lu(t)1 2 lIu2(t)112.

v

t

:, I ~ J:

Since the function t ~ lIu2(t) 11 2 is integrable, this shows that

exp (-

Ilu,(s)II'ds) • IU(l)I'

<;;0.

Integrating and applying (3.64), we find lu(t)1 2 :E;;; 0, V t E [0, T].

Thus Ul

= u2'

and the solution is unique. Remark 3.3. As a consequence of (3.48) the (unique) solution of the Navier-Stokes equations satisfies (3.66) Remark 3.4. Theorem 3.2 covers both the bounded and unbounded cases; there are no differences in the proofs. 3.4. On Regularity and Uniqueness (n = 3)

The results of Section 3.3 cannot be extended to higher dimensions due to the lack of information concerning the regularity of the weak solutions given by Theorem 3.1. Nevertheless, we shall prove some further regularity properties of a solution, which are weaker than those of the two-dimensional case. We then give a uniqueness theorem in a class of functions where the existence is not known; this result shows, however, how the information concerning

296

The evolutionNavier-Stokes equations

Ch. Ill, §3

the regularity of weak solutions are related to uniqueness. The result similar to Lemma 3.3 is lemma 3.5.

If n = 3, for any open set Q:

1/4 3/4 IIvllL4(,{l) ~ 21/2 1IvI1L2(,{l) llgrad v11 L2(,{l) V v EH1(Q o ). ,

(3.67)

Proof. We only have to prove (3.67) for v E!’j(Q). For such a v, by application of (3.48), we write

f.:’(Xldx ~ 2r~

I(

(f.,i

f.:’dx dx, ). l

(3.68) But

and hence

sup! x3

iR

2

v2dxldx2~2J

iR

3

Iv1 D3)1ldx~21 v1l2

~I D3v1l2

L (iR-j

3'

L (iR )

With this inequality we deduce from (3.68) that

f

dl 3

v4(x)dx

~41 vl

2

3

L (iR )

IID 3vllL 2 (iR) 3

(±

i= 1

IIDiVIIL22(D3») '"

en. III. § 3

Existence and uniqueness theorems (n <; 4)

~41 vlI

2

3

L (6{)

(

±

i= 1

297

3/2

IIDivllL 2 (6{ 3 ))

and (3.67) follows. Theorem 3.3. If n = 3, the solution u of (3.17)-(3.19) given by Theorem 3.1 satisfies u E L 8/3 (0, T; L4 (n))

(3.69)

u' E L 4/3 (0, T; V’).

(3.70)

Proof. For almost every t, according to (3.67),

Ilu(t)II L 4 (0) ~ Co Iu(t)1 1/ 4 lIu(t) 11 3/ 4 •

(3.71 )

The function on the right-hand side belongs to L 8/3 (0, T), and thus also the function on the left-hand side. Using the Holder inequality, we derived in Chapter II the inequality Ib(u, u, v)1

= Ib(u, v, u)1 ~

Clllulli4(0)IIvll, V u, v E V(l).

(3.72)

Therefore, if u E L 2 (0, T; V) n L (0, T; H), Bu belongs to L 4/3 (0, T; V') since (3.73) IIBu(t)II v' ~ cl II u(t)IIi4(0) 00

IIBu(t) II v' ~ c21 u(t)1 1/ 2 Ilu(t)11 3/ 2 , a.e.

D

(3.74)

In the two-dimensional case we established that any solution of (3.17)-(3.19) satisfies (3.60) and (3.66) and this was the property which essentially enabled us to prove uniqueness. For n = 3, (3.60) and (3.66) are replaced by the weaker results (3.69)-(3.70). Now we show that there is at most one solution in a class of functions smaller than that in which we obtained existence. Theorem 3.4. If n

such that

= 3, there is at most one solution ofProblem 3.2

u E L 2 (0, T; V) n L (0, T; H)

(3.75)

u E L8 (0, T; L4 (n)).

(3.76)

OO

Such a solution would be continuous from [0, T] into H. (l)This inequality with c depending on n holds for any dimension of space.

Theevolution NavieT’-Stoke, eqUiltionl

298

ell. III, §3

Proof. (i) The inequalities (3.72) - (3.73) imply that ifu satisfies (3.76)

then

Bu E L2 (0, T; V’) (at least).

(3.77)

Therefore ifu satisfies (3.75) - (3.76) and (3.18), then

u’EL2 (0, T; V’)

(3.78)

and according to Lemma 1.2, u is almost everywhere equal to a contin› uous function from [0, T] into H. (ii) By the Holder inequality and (3.67), Ib(u. u, v)1 ~colluIIL4(n)lIvIIL4(n)lIulI, Ib(u. u, v)1 ~c1IuI1/4I1uIl7/4I1vIIL4(nr

(3.79)

(iii) Let us assume that u1 and u2 are two solutions of (3.17) - (3.19) which satisfy (3.75) - (3.76) and let u u1 -"2.

=

As in the proof of Theorem 3.2 one can show that d

-!u(t)1 2 + 2vllu(t)1I 2 = 2b(u(t), u(t), u2(t)).

dt

(3.80)

We then bound the right-hand side, according to (3.79), by 2c1Iu(t)11/4I1u(t)1I7 /4 11u2 (t)II L4 (n) ~

v lIu(t) II2 + c2Iu(t)12I1u2 (t)II~4

(n)'

We get d dt

2

-lu(t)1 ~c2I1u2(t)IIL4(n)lu(t)1

Since the function t -+ IU2 (t)I~4 proof as for Theorem 3.2.

4

2 (n)

.

.

is integrable, we may complete the

Remark 3.5. The preceding proof is valid for S'l bounded or unbounded. Remark 3.6. There are many similar results of uniqueness which can be proved by assuming some other properties of regularity. For example (c.f, Lions [2] p, 84), there is uniqueness in any dimension if, in place of (3.76), u satisfies

u E L’(O, T; l!(S’l))

(3.81)

Existence and uniqueness theorems (n c; 4)

Ch. Ill, §3

with

- +-

2 s

n r

~

2 s

n

= 1 if 0

- +-

r

299

1 if 0 is bounded, (3.82)

is unbounded.

3.5. More Regular Solutions

Our purpose in this section is to prove that by assuming more regularity on the data, we can obtain more regular solutions in the two-dimensional case. In the three-dimensional case the existence of such more regular solutions is proved on arbitrary intervals of time provided we assume that the given data Uo ,I, are "small enough" or that v is large enough. 3.5.1. The Two-Dimensional Case

Theorem 3.5. Weassume that n = 2 and that

f and I’ E L2(0, T; V’),{(O) EH Uo EH2(0) n V.

(3.83) (3.84)

Then the unique solution of Problem 3.2 given by Theorems 3.1 and 3.2 satisfies u’ E L 2(0, T; V) nLOO(O, T;H). (3.85)

Proof. (i) We return to the Galerkin approximation used in the proof of Theorem 3.1. We need only to show that this approximate solution also satisfies the two a priori estimates:

u~

remains in a bounded set of L 2(0, T; V) n LOO(O, T,H). (3.86)

In the limit (3.86) implies (3.85). Since Uo E V n H2(0), we can choose uom as the orthogonal projection in V n H 2(0) of Uo onto the space spanned by wl’ ... , w m ; then 2 uom ~ Uo inH (n ), as m ~

lI uom IIH 2 (n) ~ lIuo IIH 2 (n)'

00,

I

(3.87)

Theevolution Navies-Stokes equations

300

(ii) We multiply (3.22) by

j

= I, ... , m; this gives

gjm (t) and add the resulting

lu'm (t)1 2 + v« um (t), u~ (t)))

= <{(t), u~

Ch. III, §3

equations for

+ b (um(t), um(t), u~(t))

(r)>,

In particular, at time t = 0,

lu~ (0)12 = (f(0), u~ (0)) + V (l:iUom ,u~

(0)) - b(uOm' uOm' u~ (0))

(3.89)

so that ~ I{(O)I

lu~(O)1

+ vll:iUijrnI + IBuOmI.

(3.90)

It is clear from (3.87) that

ILluO m I ~

Co lI u om " H2 (0) ~ Co lIu o" H 2(0)'

For BUom we have, by the Holder inequality, Ib(u, u, v)1 ~ c1IluIlL4(0)lgrad uIL4(0)lvl

(by (3.48) and the Sovolev inequality)

~

~ c2I1ulllluIlH2(0) lvi, V u EH 2(O), V v E L 2(O) and hence 1Buom I ~ c211uom

~

c2

IIl1uom "H 2(0) ~ (by 3.87))

11uo 1112 (0)

(3.91)

Finally (3.90) and the above estimates show that (3.92)

u~ (0) belongs to a bounded set of H.

(iii) We are allowed to differentiate (3.22) in the t variable, and since {satisfies (3.83), we get

(u~,

Wi) + v«u~,

Wi)) + b(u~,

Urn, Wi) (3.93)

j

gim

We multiply (3.93) by (t) and add the resulting equations for find (taking (3.2) into account):

= I, ... , m; we d

Cit lu~

2

(t)1 + 2v lIu~

2

(t)1I + 2b (u~

(t), u m (t), u~ (t))

=

Existence and uniqueness theorems (n '" 4)

Ch.lII, §3

301

(3.94)

(t).

= 2
By Lemma 3.4,

(t)) I ~ 2 3/ 2 1u:n (t) IlIu:n(t) 1lllum(t)11

(t), um (t), u~

2lb(u~

(t)1I 2 +

~ vllu~

ff

lIum (t)11 2 lu~ (t)12.

Thus, we deduce from (3.94) that d v - lu~(t)12+-lIu~(t)1I2~-I!’(t)I~’+lf>m(t)lu~(t)12 dt 2

2

(3.94)

v

where

If>m (t)

2

= - lIum(t)11 2 v

Then, by the usual method of the Gronwall inequality, , 2 exp(-d {lum(t)1 dt

J t

If>m(s)ds}

o

whence

2

I.;" (tll’ < (I.;" (Oll’ +;

f

~-I! 2 , (t)l 2v’

v

t

f

t

0 If’(sllj"ds)

exp

0 ¢",(sld.,.

(3.96)

Since the functions um remain in a bounded set of L 2(0, T; V) (cf. (3.31)) and because of (3.92), the right-hand side of (3.96) is uniformly bounded for s E [0, T] and m: u~

belongs to a bounded set of L (0, T; H).

(3.97)

OO

With (3.97) we infer easily from (3.95) that the u~ bounded set of L2 (0, T; V). The proof is achieved. 0

remain in a

Theorem 3.6. The assumptions are those of Theorem 3.5. and we assume moreover that n is a bounded set of class C(j2 and that

f E L (0, T; H). OO

(3.98)

Then the function u satisfies u E L (0, T;H2 (n )) . OO

(3.99)

The evolutionNavier-Stokes equations

302

Ch. JII, §3

Proof. (i) We write (3.18) in the form v «u(t), v))

= (g(t), v),

Vv E V,

(3.100)

where g(t)

=f(t) -

u'(t) - Bu (t).

(3.10 1)

The proof is now based on two successive applications of Proposition 1.2.2. (ii) Since u E LOG (0, T; V) and Ib(u(t), u(t),

v)1

~collu(t)IIL4(r1)llu(t)lIl1vIIL4(o)

(3.102)

~ c11Iu(t)1I 2 Il vIlL4(o)' we have Bu E LOG (0, T; L 4/3(0)).

Thus (i-u'E LOG (0, T; H)), g E LOG (0, T; L4/3(0)).

Proposition 1.2.2. then implies that

u ELoo(O, T, W2 ,4/3(0 )). By the Sobolev theorem, W2,4/3 (0) CLOG (0), and hence

u E LOG (Q). (iii) We can now improve (3.103). We replace (3.102) by the inequality Ib(u(t), u(t), v)1 ~

c21IuIILoo(Q)llu(t)llllvll

which shows that OO

Bu E L (0, T;H).

This implies that g E LOO(O, T; H)

and another application of Proposition I. 2.2. gives uELoo(O, T;H2(On.

(3.103)

303

Existence and uniqueness theorems (n ... 4)

Ch.I1I, §3

Remark 3.7. By repeated application of Proposition I.2. 2 it is now easy to prove, exactly as in Proposition II.l.l, that if n is of class qjeo , Uo is given in ~eo (n), and f is given in ~eo (Q), then the solution u is in ~eo (Q). By the same method, intermediate regularity properties can be obtained with suitable hypotheses on the data. 0

3.5.2. The Three-Dimensional Case. We will prove for n = 3 some regularity properties similar to those obtained for n = 2, but in the present case this will be done only by assuming that the data are "small". In the next theorem we denote by c some constant such that Ib(u, v, w)] E;;; cllullllvllllwll, V u, v, wE V. Theorem 3.7. We assume that n satisfying Uo EH2 (n )

re ir c;

= 3 and

that there are given

n V.

T;H),f'EL 1(0, T;H)

(3.104)

f and

Uo

(3.105) (3.106)

and a further condition given in the course of the proof which is satisfied if v is large enough or iff and Uo are "small enough" (1). Then there exists a unique solution ofProblem 3.2 which satisfies moreover

u’ E L 2 (0, T; V) n Leo (0, T; H).

(3.107)

Proof. (i) To begin with, we observe that uniqueness is merely a consequence of Theorem 3.4, because such a solution will satisfy

u ELeo(O, T; V)

(3.108)

and then V C L4 (n ) implies (see (3.76)) that

u E Leo (0, T; L4 (n)).

(3.109)

(ii) Some of the steps of the existence proof are the same as in Theorem 3.4: we use the Galerkin method of Theorem 3.1, and we choose the basis and urn so that (3.87) holds. The estimates (3.90), (3.91), and thus (3.92) still hold: (1) See (3.115).

TheevolutionNavier-Stokes equations

304

Ch.IIl, §3

Iu~ (0)1 ’" d 1 = 1/(0)1 + vCo IIuo IIH 2(0) + c1 IIuo 1112 (0)'

(3.110)

We derive in the same fashion equation (3.94) and we deduce from it using (3.104): d dt

- Iu~

(t)1

2

+ 2 (v - c IIum (t)11) IIu~ (t)II 2 '" 21f’(t)llu~

(t)l. (3.111)

(iii) There results from (3.28) and (3.29) that

2 1 vllum (t)II '" -1I/(t)II~’

- 2(um (t), u~ (t))

v

1

+ 21um (t)1 Iu~ (t)1

'" -1I/(t)II~’ v

(3.112) where d2 = II/IIlOO(O, T; V’)•

(3.113)

Using (3.110), we infer from (3.112) that, at time t 2 d2 vllu m (0)11 ’" -

v

Td2 1/2

2

+ 2d1(Iuol + - ) v

= 0,

= d3

(3.114)

The hypothesis mentioned in the statement of the theorem is that d4

= d2 + (l + dV (Iuci + Td2)1/2 • V

• exp (

V

I

:If'('lld' ) < ::.

(3.115)

Since d3 ’" d4 , we get as a consequence of (3.114) - (3.115) 2

vllUm (0)11 ’" d3 ’" d4

v3

< c2

and then v - cllum (0)11

> O.

We deduce from this inequality that v - c IIum (t)II remains positive

Existence and uniqueness theorems (n <; 4)

Ch.III, §3

305

on some interval with origin O. We denote by Tm the first time t such that

~

T

or, if this does not happen, Tm = T. Then v-ellum(t)II~O,

(3.116)

O~t~Tm’

(iv) With (3.116) we deduce from (3.111) that

d dt lu~

2 (t)1 ~ 21!,(t)llu~

(t)I,

d 2 dt (1 + Iu:"U)1 ) ~ If'(t)1 (1 + lu~

Thus

ft If’(s)lds)}~O,

d dt {(1 + lu~(t)12)exp(-

1+

(t)1 2 ).

0

I":" (tll’ <; (l + I":"(011') exp (

s:

1"(Slldsl) ,

and, by (3.110), I + 1u;" (Ill’ <; (l + dllexp

(

f :1f’(Sllds ). 0 .. t<;

Tm

(3.1171

From (3.112), (3.115), and (3.117) we get vllum(t)112~d4’

O~t~

v - ell"m (t)1I Then Tm

= T, and

- d lu~ dt

(t)e

~

v- e

Tm,

A>

A

0, 0

~ t ~ r;

(3.118)

(3.111) implies

+ 2(v - e

~) lIu~ v

(t)1I 2 ~ 21t’(t)llu~

and we easily deduce from this relation that

(t)I, O~t~

T,

Theevolution Navier-Stoker equations

306

u~ remains in a bounded set of L2(0, T; V)

The existence is proved.

Ch. III, §3

n LOO (0, T; H).

0

(3.119)

As in the two-dimensional case, we also have Theorem 3.8. With the assumptions of Theorem 3.7, and if we assume moreover that 0 is of class ~oo, the function u satisfies (3.120)

uELOO(O, T;H2(0)).

Proof. We write (3.18) in the form, v«u (t), v))

= (x(t), v),

V v E V,

with g(t)

=f(t) -u'(t) -Bu(t).

Since f - u’ E L" (0, T; H), Proposition 1.2.2 gives (3.120) provided we show that OO

Bu E i: (0, T; H) (and hence g E L (0, T; H)).

(3.121 )

This result is also obtained by repeated application of Proposition 1.2.2 and various estimates on the form b. By the Holder inequality, we have: Ib(u(t), u(t), v)1 ~ Co Ilu(t)IIL6(rnllu(t)lllvIL3 (0) Ib(u(t), u(t),

vi ~ clllu(t)1121IvIlL3(or

(3.122)

We deduce from (3.122) (and u E L" (0, T; V»), that Bu E LOO(O, T; L3/2(0)), g E LOO (0, T; L 3/2(0)).

Proposition 1.2.2 implies that u E L" (0,

T; W2, 3/2(0))

and, in particular (since W2 , 3/2 (0) C L 8(0) for n

= 3),

u E L"" (0, T; L 8 (0)). Using again the Holder inequality we estimate b by b(u(t), u(t), v) E;;; Co lIu(t)IILB(0)lIu(t)lIl1vIlLB/3(0).

Hence

Existence and uniqueness theorems (n <; 4)

Ch. Ill, §3

307

Bu, g E tr (0, T; £8/5 (n)) and by Proposition I. 2.2,

u E L (0, T; W 2 8/5 (n)) C tr (n x [0, TD.

(3.123)

OO

With (3.123) and u E Loo (0, T; V), the proof of (3.121) is easy, and thus Theorem 3.8 is proved.

Remark 3.8. The same remark about regularity as Remark 3.7 holds. Introduction of the Pressure (n ~ 4). To introduce the pressure, let us set

au) =

I:

u(s )ds, P(O =

f:

Bu(s)ds, F(t) =

f

>tS)ds.

If u is a solution of (3.17) - (3.19) then, for any n ~ 4, U. (j, FErtf([O", T); V’).

(3.124)

Integrating (3.18), we see that v((U(t),v))=, VvEV, VtE[O,T],

with g(t)

= F(t) -(j(t) -u(t) + uo,

g E ~([O,

(3.125)

T]; V’).

By application of Proposition 1.1.1 and Proposition 1.1.2, we get for each t E [0, T], the existence of some function p(t), p(t) E L 2(n),

such that -vb.U(t) + grad p (t)

= g(t)

or u(t) -uo -vb.U(t) + (j(t) + grad p (t)

= F(t).

(3.126)

According to Remark 1.1.4, the gradient operator is an isomorphism from L 2 (n ) /0{ into B-1 (n). Observing that grad p

=g-vb.U,

we conclude that grad p belongs to pE~([O,

T);L 2 (n )).

rc [0, T) ; B-1 (n)) and therefore (3.127)

TheevolutionNavier-Stokes equations

308

Ch. III, § J

This enables us to differentiate (3.126) in the distribution sense in Q = 0 X (0, 1); setting

ap

(3.128)

p = at’

we obtain -au at

n

vtxu +

’"

~ ;=1

u;D;u + grad p

=t. in Q.

(3.129)

The pressure appears in general as a distribution on Q defined by (3.127) - (3.128). Under the assumptions of Theorems 3.6 (n= 2) or 3.8 (n = 3), the application of Proposition 1.2.2 shows also that P E L (0, T; HI (0». OO

(3.130)

3.6. Relations between the problems of existence and uniqueness (n

= 3)

The above sections pointed out the two important problems in the three-dimensional case, for the theory of Navier-Stokes equations: - uniqueness of "very weak solutions", i.e. of solutions whose exist› ence is guaranteed by Theorem 3.1. - existence in the large and for any data of "more regular solutions" , for instance solutions whose uniqueness is guaranteed by Theorem 3.4, or the solutions whose existence is proved in a restrictive case in Theorem 3.7. For convenience, until the end of this Section we will call, - weak solutions, the solutions u of (3.13), (3.14), such that u E L 2(0, T; V)

n i : (0, T; H)

(3.131)

and thus (Theorem 3.2)

u E L8 /3 (0, T; £4 (0», u' E £4/3 (0, T; V’)

(3.132)

- strong solutions, the solutions v of (3.13), (3.14), such that (3.131), (3.132) hold, and moreover

v E £8 (0, T; £4 (0».

(3.133)

According to Theorem 3.1, 3.2, 3.4 and 3.7, we know the existence but not the uniqueness of weak solutions, and we know the uniqueness

Ch. III, §3

Existence and uniqueness theorems

309

(n " 4)

but not the existence of strong solutions (except in some very restrictive cases). We observe that the weak solutions given by Theorem 3.1 satisfy the energy inequality (3.47) (see remark 3.2):

lu(I)I' + 2"

f~

Ilu(s)II'ds

.. luoI' + 2

f >r s ), u(s»ds, V t

E [0, T].

(3.134)

According to Theorem 3.4, (3.78), and Lemma 1.2, the strong solu› tions (if they exist) satisfy an energy equality instead of (3.134):

1"(1)1' + 2"

f~

!vIs) Il'ds

= IUo I' + 2

f~

(f(s), "(s»ds, V t E [0, T] .

(3.135)

The problems of the uniqueness of weak solutions and of the existence of strong solutions are related as follows: Theorem 3.9. We assume that n given.

IE £2(0, T; H),

Uo EH.

= 3 and that I and Uo are arbitrarily (3.136)

If there exists a solution v of (3.13), (3.14) satisfying (3.131), (3.134) (i.e. a strong solution), then there does not exist any other solution u of (3.13) (3.14) satisfying (3.131), (3.132) and (3.134) (i.e. a weak solution satisfying the energy inequality.). This result was proved by J. Sather and J. Serrin (see Serrin [3]). Proof. Let u and v be the two solutions mentioned in the statement of Theorem 3.9;u satisfies (3.134), v satisfies (3.135). We show in the next Lemma that

cn.rn. s1,

The evolution Navier-Stokes equations

310

f: f:

(u(t), v) + 2v

+

=luol’

«U(S), v(s»)ds

(f(s), U(S)

+ v(s)lds

(3.137)

- f>(w(s), w(s), v(s»ds

where w = u - v. We now add (3.134) to (3.135) and subtract two times (3.137) from the corresponding inequality. After expanding we get

Iw(tl!’ + 2v

f:

Ilw(s)II’ds

"2 f>(W(S), w(s),

vIs»~

(3.138)

ds.

Using the Holder inequality, we derived in Chapter II the bound Ib(w(s), w(s), v (s» I ~ cl "w(s)IIL4(o)"w(s)lIllv(s)IIL4(o)'

Because of Lemma 3.5 we can maiorize this expression by c2Iw(s)ll/4I1w(s)II7/4 II v(s)II L4(0) ~ vlllw(s)1I 2

+ C3 Iw (s) 12 IIv(s)II~4

(0)'

Using this maioration, we deduce from (3.138) that

Io Iw(s)I’ Iv(s) I~. t

Iw(t)I’ "2c,

(n)dx

Since the function t -+ oCt) = 2c3I1v(t)IIl4 (0) is integrable (v satisfies (3.133», the Gronwall inequality gives

Ch. III, § 3

f

Existence and uniqueness theorems (n

0;;;

4)

311

t

Iw(OI' <; and thus w = U,

-

0 0 (’ )

V

Iw(')I' ds <; 0,

= 0.

It remains to prove (3.137).

Lemma 3.6. With U and v as in Theorem 3.9, the relation (3.137) holds.

° !/(O

Proof. Let P E~(tR) be a regularizing function, P ~ 0, p(-t) = for ItI ~ 1, and

p(t)

= p(t),

dt = 1;

let Pe be defined by Pe(t) = lie p(tle). We associate with any function w defined on [0, T] the function defined on tR, equal to won [0, T] and to outside this integral. Using (3.18) we write,

°

u ’ + vAu + Bu =1 on ] 0, T[, v’ + vAv + Bv =1 on ] 0, T[,

w

(3.139) (3.140)

and after regularization, (3.139) gives:

d

dt (u•Pe *P e )

= if- -

Bu - vAu)*P e •Pe , on ] e, T - e[.

(3.141)

Due to (3.139) and (3.141), the following relations hold on ] e, T - e [:

; (v, u*P e *P e)

= (v’, u*P e *P e) + (v, U’*P e *P e) = {I -

vAv - Bv, u*P e *P e

+ {v, (j - Bu - vAu)*P e *P e), Since the function Pe is odd, we also have d -(v u )={/-vAv-Bv u )+(v I-Bu-vAu) dt ’ e ’ e e’ , where u e = u*P e *P e, v e = v*P e .P e

(3.142)

312

Theevolution Navier-Stokes equations

<s < t < T-

We now integrate (3.142) from s to t, (v(I), u,(I» - (v(s), u,(s»

- v

J:

=[

Ch. III, §3

e; we get

(((o), u,(o) + v,(o»

(«v(o), u, (0»))

+ «v, (0), u(o»))) do

- ft {b(v(a), v(a), ue(a)

js

+ b(u(a), u(a), ve(a))} da.

Due to (3.131), (3.132) and (3.133),

ue~uinL?oc(]O,T[;

V)andL~~(JO,T[;L4(0»),

v e ~vinL?oc(]O,

T[; V)andL~c(]O,

4

T[; L (0 »).

I

(3.143)

(3.144)

The passage to the limit in (3.143) is then legitimate; we see that for almost all sand t, 0 < s < t < T: (v(l), u(I» - (v(s), u(s»

=

f

+ 2v

J:

r

«u(o), v(o»)) do

(((o), u(o) H(O» do -

+ b(u(a), u(a), v(a)} da.

(b(v(o), v(o), u(o» (3.145)

As the function u is weakly continuous in H (Theorem 3.1) and the function v is strongly continuous in H (Theorem 3.4), the function t ~ (u(t), v(t»

is continuous and therefore the relation (3.145) holds for all s and t, 0’" s < t '" T. Setting s = 0 in (3.145) we obtain precisely (3.137) if we moreover observe that b(v, v, u) + b(u, u, v) = b(u - v, u, v) = b(u - v, u - v, v).

Existence and uniqueness theorems

Ch. III, § 3

313

3.7. Utilization of a special basis. If 0 is a bounded Lipschitz open set in \Rn (n = 2, 3), one can use as a special basis for the Galerkin method (§ 3.2), the basis of eigenfunctions Wj introduced in Chapter I, §2.6. This will enable us to obtain further a priori estimates on the solution and existence results of regular solutions, slightly different from that of Section 3.5.

3.7.1. Preliminary Results. The following Lemmas will be useful. Lemma 3.7. Let n. be a bounded open set of class ~2 in \Rn (arbitrary n). Then IAu / is a norm on V n H2(0) which is equivalent to the norm induced by H 2 (0).

Proof. For u E V,/E L 2(0), Au =I is equivalent to ((u, v))

=(f, v),

V v E V.

(3.146)

The interpretation of (3.146) as a linear Stokes problem and Proposition 1.2.2. give that

IIullH,(o) =r;; ; Co ltl = Co IAul. The inverse inequality is easy and the Lemma is proved. Lemma 3.8. Assume that 0 C \Rn (n = 2, 3) is bounded and of class CC2 • Ifu E V n H 2 (0 ), then B u E He L 2 (0) and

IBul =r;;;;c1/ull/2 IIull IAu1 1/2 if n =2 IBu I =r;; ; C2 IIull 3/ 2 IAu 11/2 tf n = 3. Proof. If n follows: I

= 2, we apply Holder’s inequality

(3.147) (3.148) with exponents 4,4, 2, as

f

n ». (D,uj) v/ dxl "'Iu, IL , (n) IDt"j IL • (n) Iv/IL , (nJ'

By (3.48)(1), the right hand side is bounded by (1) Relation

(3.48) established for v EH ~(o) is also valid if v ENI (0), the coefficient 2 1/ 4 being replaced by some other constant c =c(O). See Lions & Magenes [1].

314

The evolution Navier-Stokes equations

Ch. III, §3

c31ui11/2 [grad ujllgrad Diujl1/2lvjl

Whence Ib(u, u,

v)1 ~ c4IuI1/2I1ull lulI~~({1)

Ivi

which implies (3.147) after application of Lemma 3.7. If n = 3, we apply Holder’s inequality with exponents 6, 4, 12 and 2:

By the Sobolev imbedding Theorems, this is less than 12

/ 13/2 IDiUj 1H1({1)lvjIL2({1) Cs IUiH1({1)

whence Ib(u, u, v)1 ~ c611u1I3/2 lIulI~:({1)

lvi,

and (3.148) is proved.

3.7.2. The Two-Dimensional case Theorem 3.10. We assume that 0 is a bounded open set of classCC 2 in cR 2 .

Let f and Uo be given such that UOEH,

(3.149)

fEL 2(0, T;H).

(3.150)

Then there exists a unique solution to Problem 3.2., which satisfies moreover

v'tU E L2 (0, T; H2(0 )) n L- (0, T; V), -rr u’ E L2 (0, T; H)

(3.151)

If Uo E V, then U

E

L2 (0, T;H2(0)) n L- (0, T; V), u’ E L2(0, T; H)

(3.152)

Ch.III, §3

315

Existence and uniqueness theorems

Proof. We begin with the case Uo E V. We consider again the Galerkin approximation used in the proof of Theorem 3.1. This time the w/s are those given in Chapter 1, § 2.6, and we assume that uO m E Sp [wI' ... , wn ] is chosen so that uO m .... uo, strongly in V, as m .... 00.

Relation (3.22) can be written (u~,

Wj) + (v A"m + BUm’ Wj~

= (f, Wj), 1 <.j <. m,

(3.1 53)

By I (2.64), «w j, v))

= (Awj’

v)

= "l\j(Wj’ v),

V v E V.

(3.154)

Hence after multiplication by }.’i’ we can write (3.153) as follows

«u:n, Wj)) + v (Au m, AWj) + (Bum’ AWj) = if, AWj)’

We multiply the relation by gj (see (3.21)) and add for j = 1, ... , m; we obtain 1 d 2 2 (3.155) - -liu mII + vlAumI + (Bum’ Aum) = if, Aum)• 2 dt We deduce from Lemma 3.8 that, 1 d

- -1I"m 11 2 + vlAum12 2 dt

<’lfllAuml + cl1uml1/211umiliAuml3/2

v

1

v

v

4

<. -41Aum12 + -ltl 2 + -IAum12 + c71uml211umll4 Whence

d 2 -liu mIl 2+vIAu mI 2,.;;;; -lfI2+2c7IumI2I1umIl4. dt v In particular, setting om (t) = 2c7 IUm(t)1 2I1um(t)1I 2 , we get d 2 -lium ll2 <. -lfl 2 + om II Um 11 2 . dt v From the estimates (3.30), (3.31),

f:

am (t) dt" canst = "s.

(3.156)

316

Ch.I1I, §3

The evolution Navier-Stokes equations

and by Gronwall’s method and (3.153), we find that the sequence um remains bounded in LOO(O, T; V).

Back to (3.156) we get now,

II um (7)11' + v [IAu", I' dt <;;lIuO m II' +

~

f:

(3.157)

If I' dt

+ 2"1 [1u",I'IIu",II' dt and from (3.157) and Lemma 3.7, the sequence u m remains bounded in L2(0, T; H 2(0 » .

(3.158)

It is easy to conclude that um converges to u, with U in L" (0, T; V)

n L2(0, T;H2(0 » . Lemma 3.8 implies thatBu E L4(0, T;H); on the

other hand, Au E L2 (0, T; H), and by (3.18), u' =1 - Bu - v A u E L 2 (0, T; H). The theorem is proved in the case Uo E V. If Uo E H, we multiply (3.156) by t and we obtain d

2t

- (t1lu",1I 2 ) + vtlAum l2 ~ 1Iu",1I 2 + -1/1 2 + tUm lIumll 2 dt v and we obtain the result that...;tu", remains bounded in Loo (0, T; V) and L2 (0, T; H2 (0». At the limit m -+ 00, we obtain the first part of (3.151); the resultv’t u’ E L 2 (0, T; H) is obtained with a proof similar to the preceding one: by (3.147), ...;tu'=...;t(f -Bu - vAu)

clearly belongs to L2 (0, T; H).

3.7.3. The three-dimensional case. Theorem 3.11. We assume that 0 is a bounded open set of class CC2

in

6{3.

Let there be given Uo and 1 such that

uoE V,fELOO(O, T;H).

(3.159)

Existence and uniqueness theorems

Ch. III, §3

317

Then there exists T. : : z min (T, T1 ) , T1

3

(3.160)

: : z 4c9 M2

2 M::::z4 max (llu oIl 2 , - - 2 N(f)2), N(f)

clOv

such that there exists a unique solution moreover U satisfies UE

t: (0, T.; V)

u'EL2(0,

U

= I/lroo(o, T;H)'

(3.161)

of Problem 3.2 on (0, T.);

n L2(O, T.; H 2 (!2»

(3.162) (3.163)

T.;H).

Proof. We proceed as in Theorem 3.10, until the equation (3.155). Then Lemma 3.8 gives

1 d

2" dt

lIumll2 + vlAuml 2:E;;; IfllAum1+ c2I1umIl3/2IAumI3/2

Whence

d 2 - liumll2 + vlAuml2:E;;; -1/1 2 + 2c911um1l 6. dt v By Lemma 3.8, IAvl :E;;;
d

(3.164)

2

- liumIl2+ clO v llumIl2:E;;; -N(f)+ 2c911um1l6. v dt Now we claim that

lIum (t)1I 2 :E;;; M, 0 :E;;; t

:E;;; T1 '

(3.166)

provided Ilu omII :E;;; lIuo II and this is satisfied for instance if uO m = PmUO is the projection of Uo (either in V or H) on the space spanned by .,w m • Indeed let z = max (M/2, lIum 11 2 ). Then, by a well known result of G. Stampacchia [1], the function z is almost everywhere differentiable and wI'"

dz

- : : z 0 if lIum 11 2 :E;;; M/2 dt

The evolution Navier-Stokes equations

318

Ch. Ill, § 3

and dz d dt - dt

lIum 11 2 otherwise.

Whence dz - ~ 2 c9 z3, z(O) dt

=IJ./2,

so that

~ ~ 4 (l

IJ.2

- c9 t IJ.2) IJ.2

~

2.

z2

for t ~ T1 We deduce that the sequence um remains bounded in L oo (0, T.; V) and then, as before, in L2(0, T.; n2 (n». Also u~ remains bounded in L2 (0, T.; H). The limit U is the unique solution to Problem 3.2 on [0, T.]. Whence the result.

Remark 3.9. Either directly or by passing to the lower limit in (3.156), (3.164), we see that u satisfies d

2

-lIull 2 + vlAul 2 ~ -111 2 + 2C71ul2 lIul1 4 dt v (n

= 2, 0 < t < T)

d

2

dt

v

(3.167)

-lIull 2 + vlAul 2 ~ -1/1 2 + 2C911ul16 (n

= 3, 0 < t < T.).

3.8. The special case /

(3.168)

=0

We are going to show that for f= 0, the fluid smoothly tends to the equilibrium, as t -+ 00.

Theorem 3.12. We assume that n is a ~2 open bounded set in ~n (n = 2, 3), and that Uo E V and / = O. If n = 2, then u E L"" (0, 00; V) and tends to 0 in Vast -+ + 00. If n = 3, then

u E L"" (0, T2 ; V), u E L""(T 3 , 00; V)

319

Existence and uniqueness theorems

Ch. III, §3

for some T2 and T3 estimated below, (0 Vast -+ + 00.

< T2 <; T3 ) and u tends

to 0 in

Proof. Let u be some solution to Problem 3.2 given by Theorem 3.1 (the solution if n = 2). From (3.47)

lu(I)1 2 + 2.

J:

lIu(9)1I 2 ds <;;lu ol 2 , V t> 0,

(3.169)

and (3.167), (3.168) give

~ I/ul/2 + clO vI/ul/ 2 ~ cn I/ul/2n, dt cn = 2c71uo/2 if n = 2, cn = 2c9 if n = 3.

(3.170)

If for some t1 , u(t1) E V and I/u(t1)1/ is sufficiently small so that:

I/U(t1)I/2(n-1)~

7cn 2cn

(3.171)

then d

-

dt

2

I/u(t1)1/

+

VClO

2

-I/U(t1)1/ ~O,

2

so that I/u(t)I/ will decay after t1 ; more precisely (3.171) will remain valid for any t;;;;: t 1 , u E L" (t1’ 00; V), and I/u(t)I/ -+ 0 exponentially as t -+

+ 00.

The existence of some t 1 for which (3.171) holds is guaranteed by (3.169). Otherwise we must have 2

luol ;;;;:2vt1

VClO ( 2cn )

1/(n-1) ,

and this is impossible for t 1 sufficiently large. If n = 3 we can take T2 = T1 given by Theorem 3.11. Or more directly d

gives

- I/u1I 2 ~cn dt

I/uI1 6 ,

I/u(t)1I ~ (1 - 2cn t lIu ol/ 4 ) 1/4 ’ for t < T2 = (2cn lIu ol/ 4 r 1.

The evolution Navier-Stokes equations

320

Ch.III, §4

§4. Alternate Proof of Existence by Semi-Discretization Our goal now is to give an alternate proof of the existence of weak solutions of the Navier-Stokes equations which will be valid in any number of space dimensions. An approximate solution is constructed by semi-discretization in t, and we then pass to the limit using compactness arguments. In Section 4.1 we reformulate the problem in a way which is appro› priate in any dimension and we state the existence results; Section 4.2 describes the construction of the approximate solution; Sections 4.3 and 4.4 deal with the a priori estimates and the passage to the limit. 4.1. Statement of the Problem

Before giving the existence theorem in higher dimensions we must reformulate the problem of weak solutions. As in the stationary case, if n > 4, the form b is not trilinear continuous on V and a statement such as (3.10)-(3.14) does not make sense since the b(u(t), u(t), v) term in (3.13) may not be defined. For this purpose we introduce again (see Chapter II, Section 1.2) the spaces Vg:

Vg = the closure of ’f"in HA(n) n !p(n), S ~ 1.

(4.1)

The spaces HA(n) n HS(n) and Vs are endowed with the usual Hilbert norm of HS(n): 1/2

(s integer).

Obviously (s

~

(4.2)

1),

Vg C V

(4.3)

with a continuous injection and Vg is dense in V. The form b is defined on V X V X Vg, provided s ~ n12; more precisely: Lemma 4.1. The form b is trilinear continuous on V X V X Vg if s ~n12 and Ib(u, v, w)/ ~clul IIvll IIwll vs (1). (4.4) (1) Any dimension, n bounded or not.

321

Alternate proof of existence by semi-discretization

Ch. III, §4

Proof. For u, v,

E

W

Y, the Holder inequality gives n

Ib(u, v,

= Ib(u,

w)1

~

w, v)1

Ilv; IIL 2n/n -

2

i,t 1 lIuiIlL 2 (n ) IIDiWjllLncnr

(n)

H6(n) C L 2n /n- 2 (n»

~ (by the Sobolev inequality n

~

Co lui IIvll

L

i.]» 1

IIDiw;1I

n . L (0)

Since s ~ n12, EP- 1 (0.) is included in L q (0.) where lis - 1 ,q q 2 n

- =- - -

~

n.

(4.5)

If W E: ~ then Diw; belongs to Hs-1 (n) and to Lq (0.); Diwj belonging to Lq(n) n L 2(n) then Diw; E Ln(n) also and IIDiwJ.1I n

L (on)

~ c1 !Iwll v s

so that Ib(u, v,

w)1 ~ c21ul IIvll IIwllv..

(4.6)

s

This estimate shows that we can extend by continuity the form b from r’X::Y X cy onto V X V X ~, and even H X V X ~, by (4.6). Lemma 4.2. If u belongs to L 2 (0, T; V) n L (0, T; H) then Bu belongs to L 2 (0, T; ~’) for s ~ n12. 00

Proof. By the definition of B and because of (4.4), I<Bu(t),

hence

v>1

= Ib(u(t), u(t), v)] ~ clu(t)1 lIu(t) I IIvllv, s

IIBu(t)IIv.' ~ clu(t)1 s

and the lemma is proved.

V vE

~;

lIu(t)1I for a.e. t E [0, T]

(4.7)

0

In all dimensions of space, we can give the following weak formula› tion of the Navier-Stokes problem:

Ch.III, §4

Theevolution Navier-Stokes equations

322

Problem 4.1. For

I and Uo

given such that

IE L 2(0, T; V’),

(4.7)

Uo EH,

(4.8)

to find u satisfying

u

E

L 2 (0, T; V)

n L (0, 00

d -(u. v) + v«u, v)) + btu, u, v) dt

= (f, v>, V v E

(4.9)

T; H)

Vg ( s

~

I)

(4.10) (4.11)

u(O)=Uo·

If u satisfies (4.9) and (4.10), then d - (u, v> dt

with g

=I -

= (g, v>, V v E Vg

Bu - vAu.

Due to Lemma 4.2, Bu belongs to L 2 (0, T; Vg’) and since 1- vAu belongs to £2 (0, T; V’), gEL 2(0, T; Vg’).

(4.12)

Lemma 1.1 then implies that u’ E L 2(0, T; Vg’)

u’

=I -

\

(4.13)

vAu - Bu;

therefore u is almost everywhere equal to a continuous function from [0, T] into Vg’ and (4.11) makes sense. An alternate formulation of Problem 4.1 is the following one. Problem 4.2. Given I and uo, satisfying (4.7)-(4.8), to find u satisfying u E L 2(0, T; V) n L"" (0, T; H), u’ E L 2(0, T;

V;')

(s ~ i ), (4.14)

Ch. III, §4

323

Alternate proof of existence by semi-discretization

u ’ + vAu + Bu 11(0) = "

0

= f on (0, T),

(4.15) (4.16)

I

The formulations (4.9)-(4.11) and (4.14)-(4.16) are equivalent. The existence of solutions of these problems is given by the following theorem which implies Theorem 3.1: Theorem 4.1. Let there be given f and uo which satisfy (4.7)-(4.8). Then, there exists at least one solution u of Problem 4.2. Moreover u is weakly continuous from [0, T] into H This theorem is proved in Sections 4.2 and 4.3; the weak continuity

in H is a direct consequence of (4.14) and Lemma 1.4.

4.2. The Approximate Solutions Let N be an integer which will later go to infinity and set (4.17) k = TIN. We will define recursively a family of elements of V, say o, u u 1, ... , uN, where u" will be in some sense an approximation of the function u we are looking for, on the interval mk < t < (m + 1)k. We define first the elements l, ..., of V’:

tv

t" = ~J

mk

f(t)dt, m = 1, .. . ,N;

t"

E V'.

(4.18)

(m -l)k

We begin with

uo

= uo- the given initial

data;

(4.19)

then when uo , ..., u" - 1 are known, we define u" as an element of V which satisfies

um

_

um -

1

- -k - - + vAu m. + Bum

= fm ,.

(4.20)

u" depends on k; for simplification we denote it um in lieu of u’{;. The existence of such a u" is asserted by Lemma 4.3, whose proof

is postponed to the end of this Section.

Lemma 4.3. For each fixed k and each m ;;;, 1, there exists at least one u" satisfying (4.20) and moreover

The evolution Navier-Stokes equations

324 I

u" 12

-

I

Ch. III, §4

u" - 11 2 + I u" - u’" - 11 2 + 2kv lIum U2

,.;; 2k(fm, u" ).

(4.21)

For each fixed k (or N), we associate to the elements u 1 , the following approximate functions: uk: [0, T] ~ V, uk(t) m = 1, .. .,N Wk:

[0, T]

~

=u'",

••. ,

uN,

t E [em - 1)k, mk],

(4.22)

H, Wk is continuous, linear on each interval

[em - 1)k, mk] and wk (mk) = u": m = 0, ..., N.

(4.23)

°

In Section 4.3 we will give a priori estimates of these functions; we will then pass to the limit k ~ (Section 4.3). D Proof of Lemma 4.3. The equation (4.20) must be understood in a space larger than V’, for example in a space Vg’, s ;;;;, n/2. It is equivalent to (um, v) + kv«u m, v)) + kbtu", u"', v)

= (u m-1

+ kf’", v), Vv

E

Vg.

(4.24)

We proceed by the Galerkin method, essentially as for Theorem II.l.2. We choose a sequence of elements w1 ' ..., wi’ ..., of ~ which is free and total in Vg and thus in V. For each r, by application of Lemma 1.4, we prove the existence of an element 4J, (depending on r, k, m): r

4J,

= ~ 1 ~~ 1=

(4.25)

,wi’

(4J" v) + kV«4J" v)) + kb(4Jr, 4Jr, v) = (u m- 1 + kim, v), V v E Sp(w1' ..., wr).(l)

(4.26)

We must then get an a priori estimate independent of 1, and pass to the limit r ~ 00 (k and m are fixed in this proof). Taking v = 4J, in (4.26) we get C4Jr - u" - 1, 4Jr) + kvll4Jr 112

Now (1)

= k(fm, 4Jr).

2(a- b, a)= lal 2 -lbl 2 + la- b1 2 , Va, b EH,

Sp(Wt, .

0

0,

_ wr ) - the space spanned by Wt,

0

0

0’

w r.

(4.27)

(4.28)

Ch. III, §4

Alternate proof of existence by semi-discretization

325

so that (4.27) gives lrl 2 + Ir - u" -11 2 + 2kvllrl12 = lum-1!2 + 2k(fm, r> ~ lu m-11 2 + 2k IIfmllv,lrl ~ lu m-11 2 +vkllrll2

+~ I fml ~,.

(4.29)

v

Hence

The inequality (4.30) shows that the sequence r remains bounded in Vas r ~ 00. Therefore we can extract from r a subsequence r' such that r' ~in

(4.31)

Vweakly,asr’~oo.

By standard arguments we then pass to the limit in (4.26) and prove that = u" satisfies (4.24). It remains to establish (4.21). This would be obvious if we could take v = u" in (4.24); since u" $ ~ in general we proceed instead by passage to the limit. We pass to the lower limit in (4.29), noting that the norm is lower semi-continuous for the weak topology: 11 2 ~ {im lr’1

2

r-+-oo

,

The proof is complete.

1111 2 ~ lim IIr' 11 2 . r-+ oo

0

4.3. A Priori Estimates

Lemma 4.4.

m = I, .. .,N,

luml 2 ~dl’

(4.32)

N

k

~

(4.33)

m=1

N

~

m=1

lu m

-

u" -

11

2

~ d1’

where d l depends only on the data:

(4.34)

The evolution Navier-Stokes equations

326

d1

= luol

T 12

2

f

+ -;

(4.35)

IIt(s) II v,cIs.

0

Ch.III, §4

Proof. As mentioned in the proof of Lemma 4.3, we cannot take the scalar product of (4.20) by u’" , at least for n > 4 (Bum EI: V’). But (4.21) will play the same role as the equation we would obtain by this procedure. We majorize the right-hand side of (4.21) by

2k Iltm II v' lIu m II E;;; kullum 11 2 + ~ Ilfm I ~I, V

and we obtain

lu m 12 _ lum-112 + lu m _u m-1 12 + kvllu mll 2 E;;; ~ IItm I ~, v

m = 1,... , N.

(4.36)

Summing the equalities (4.36) for m = 1, ... , N, we find

luNl 2 +

N

L:

m=l

N

lu m-um- 11 2 + vk

L:

m=l

lIumll2~

N

2 1Uo 1

"’" litm II 2 " + -k L., v

v m=l

(4.37)

Summing the inequalities (4.36) for m = 1, ... , r, and dropping the term lIu m 11 2 , we get r

lu'I

2

E;;;

lu ol + ~ 2

v

L:

m=l

Ilfm

lit, ~

N

IUoI2+~

v

L:

m=l

IItm ll t " r = l , ... ,N.

(4.38)

The lemma is now a consequence of (4.37) - (4.38) and of a major› ation of the right-hand side of these inequalities given in the next lemma. Lemma 4.5. Let f'" bedejlnedby(4.l8), Then

N

k

327

Alternate proof of existence by semi-discretization

Ch.III, §4

m2;]

J.

T

IIf"' 1Ij,. <;

(4.39)

1(f(t)IIj,. dr,

Proof. Due to the Schwarz inequality,

Ilf II}, =. k~ II m

f

mk (m-l)k

f(t)dtll}' ~

~

J

mk

11f(t)II 2 dt.

(m-l)k

Then (4.39) follows by summation of these inequalities for m = I, ... ,N. 0 The last a priori estimate is the following: Lemma 4.6. N

""

The sum k L.. II

u’" _ u m -

1

k

m=l

II}, is bounded independently of k.

Proof. Taking the norm in (4.20) we obtain

II

u" _ u m k

1

II v's ~ Ilfmll v's + vllAunll v's + IIBumllv's ~ Cl

{i1f m llv' + Ilunn v } + II Bunll vs"

From (4.4) and (4.32) we get

IIBun I ~’ s ~ C31un 12

nun 11 2 ~

c411u m 11 2

We finally have

and we finish the proof using (4.33) and Lemma 4.5.

0

328

The evolution Navier-Stokes equations

Ch. III, §4

It is interesting now to interpret the above in terms of the approx› imate functions:

Lemma 4.7. The functions uk and wk remain in a bounded set of L 2 (0, T: V) n Lao (0, T: ll); w~ is bounded in L 2 (0, T: V;) and uk - wk -+

°

(4.40)

in L2(0, T; H) (IS k -+ 00.

Proof. The estimations on uk and wk are just interpretations of (4.32)› (4.33) and Lemma 4.6; (4.40) is a consequence of (4.34) and the next lemma. Lemma 4.8.

(4.41 ) Proof. wk(t) - uk(t)

m = (t-mk) k (u -

U

m-l

)

for (m - 1) k :E;;; t :E;;; mk,

mk

S

IWk(t)-uk(t)12dt=

(m-l)k

and we find (4.41) by summation.

4.4. Passage to the Limit Due to Lemma 4.7, we can extract from uk a subsequence uk’ such that Uk’-+U

in L 2(0, T; V) weakly, in Lao (0, T; ll) weak-star

(4.42)

We want to prove that U is a solution of (4.14) - (4.16); we need a strong convergence result for the uk’ in order to pass to the limit in (4.20). The functions wk will play, for this, a useful auxiliary role. We can choose the subsequence k’, so that wk’ -+ U. in L2(0, T: V) weakly,

in Lao (0, T: ll) weak-star,

(4.43)

A lternate proof of existence by semi-discretization

Ch. III, §4

dwk' -+ dt

I··

U* In

L 2 (0, T,. VsI ) weakly.

329

(4.44)

Because of (4.40), U = U*. Theorem 2.1 shows us that wk'-+'u in L 2(0, T; H);

(4.45)

thus by (4.40), uk' -+ U in L 2(0, T; H).

(4.46)

The equations (4.20) can be interpreted as dWk +Auk +Buk =fk, dt

(4.47)

with fk defined by

h (t) =f

m,

(m - 1) k :s;;;, t < mk, m = I, ... , N.

Because of (4.42), (4.46) and Lemmas 3.2 and 4.2,

BU/t -+ Bu in L 2 (0, T; V;) weakly. By Lemma 4.9 below,

!k -+ fin L 2(0,

T; V');

therefore we can pass to the limit in (4.47), and we find u' + vAu

+ Bu = f.

Due to (4.43), (4.44) and Lemma 4.1, (wk' (t), a) -+ (u(t), a},

v

oE

since wk'(O) = uo, we get

V;,

V t E [0, T];

u(O)=uo·

We have proved that u satisfies (4. I 4) - (4. I 6); the proof of Theorem 4.1 will be complete once we prove Lemma 4.9. fk -+ fin L 2(0, T; V'), as k -+

..

o.

(1) Strictly speaking if "0 E V, wk(t) E V for t .. k; in this case we simply replace L' (0, T; V) by oc ([0, T); V) whenever we are considering the functions wk'

Ll

(4.48)

330

Ch. III, §4

The evolution Navier-Stokes equations

Proof. We observe that the transformation

f-+fk is a linear averaging mapping in £2(0, T; V’); this mapping is continuous by Lemma 4.5 which enables us to assert that:

IIfk ilL’

(0, T; V’) E;;;;

IIfIl L , (0, T; V).

(4.49)

Therefore, instead of proving (4.48) for any f in £2 (0, T; V’) we need only to prove it for fin a dense subspace of £2(0, T; V'); for anfin ~([O, T]; V’) the result is elementary and we skip its proof. Remark 4.1. Summing the equations (4.21) for m = I, ... , r, and dropping the terms lu m - u m - I I2 , we get r

r

lu'I 2 + 2kv

L

m=I

lIum 11 2 ,.;;;;

lu ol 2 + 2k

The relation (4.50) can be interpreted as

f

tk

1,,>(1)1' + 2.

(4.50)

f

tk

0 II,,> (s)II'lIs "Iuol' +

where tk

L ir: u m >.

m=I

= (m + I )k, for

mk E;;;; t

0 (fk(s), u,(s»

< (m + I )k.

lis,

(4.51)

(4.52)

For each fixed t, Uk' (t) is bounded in H independently of k and t; as k ' -+ 0, uk'(t) converges to u(t) in V; weakly; therefore (1) uk'(t) -+ u(t) in H weakly, as

k’-+ 0, V

t E [0, T].

(4.53)

We then pass to the lower limit in (4.51) (t fixed, k ' -+ 0), using (4.42) and (4.53). This leads to the energy inequality:

J t

lu(l)I' + 2.

0 lIu(s)11' lis

f

..

t

luol' + 2 (1) Proof by contradiction.

0 (/(s),

(~5~

u(s» lis, V t E [0, T].

Discretization ofthe Navier-Stokes equations

Ch. III, §5

If n

= 2, using (3.62),

f

it is easy to prove directly the energy equality:

t

lu(l)I' + 2"

0 lIu(,)II'ds

f:(/('),

luol' + 2

331

= (4.55) u(,) ds, V t E [0,

11.

§5. Discretization of the Navier-Stokes Equations: General Stability and Convergence Theorems This section is concerned with a general discussion of the discretiz› ation of the evolution Navier-Stokes equations. We study here a full discretization of the equations, both in the space and time variables: I) The discretization in the space variables appears through the introduction of an external approximation of the space V; for example, one of the approximations (APXI) to (APX4), corresponding either to fmite differences or finite elements. Actually these particular examples will be discussed in more detail in reference [9]. 2) For the discretization in the time variables, we propose, among many natural and classical schemes, four schemes with two levels in time (fully implicit scheme, Crank-Nicholson scheme, a scheme implicit in the linear part and explicit in its nonlinear part, and a scheme of explicit type). After the description of the scheme under consideration we proceed to study the stability of these schemes. The problem of stability is the terminology in Numerical Analysis for the problem of getting a priori estimates on the approximate solutions. Classically, discretization in both space and time of evolution equations can lead to unstable or conditionally stable schemes: the approximate solutions are unbounded unless the discretization parameters satisfy some restriction. We discuss in full detail the numerical stability of the four schemes considered. To our knowledge the methods used here are non-classical methods for studying the stability of nonlinear equations. The study of nonlinear instability is a difficult problem; our study here, based on the energy method, leads only to sufficient conditions for stability; the stability

332

The evolution Navier-Stokes equations

Ch. III, §5

conditions which are obtained seem close to being necessary, but the problem of necessary conditions of stability is not studied at all in the text. The last subject treated in this section is the convergence of the schemes. Two general convergence theorems in suitable spaces are proved for the different schemes. The proof of convergence depends on discrete compactness methods. Owing to the lack of uniqueness of weak solutions in the three-dimensional case, the convergence results obtained in the two- and three-dimensional cases are different, and better, of course, if n = 2. The division of this material throughout subsequent subsections is as follows: In Subsection 5.1 we describe the general type of discretization and the numerical schemes which will be studied. In Subsections 5.2, 5.3, and 5.4, we successively study the stability of Schemes 5.1 and 5.2 (Subsection 5.2),5.3 (Subsection 5.3) and 5.4 (Subsection 5.4). Subsection 5.5 deals with auxiliary a priori estimates of a rather techni› cal character (involving fractional derivatives in time of the approxi› mate functions). Subsection 5.6 contains the description of the consistency hypotheses, the statement of the general convergence theorems, and the proofs of these theorems. The application of these results to specific approximations of the space V will be treated in Section 6. There we will also study practical methods for the resolution of the discrete problems, and the Appendix contains the description of practical examples. Other methods of approx› imation of the nonlinear evolution Navier-Stokes equations are given in Sections 7 and 8, including the fractional step or proj ection method and the artificial compressibility method. From now on we restrict ourselves to the "concrete" dimensions of space, n = 2 and n = 3. 5.1. Descrip tion of the Approximation Schemes

From now on we will be concerned with the approximation of the solutions of the Navier-Stokes equations in the two- and three› dimensional cases exclusively, n being bounded. For simplicity we suppose that the given data, uo, f, satisfy

t E L2 (O, T; H),

(5.1)

and, as before, UOEH.

(5.2)

Discretization of theNavier-Stokes equations

Ch.Ill, §5

333

Theorems 3.1 and 3.2 tell us that there exists a unique solution of Problem 3.2 if n = 2, and that there exists at least one such solution if n = 3. Let there be given a stable and convergent external approximation of the space V, say {( lJz, Ph’ ’h)h E :K , (W, F) }; the lJz are assumed to be finite dimensional spaces. This approximation could be any of the approximations (APXl), ... , (APX5), that were described in Chapter 1. For simplicity we assume that

lJz

C L 2(0), V hE ’3(,

(5.3)

a condition which is realized by all the previous approximations. The space lJz is therefore equipped with two norms: the norm I-I induced by L2 (0) and its own norm II-Ilh' Since fh is finite dimensional, these norms must be equivalent; the quotient of the two norms is bounded by a constant which may depend on h. Therefore we assume more precisely that IUh I ~ do

Ilullh' V uh

E

fh,

(5.4)

do independent of h, and lIuhllh ~ S(h)

IUhl, V uh E T-h.

(5.5)

The constant S(h), which usually depends on h, plays an important role in the study of the stability of the numerical approximation; for this reason S(h) is sometimes called the stability constant. Usually S(h) -+ +00, as h -+ O. Let there be given a trilinear continuous form on lJz, say bh (uh’ vh’ wh)’ which satisfies bh(uh,vh,vh)=O VUh,vhET-h, Ibh(Uh’ vh’ wh)1 ~dllluhllh

IIvhllh

V uh’ vh’ wh E

(5.6)

Ilwhllh'

lJz,

(5.7)

(d i independent of h),

and some further properties which will be announced when needed (i.e., when discussing the stability and the convergence of the schemes). Let us divide the interval [0, T] into N intervals of equal length k: k

= TIN.

(5.8)

As in Section 4, we associate with k and the functions f, the elements

t’, ....I":

334

The evolution Navier-Stokes equations

fm =

~J

Ch.III. §5

f(t)dt, m = 1, ... , N; fm E L2 (n ).

mk

(5.9)

(m-l)k

We will describe and study four basic schemes chosen from among a large class of interesting and sometimes classical schemes which have been proposed for the Navier-Stokes equations. For all the four schemes we define recursively for each hand k a family of elements u2, ..• , uf, of v". Actually these elements depend on h, k (and the data), and should be denoted uJ:l; nevertheless, for simplicity we do not emphasize this double dependence. In each of the four schemes, we start the recurrence with u2 = the orthogonal projection of Uo onto

v" , in L2 (n);

(5.10)

this definition makes sense by (5.3) and we immediately observe that

lu21 ~ luol, V h.

(5.11)

Scheme 5.1. When u2, ... , uJ:!-l, are known, uJ:! is the solution in

v"

of

1

k (uJ:!- uJ:!-l, vh) + v ((uJ:!, vh»h + bh(uJ:!-l, uJ:!, vh) (5.12) Scheme 5.2. When u2, ... , uW- 1, are known, uW is the solution in

k1 (uhm -

ml

ml

uh - , vh) + 2" ((uh V

v"

of

m + uh' vh»h

m-l + m ) + 2"1 bh (m-l uh ' uh uh' vh

(5.13) Scheme 5.3. When u2, ... , u’{!-l, are known, uW is the solution in 1

k (uW- u’{!-l,

v" of

vh) + v ((uW, vh»h + bh(u'{!-l, u’{!-l, vh)

= (1m , vh)'

V vh E

v".

Scheme 5.4. When u2, ... , u’{!-l, are known, u'{! is the solution in

(5.14)

v" of

Ch. III, §5

Discretization ofthe Navier-Stokes equations

335

(5.15) For all the schemes the equation defining u,!! is equivalent to a linear equation of the form (5.16) L h depends on m, ah depends on m for Schemes 5.1 and 5.2, but not in the case of Schemes 5.3 and 5.4. We observe, in all cases, that

(5.17) and therefore the existence and uniqueness of the solution of (5.16) is a consequence of the Projection Theorem (Theorem 1.2.2). Remark 5.1. (i) The computation of u,!! requires the inversion of a matrix; - the matrix is positive definite, nonsymmetric and depends on m for Schemes 5.1 and 5.2, - the matrix is positive definite, symmetric, and does not depend on m for Schemes 5.3 and 5.4. (ii) Scheme 5.1 is the standard fully implicit scheme; Scheme 5.2 is an interpretation of the classical Crank-Nicholson scheme. Scheme 5.3 is a partially implicit scheme, implicit only in the linear part of the operator. (iii) Scheme 5.4 is an explicit scheme or more precisely an interpre› tation of the so-called explicit schemes; this terminology is justified by the fact that this type of scheme usually gives u,!! explicitly, that is to say without inverting any matrix. In the present case, due to the discrete condition div u = 0 built in the space J-h, the determination of u,!! necessitates the inversion of a matrix. This restricts considerably the interest of this scheme, but we considered it of interest nevertheless. (iv) Besides this discussion on the type of scheme, the reader is referred to Section 6 for practical methods of computation of the u'!!.

The evolution Navier-Stokes equations

336

Ch. Ill, §5

Remark 5.2: Related Schemes. (i) A related form of Schemes 5.1 and 5.2 is a nonlinear form of these schemes: Scheme 5.1’.

~

(u7:- u7:- 1, vh) + v((u7:, vh))h + bh (u7:, u7:, vh)

= (fm, vh)’

V vh E

fh.

(5.18)

Scheme 5.2’.

(5.19) A related form of Scheme 5.3 is a Crank-Nicholson scheme, implicit in its linear part: (ii)

Scheme 5.3’.

kI (m uh -

m )) + b (m-1 m-1 2"V ((uhm-1 + Uh’Vh h hU ,U ’Vh ) = (fm, Vh)’ V Vh E fh. (5.20)

uhm-1 ,vh ) +

(iii) These Schemes could be studied by exactly the same methods as Schemes 5.1 - 5.4. ’

5.2. Stability of Schemes 5.1 and 5.2

The problem is to prove some a priori estimates on the approximate solution. 5.2.1. Scheme 5.1

Lemma 5.1. The solution u7:of(5.12) remain bounded in the following

sense:

lu7:1 2 ~d2’ N

m = 0, ... , N,

~ lu7: - u7:- 112 ~

m=1

d2 ,

(5.21) (5.22)

Ch. III, §5

Discretization of the Navier-Stokes equations

337

N

(5.23) where

a, = lu.I' + Proof. We take

d! J~ If(s)I'

J’h = u'{(

(5.24)

ds.

in (5.12). Due to (5.6) and the identity

2(a-b, a) = lal 2 - Ibl2 + la-bI 2 , Va, bE L2(U),

(5.25)

we obtain

lu'{( 12 - lu'{(-lI2 + lu'{( - u'{(-lI2 + 2kv lIu'{( I ~

= 2k (1m, u'{() :E;;; 2k Ifm Ilu'{(I :E;;;

(by (5.4))

:E;;; 2kdo Ifm I lIu'{( II h

(5.26)

kd 2

lIu'{( I ~ + _ 0 Ifm 12.

:E;;; kv

V

Hence

lu'{( 12 - lu'{(-lI2 + lu'{( - u'{(-lI2 + kv lIu'{( I ~ kda

:E;;; -

v

m 2

_

If I ,m - 1, ... , N.

(5.27)

Adding these inequalities for m = 1, ... , N, we get N

e

ImY + m=l L h

N

lu m h

um- ll 2 + kv h

2

:E;;;

kd lu21 2 + _0 V

L

m 2 m=l lIu h 11 h

N "" L..

m=l

lfml2.

(5.28)

One can check as in Lemma 4.5 that (5.29)

338

The evolution Navier-Stokes equations

Ch. III, § 5

thus, by (5.11) and (5.29), it follows that the right-hand side of (5.28) is bounded by

d!

d,. = luol' +

1:

If(s)I' ds.

(5.30)

This proves (5.22) and (5.23). We then add the inequalities (5.27) for m positive terms, we get

~

(due to the above)

= 1, ... , r; dropping some

d2 ;

~

(5.21) is proved too. 5.2.2. Scheme 5.2

Lemma 5.2. The solutions u;:’ of (5.13) remain bounded in the follow›

ing sense:

m = 1, ... ,N,

lu;:'1 2 ~d,., ,

N

k L..

m=1

II

u m + u m- 1 2 h

2

h

II h

(5.31)

d

_2,

~

(5.32)

v

with the same d,. as (5.24).

Proof. We take vh = u;:' + u;:,-1 (in 5.13). Due to (5.6) we fmd

lu;:'1 2

_

lu;:,-11 2 + 2kv II

um + um- 1 h

2

h

II ~

= k (1m , u;:' + u;:,-1) ~ 2kd o 11m 1II

um +u m- 1 h

m + m-1

-c kv II uh Therefore

Uh

2

2 h

IIh d2

11 2 + k ....Q 11m 12 h

V

(5.33)

Ch. III, §5

Discretization of the Navier-Stokes equations

u m +u m - 1

luW /2 + ku II

2 h

h

II ~ ~ luW- 1 12 +

kd 02 V

339

lfm 12

(5.34)

We add these relations for m = I, ... , N and get N

lu/Y1 + kv 2

~

’"

L.

m=l

m

m 1 II u h + uh - I ~

(as before)

2

~

d2

This proves (5.32); adding then the relations (5.34) for m = I, ... , r, and dropping the unnecessary terms, we find

this implies (5.31).

5.2.3. Stability Theorems We recall first a definition: Definition S.1. An infinite set of functions E is called 0(0, T; X) stable if and only if E is a bounded subset of LP (0, T; X). It is interesting to deduce from the previous estimations some stability results. In order to state these results, we introduce the approximate functions uh Uh:

[0, T] 1-+

v" ,

(5.35)

uh(t)

=uW, (m-I) k ~ t < mk (Scheme

uh(t)

=

u m +u m- 1 h 2 h , (m-l) k ~ t

m = I, ... ,N. Due to Lemmas 5.1 and 5.2,

5.1)

< mk (Scheme

5.2), (5.36)

The evolution Navier-Stokes equations

340

Ch. III, §5

(5.37)

(5.38) Since the prolongation operators Ph E l(Vh, F) are stable, we have

r

IIPhuh lip"" d 3 lIuh IIh, YUh E Vh, (d 3 independent of h). (5.39)

We infer from (5.38) that

UP.". (I) Aid! <; d~:2

.

These remarks enable us to state the stability theorem: Theorem 5.1. The functions u h' hEX, corresponding to Schemes 5.1 and 5.2 are unconditionally L"" (0,- T; L 2 (n)) stable; the functions Phuh are unconditionally L 2(0, T; F) stable. Remark 5.2. The majoration (5.22), and similar majorations for the other schemes which we will give later on, does not correspond to stability results but will be technically useful for the proof of the con› vergence of the scheme. For the same majoration for Scheme 5.2, see Subsection 5.4.3. 5.3. Stability of Scheme 5.3

We infer from (5.5) and (5.7) that Ibh (u h' u h' vh)1 "" d 1 lIuh I ~ IIvh IIh "" d 1 S2(h)lu h I lIuh IIh IV h I,

V Uh' Vh E

Vh. (5.40)

Sometimes this relation can be improved and this means an important improvement of some restrictive conditions of stability which will appear later on in this section; for this reason we will assume that Ibh(u h, u fI ' vh)1 ""SI (h) Iuhlllu h II h Ivhl, YUh' Vh E Vh'

(5.4l)

where at least (5.42)

Discretization of the Navter-Stokes equations

Ch. III, §5

5.3.1. A Priori Estimates Lemma 5.3. We assume that k and h satisfy kSi(h) ~ d’, kS 2(h) ~ d" (l),

341

(5.43)

where d’ and d" are some constants depending on the data and are estimated in the course of the proof Then, the u’!: given by (5.14) remain bounded in the following sense: ... ,N,

luWI~d4,m=O,

(5.44)

N

E

m=1

Iuhm - uhm- 112 ...... «« 4’

(5.45)

N

L:

(5.46) Ilu,!: I ~ ~ d4, m=1 where d4 is some constant depending only on the data, d', and d".

k

Proof. We write (5.14) with vh = u’!:. Using again (5.25), we obtain the relation m 11 2 luhm12 _ luhm-11 2 + lu«m r>»u m-11 2 + 2kvlluh h (5.47) = -2kbh (u’!: - 1 , u,!: - 1, u’!:) + 2k(fm, u’!: ). Due to (5.6) the right-hand side of (5.47) is equal to

-2kbh (u’!: - 1, u’!: - 1, u’!: _ u’!: - 1) + 2k(fm, u’!: ); this expression is less than (cf. (5.4) and (5.41): m um- 11 2kS 1 (h)/uhm- 11 lIu hm- 111 hluh -h + 2kdO11mI Ilu,!: II h , kvllumll 2 +2k2S2(h)lum-112 lIum - 1 112 h

h

1

1

h

+ _ lu,!: - u,!: 2

h

kd2 112 + _0 V

h

Ifm 12.

{l)In practice, one of these relations should be a consequence of the other (this depends on the explicit values of 8 and 8 1) ,

342

The evolution Navier-Stokes equations

Ch. III, §5

(5.48) We add these inequalities for m

= 1, ..., r;

r

III: 12 +.!.2

h

r

L

lu m

m=1

_

h

um - 11 2 +·kv h

.E

m=1

lIu m 11 2 h

h

r

L

(5.49)

m=2 r

L

m= 1

Let us assume that for some fixed S, 0 < ~

2kSi(h)AN ~ v -~,

< v.

(5.51)

If this inequality holds, it is easy to show recursively that

1 lu rh 12 +2

r

L

m= 1

\u;:' - u;:' - 11 2

r

+k~

L

lIu;:’lI~

m=1

~Ar’

r= 1, .. .,N.

(5.52)

Indeed the relation (5.48) written with m = 1, shows us that (5.52) is true for r = 1. Let us assume then that (5.52) is valid up to the order r - 1, and let us show this relation for the integer r. We observe that, by assumption,

lu;:' 12 ~ Am ~ AN'

m

= 1, ..., r -

1;

therefore, by (5.51), r

2k 2 S2 (h) 1

L

m=2

lu hm - 11 2 lIu hm - 111 h2 r

L

m=2

(5.53)

Ch.III, §5

Discretization of the Navier-Stokes equations r

El

<:. 2k2Si(h)~

343

lIu~ I ~

r

<:'k(v-6)

El

lIu~II~.

Putting this majoration into (5.49), we get (5.52) for the integer r. The proof is complete if we show that a condition of the type (5.43) ensures (5.51). According to a majoration used in Lemmas 5.1 and 5.2 (see (5.11), (5.29» AN’;; luol2 + ~

2JT If(sll2ds + 2k 2S; (h)luoI2 Ilu211~ 0

<:. (see (5.5) and (5.24» <:. d 2 + 2k2Si(h)S2(h)!uo 12 . Hence, if (5.43) is satisfied, 2kSi (h»'N

<:. 2d'(d2 + ia' d"lu o 12 ) .

and this is certainly bounded by v 2d'(dz

~

if d’ and d" are sufficiently small:

+ 2d'd"luoI2) <:. v -~.

(5.54)

The proof is complete. 5.3.2. The Stability Theorem We define for the Scheme 5.3 the approximate functions uh by: uh: [0, T) -+

uh (t) = u~,

fh

(m - 1)k

<:. t < mk,

m = I, ..., N.

We infer from (5.39), (5.44), (5.45) that if (5.43) holds then Sup

tE

[tY, T]

IU h (t)1 <:.

v’"d;"

(5.55)

The evolution Navier-Stokes equation,

344

Ch. 111, §S

and thus Theorem 5.2. The functions uh and p uh ’ h E ’3f, corresponding to the Scheme 5.3 are respectively tr (0, T; £2(0) and L2(0, T; F) stable, pro› vided k and h remain connected by (5043). Definition 5.2. Conditions such as (5043) are called stability conditions. They are sufficient conditions ensuring the stability of the scheme. A scheme is called conditionally or unconditionally stable according to whether such a condition occurs or not in proving stability.

5 A. Stability of Scheme 5.4 504.1. A Priori Estimates Lemma SA. Weassume that k and h satisfy

1- 8 4v

2

kS (h) E;;; - - , for some 8, 0< 8 < 1,

(5.56)

and

(5.57) where

ds = luol2

+( ~~ +4T) JT Ifts)1

2

ds.

(5.58)

o

Then the uf: given by (5.15) remain bounded in the following sense:

luZ' 12 E;;; ds , m = 1, ... ,N,

(5.59)

N

k

L

(5.60)

m=1

N

L

m=1

lu m h

_

um h

11 2 E;;;ds (2 - 0)

L

0

T

+ 4T

If(s)1 2 ds.

(5.61)

Discretization of the Navier•Stokes equations

Ch. III, §5

Proof. We replace vh by u~

=

2(a - b, b)

we find

in (5.15); due to the identity

-I

Ibl2 -

101 2 -

12 - Iu~ - 11 2 - Iu~

Iu~

= zur«,

lu~12

hI ~

-

III~

IIh

+ kd~

Ifm 12 ,

-Iu~

-u h - 112 +kvllu~-III~

V

_lu~-112 kd 2

11 2 + 2kv II U~

-

I/mi.

V

(5.62)

U~)

vk lIu

~_O

b1 2 , Va, b E L 2 (S"2 ),

la -

- U~

~ 2kdO11m I Ilu~

~

345

(5.63)

The above differs from (5.26) in that the term lu~ - u~-112 on the left-hand side is affected with a minus sign and so, we must majorize it. In order to majorize lugz - ugz- 112, we write (5.15) with . "« = uhm - uhm-l . Th’is gives 21u~

- u~ -

11 2

= -2kv«u~

-2kb (u m -

I u m- I

- I , u~ - u~ - I ))h (5.64) u m _ u m - I) + 2k(fm u m _ u m - I)

hh’h’h

h

’h

h’

We successively majorize all the terms on the right-hand side, using repeatedly (5.5), (5.41), and the Schwarz inequality:

-I, u h - u~ -I ))h ~ 2kvllu~

-2kv«u~

-Ili h lu~

~ 2kvS(h) lIu~

1

~ -Iu m _ u m -112 4 h h h

- u~ - Il h

-Ili h lIuh - u~ -lll h

+ 4k2v2S2(h)lIum-11I2. h h»

-2kb (u m- I u m- I um-I) h

h

’

h

’

h-

h

~ 2kS I (h)lu~ - IllIu Ili h lu~ 1 ~ 4"lu~ - u -112 + 4k2Si(h)lu~

- u~ - II

h

2k(fm , u~

~

1

4" lu~ -

- u~ - I ) ~ 2klfm

u~ -112 + 4k 2 11m 12 .

-112I1ugz-III~;

I Iu~ _ u~ - II

346

The evolution Navier-Stokes equations

Ch.I1I, §5

Therefore (5.64) becomes

lu;:' - u;:,-11 2

E;;; 4k2v2S2(h)lIur-111~

+ 4k2S;(h)lu;:' -11 2 Ilu;:' - 111~

+ 4k 21fm 12

E;;; (by (5.56))

+ 4k2Sf (h)lu;:' - 11 211u;:' - 111~

E;;; kv(l - lJ) lIu;:' - 111~

+ 4k 21fm 12 ;

(5.65)

for (5.63) we then have

lu;:'1 2 -lu;:'-11 2 +k(vlJ -4kSi(h)lu;:’-112)lIu;:’-111~ E;;; k(

~~

+ 4k ) Ifm 12

E;;; (since k E;;; E;;; k (

~~

n

+ 4T) Ifm 12

(5.66)

Summing these inequalities for m = 1, ..., r, we arrive at r

lu~

2 1

+ k m~1

(vlJ - 4kSi (h)lu;:' -11 2 ) lIu;:' -111~

where

d~

Il = luhOl2 + k( r

V

+ 4T)

i:

m=1

IF 12

E;;; Il r , (5.67)

(5.68)

Using (5.57) we will now prove recursively that r

r 12

Iuh

+ kvlJ -2

"L...J m=1

(5.69)

We observe first that Il, <'IlN =

lu21 2 + k (

.. lu oI’ +

~~

N

f:

(d: + 4T)

+ 4T )

L:

m=1

If(’
d,.

(5.70)

Ch. III, §S

Discretization of the Navier-Stokes equations

The relation (5.69) is obvious for r using (5.57) we get

= 1; writing

< lu21’ + k(d:

lull’ + kuIJ Du21~ + 4k2Si(h)lu212Iu2"~

;0;; ILl

(5.66) for m

347

= 1 and

4T) If'l' + v; lIu21 ~, +

which is (5.69) for r = 1. Assuming now that the relation (5.69) holds up to the order r -1, we will prove it at the order r. In fact by the recurrence hypothesis,

lu’;- 11 2

1 ;0;; ILN ;0;; (by (5.70» ;0;; d s ’

;0;; ILr -

Hence (5.67) gives

(5.71)

r

lur 12 h

L:

+ kv6

m=l

Iluhm -111 h2 r

L:

m=l

r

;0;; IL

r

kv6

+-

2

" LJ

m=l

(5.72)

IIU~-ll1~,

and (5.69) at the order r follows. It remains to prove (5.61). For this we return to (5.65); using (5.56), (5.57), we get

lu~ _u~-112;o; kV(

1-~)

lIu~-111~+4kTlfmI2.

(5.73)

By summation and using (5.29), we find (5.61). 5.4.2. The Stability Theorem We now set

uh : [0, T] -+ Vh u h (t) = u~ - 1, (m - 1)k;o;; t

and we have

< mk,

(5.74) m

= 1, ..., N,

(5.75)

The evolution Navier-Stokes equations

348

Ch. III, §S

Theorem 5.3. The functions u h and P"u h ’ hE ’3C, corresponding to

Scheme 5.4 are respectively tr (0, T; L2 (n)) and L 2 (0, T; F) stable. provided k and h remain connected by (5.56)-(5.58). 5.5. A Complementary Estimate for Scheme 5.2

Using the techniques extensively applied in Sections 5.3 and 5.4, we can complete Section 5.2 by giving, in the case of Scheme 5.2, an esti› mation similar to the estimation N

~

m=1

lum _ u m -112 h

h

~ Const.

(5.76)

that we proved for Schemes 5.1, 5.3, and 5.4. As mentioned in Remark 5.2, these estimations will be useful for the proof of the convergence. Lemma 5.5. The u’h defined by (5.13) (Scheme 5.2) satisfy: N

~

m=1

lu’h - u’h- 112 ~ d(l + kS 4(h)),

where d denotes a constant depending only on the data.

Proof. We take v h

= 2k(u’h

- u’h - 1) in (5.13) and obtain

21u’h -u’h- 112 =-kvllu’hll~ +kvllu’h-ll1~ _ kb (u m - 1 u m + u m - 1 u m _ um - 1) hh

’h

+ 2k(fm, u’h - u’h ~

h’h

h

1)

(by (5.40) and (5.5))

~ -kvllu’h I ~

+ kvllu’h -111~

+kd1S2(h)lu’h-lll1u’h +u’h- 1I1 h lu’h -u’h- 1 1 + 2klfm I lu’h - u’h - 11

~-kvllum

12 hh

2 + ~ lum + kvllu hm -111 h u m -11 2 2h-h

k2

4(h)lu m -11 211u m + u m -111 2 + _2 d 2S 1 h h h h

1

+ 2" lu’h - u’h - 11 2 + 2kTlfm 12 .

(5.77)

Discretization of the Navier-Stokes equations

Ch. III, §5

Thus

N

L m=l

349 N

lu: - u~ -11 ~ kvllu211~ 2

~

L

m=l

N

2

k +"2

+ 2kT

dfS4(h)

" mL: 1

lu~

f:

-11 2 lIu!:, + u~ -111~

(by (5.5), (5.11), (5.29), (5.31), (5.32))

.; kvS 2 (h) lu o I2 + 2T

1/(,)[2&

2

+ - did~kS4(h). V

The proof is complete. 5.6. Other A Priori Estimates In order to prove strong convergence results we will establish some further a priori estimates concerning the fractional derivatives in t of approximate functions. This Section 5.6 is essentially a technical section which is used in Section 5.7 where the convergence of the schemes is proved. For all the four schemes we define wh ’ a function from lR into ~, by: Wh is a continuous function from lR into ~, linear on each interval [mk, (m + l)k] , and wh (mk) = u~ , m = 0, ..., N -1; wh = outside the interval [0, T] .

°

(5.78)

Lemma 5.6. Assuming the same stability conditions as in Theorems 5.1, 5.2, 5.3,(1) the Fourier transform Qh ofwh satisfies (5.79) where the constant depends on 'Y and on the data. (l)No condition for Schemes 5.1, 5.2; conditions (5.43) for Scheme 5.3; conditions (5.56)› (5.57) for Scheme 5.4.

Theevolution Navier-Stokes equations

350

Ch.III, §5

Proof. The four equations (5.12)-(5.15) can be interpreted as d dt (wh(t), Vh)

= «(gh(f), Vh))h'

VVh E

fh'

t E

(5.80)

(0, T),

where the function gh satisfies

f: "g.

(5.81 )

(1)0. dt <;; Const.

For example, for Scheme 5.1, gh is defined by «(gh (t), vh ))h VV h E

fh,

= (1m, Vh) -

(m

-I)k~

bh (u~

- 1, u~ , vh) - v«u~

, vh ))h'

t c.mk.

Inequality (5.81) follows from (5.4), (5.7), and the previous a priori estimates:

Ilgh (t)lI h ~ do Ifm I + d l lIu~

J T

0

II h + v lIu~

- llih Ilu~

I h,

N

Ugh (I)I.dt <;; k

m~1

(do II"'I

+ d t IluZ' -I D. lIuZ' ". + vllu~lIh);

the right-hand side of this relation is bounded according to Lemma 5.1. Let us infer (5.79) from (5.80)-(5.81). Extendingg h by 0 outside [0, T] we get a function Kh such that the following equality holds on the whole t line: d )_ _ 0 ~ dt (wh (f), v h - «ih (t), "» ))h + (U h ' v h )u o - (U:' vh )5 T , VV h E

fh,

(5.82)

where 50’ 5T denote the Dirac distribution at 0 and T. By taking the Fourier transform, we then have -2i1TT(Wh(r), vh)

= (
Vh) -

(tit, Vh) exp( -2i1TrT);

(fh = Fourier transform of Kh)' Putting vh = ~h (r) and then taking absolute values we get 21Tlrll~h

(r)1 2 ~

11th (r)lI h lI~h

(r)lI h

+ cll~h

(r)l,

Discretization of the Navier-Stokes equations

Ch. III, §5

351

tJt

since uZ and remain bounded. Due to (5.81) we also have

J

T

Dg. (T)II•

.;;

0

11gb (1)1. dt’;; Const. =

c"

and, finally,

ITII~h

(T)1

2

~ c311~h

(T)lIh ·

(5.83)

For fixed 'Y, 'Y < 1/4, we observe that h ~

IT I

""""C4 ('Y)

I + ITI I + ITI1 - h ,¥TE6t

Hence

C~

I

ITI" IV. (T)I'dT .;; ~

c, h)r~

+\:I:~2>

IIV.(T)I'dT

(by (5.83»

.; c, h)

~

1

J

+00

_

IIV. (T)1'dr 00

(by the Schwarz inequality)

.; c, J_IIV. +00

ee

(T)I' dr

352

The evolution Navier-Stokes equations

Ch. m, §5

The integral

J

+ co

_ co

dr (l + Ir1 1 - 2’}’)2

is finite for 'Y < 1/4. Therefore the right-hand side of the last inequality is finite and bounded due to the Parseval relation and the previous estimations:

=d~

J T

0

Ilw. (t)l ~dt"

Const.

(5.85)

The lemma follows.

5.7. Convergence of the Numerical Schemes Our aim is to prove the convergence of Schemes 5.1 to 5.4, in some sense which will be described later on. We first state the consistency and compactness properties on the discretized data which are required to ensure the convergence. We then state and prove the convergence results.

5.7.1. Consistency and Compactness Hypotheses The subsequent hypotheses will be easier to state after this lemma: Lemma 5.7. Let {(Vh , Ph’ r h )hE:H:’ (w, F)} denote a stable and con› vergent external approximation of v. Let us assume that for some sequence h’ ~ 0, a family offunctions

uh ’ : [0, T]

~ Vh "

satisfies Ph,uh' ~ t/J in L 2(0, T; F) weakly, as h’ ~ O.

Then for almost every t, t/J(t) u

(5.86)

= wu(t), and

= w-1t/J EL 2(O, T; V).

(5.87)

Discretization of the Navier-Stokes equations

Ch.III, §5

Proof. Let us denote by checked that

J:

(J

353

some function in E»«O, T»). It is easily

Ph'Uh,(t)8(tldt -+ [

~(t)8(t)dt,

as h’ goes to O. But condition (C2) of Definition 1.3.6 (1) shows us that, under these circumstances,

~(t)6(tldt

[

E wV,

w

Since by definition, WV is isomorphic to V, V is a closed subspace of F; taking now a sequence of functions (J E converging to the Dirac distribution at the point s, s E (0, T), we see that for almost every sin [0, TJ,

J: ~(t)8,(t)dt

and hence

f/>(s) E Then, as

-+

~(s)

in F,

wV a.e.

wis an isomorphism, w- 1 f/> is defined and belongs

L 2(0, T; V).

D

to

The preceding lemma was quite general, but, in the present situation we assumed that Vh C L 2 (n ), V h;

(5.88)

therefore it can happen that for some sequence h’ -+ 0, "h’-+"

in L2(O, T; L 2 (n )) weakly,

Ph’"h’ -+

f/> in L 2(O, T; F) weakly.

By Lemma 5.6, f/> = W"... , u.. E L 2(O, T; V). Without further informa› tion we cannot assert that"
The evolution Navier-Stokes equations

354

Ch. III, §5

Let uh' be a sequence offunctions from [0, T] into Vh, such that, as h’ -+ 0. uh'-+u in L2(0, T; L 2 (0 )) weakly, Ph,uh' -+ rp in L 2 (0, T; F) weakly.

Then UE

L 2 (0, T; V) and rp

= Iou.

(5.89)

Besides (5.89), the consistency hypotheses are now the following

(1):

Let vh', wh', be two sequences of functions from [0, T] into Vh " such that, as h’ -+ 0, Ph,vh' -+ WV in L 2 (0, T; F) weakly, Ph'Wh' -+ WW in L 2 (0, T; F) (strongly).

Then, as h’ -+ 0,

J:

«v.,(I),

w.,(I)))., dr ....

J:

«v(f), w(f»)df.

(5.90)

Let uh', vh', be two sequences of functions from [0, T] into Vh , such that, as h’ -+ 0, Ph,uh' -+ WU in L 2 (0, T; F) weakly, uh' -+ U in L2 (Q) strongly, Q = 0 X (0, T),

and Ph,vh' -+ WV in L 2 (0, T; F) weakly.

Then as h’

J: J:

-+

0,

b.,(u.'(f), v.,(t), "'(fp..,w., )df ....

b(u(l), v(f), "'(f)w)df,

(1)Compare with the stationary case, (3.7), (3.8), Chapter I; (3.4), (3.5), (3.7), Chapter n.

Ch. III, §5

Discretization of the Navies-Stokes equations

355

for each scalar valued function l/I E L (0, 1) and each w E -r: If moreover a sequence of functions wk' is given with OO

l/Ik'

-+

l/I in L" (0,

then, as h’

f

-+

0, k’

1), as k’ -+ 0,

-+

0,

T

(5.91)

0 btutt), v(f), 1/!(f)w)df.

In order to prove strong convergence results as required by (5.91) (uh’ -+ U in L2 (Q ) strongly) we will assume the following: Let vh’ denote a sequence of functions from tR into Vh" with support in [0, T] and such that

f: "Vh’(f)"~.df

<; Const.,

+00

f __ITI" I~h’

(T)1' dr <; Const., for

some 0 < ’Y,

where vh’ is the Fourier transform ofvh’. Then such a sequence vh’ is relatively compact in £2 (Q). In particular, one can extract from Vh’ a subsequence (still denoted vh’) with Ph,vh’

-+

WVin L 2(0, T; F) weakly,

vh’ -+ v in £2(Q) strongly.

(5.92)

5.7.2. The Convergence Theorems The convergence theorems are stated differently according to the dimension of the space (n = 2 or 3) and to the scheme considered. We recall that we associated with the elements u’l! a function uh uh: [0, T] -+ Vh ,

356

The evolution Navter-Stokes equations

Ch.II1, §5

defmed slightly differently for the four schemes (see (5.36), (5.55), (5.75)) (1) for (m - 1)k <, t < mk (m

uK’ uh (t)

= 1, ..., N) (Schemes 5.1 and 5.3)

1

= -2 (uhm + uhm -1)

(Scheme 5.2) (Scheme 5.4)

(5.93)

We have: Theorem 5.4. The dimension of the space is n = 2 and the assumptions are (5.1) to (5.7), (5.9), (5.10), (5.41), and (5.89) to (5.92). We denote by u the unique solution of Problem 3.1. The following convergence results hold, as hand k -+ 0, uh -+ u in L 2(Q) strongly, t: (0, T; L2 (!2)) weak-star, (5.94) Phuh -+ WU in L 2(0, T; F) weakly, provided: (i) (ii)

(5.95)

Scheme 5.1: no condition, Scheme 5.2, kS 2(h) -+ 0,

(5.96)

(iii) Scheme 5.3: (5.43) is satisfied, (iv) Scheme 5.4: (5.56)-(5.57) are satisfied.

. Remark 5.3. (i) For Schemes 5.1 and 5.2 it can be proved, without any further hypotheses, that Phuh

-+ WU

in L2(0, T; F) strongly, as h, k -+ 0.

(5.97)

(ii) The same results hold for the other schemes provided we also assume that

kSf(h) -+

°

and kS 2(h) -+

°

(Schemes 5.3 and 5.4).

(5.98)

(iii) The hypotheses (5.96) used in the proof of the convergence of Scheme 5.2 is probably unnecessary since the scheme is unconditionally (1)We emphasize that uh depends on h and k; only for reasons of simplicity have we denoted this function by uh instead of uhk’

357

Discretization of the Navier-Stokes equations

Ch. III, §5

Theorem 5.5. The dimension of the space is n = 3 and, otherwise, the assumptions are the same as in Theorem 5.4. Then, there exists some sequence h’, k’ ~ 0, such that uh' ~ U

in L2 (Q) strongly,

uh’~u

in LOO(O, T; L 2 (n )) weak-star,

Ph,uh' ~ WU

in L 2(0, T; F) weakly,

(5.99) (5.100) (5.101)

where u is some solution of Problem 3.1. For any other sequence h’, k’ ~ 0, such that the convergences (5.99) to (5.101) hold, u must be some solution of Problem 3.1.

Remark 5.4. We are not able to prove that the whole sequence con› verges due to lack of uniqueness of solution for Problem 3.1. We also cannot prove strong convergence in L 2 (0, T; F) due to the lack of an energy equality for the exact problem (Problem 3.1) (for n = 3 we only have an energy inequality; see Remark 4.1). D The two theorems are proved in the remainder of this Section 5.7; we will prove Theorem 5.4 with full details for Scheme 5.1 (including (5.97)) and in the other cases we will only sketch the proofs which are actually very similar. 5.7.3. Proof of Theorem 5.4 (Scheme 5.1) According to the stability theorem (Theorem 5.1), and to (5.89), there exists a sequence h’, k’ ~ 0, such that uh' ~ u

in L" (0, T; L 2 (n)) weak star,

Ph,uh' ~ WU

in L 2(0, T; F) weakly,

(5.102)

for some u in L 2 (0, T; V) n L (0, T; H). Let us consider the piecewise linear function wh introduced in Section 5.6 (see (5.78»). By Lemma 5.6 and the estimations on the u~, we have 00

1Wh ILoo(O, T; L2(,o)) ~

Const.,

Ilph wh IIL 2 (0,

Const.,

T; F)

~

358

f__

The evolution Navier-Stokes equations

Ch. III, §5

+00

ITI "lilI. (T)I'dT <; Const.

Hence, according to (5.92), the sequence h’, k’ that Wh' -+

W

-+

0 can be chosen so

in L" (0, T; L 2 (0)) weak-star,

wh'-+w in L 2(0, T; L 2(0)) strongly, Ph,wh' -+ WW in L 2(0, T; F) weakly,

where

WE

(5.103)

L 2 (0, T; V) n L (0, T; ll). 00

We now observe that: Lemma 5.8. Uh-Wh-+OinL2(0,T;L2(0)) strongly as h and k-rts. (5.104) Thus W=U

(5.105)

Uh'-+U in L 2(0, T; L 2(0)) strongly,

(5.106)

and

as h’ and k’ -+ O. Proof. Exactly as in Lemma 4.8, we check that IUh - wh IL 2(O, T; L 2(O » =

A (.E ~

3

N

m=1

Ium _ u m h

h

11 2

)

1/2

(5.107)

Then (5.104) follows from the maioration (5.22); (5.105), (5.106) are obvious consequences of (5.102), (5.103), and (5.104). The next point is to prove that U is a solution of Problem 3.1. Lemma 5.9. The function uappearingin (5.102), (5.103), (5.105)isa solution ofProblem 3.1. Proof. We easily interpret (5.12) in the following way:

Ch.lII, §5

Discretization of the Navier-Stokes equations

+ bh (Uh (t - k),

Uh (t), Vh)

= (fk(t), Vh)’ V

t E [0, T], VVh E Vh .

359

(5.108)

where fk(t)=f m , (m -1)k<.t<mk.

(5.109)

Let v be any element in "Y and let us take vh =’h v in (5.108). Let I/J be a continuously differentiable scalar function on [0, T], with I/J(T) = O.

(5.110)

We multiply (5.108) (where vh = ’hv) by I/J(t), integrate in t, and integrate the first term by parts to get:

~

L T

(w.(t), \/i(l)r.v)dl + v

fob. T

+

1.

(u. (t - k), u. (I),

T

+

f.

T

(i,(t),

~(I)r.

«u.(t), '/J(t)r.v)).dl

v)dt = (u2, r. v),p(O)

~(I)r.v)dt.

(5.111)

We now pass to the limit in (5.111) with the sequence h’, k’ ~ 0 using essentially (5.90), (5.91), (5.102), (5.103), and Lemma 5.8; we recall also that ’h v ~

u2 ~

A

WV in F (strongly), Uo

in L 2 (n) (strongly)(l),

~ fin L2(0,

T; L2 (n )) (see Lemma 4.9).

(5.112) (5.113) (5.114)

We find in the limit (l)We recall the proof of (5.113); due to (5.1l) it suffices to prove this for uQ E ?’

case

a nd in this

360

-f f L

The evolution Navier-Stokes equations

T

J

Ch. III. §S

T

0 (u(I), 1/i(l)v)dl

+V

0 «u(I), 1/I(1)v»dl

T

+

0 b(u(t), u(I), 1/IV)dt = (u O• V)1/I(0)

T

+

(5.115)

({(I), v)1/I(1)
We infer from this equality that u is a solution of Problem 3.1, exactly as we did in the proof of Theorem 3.1 after (3.43). Since the solution of Problem 3.1 is unique (see Theorem 3.2), a contradiction argument that we have already used very often shows that

The convergences (5.102), (5.103) hold for the whole family h. k -+ 0.

(5.116)

This completes the proof of Theorem 5.4. 5.7.4. Proof off 5.97)

For the sake of completeness we will also prove (5.97). In order to prove this point we need a preliminary result which is quite general and interesting by itself.

Lemma 5.10. Let {(Vh. Ph’ rh)jf, (w. F)} be a stable and convergent external approximation of V. For a given element v of L 2 (0, T; V), one can define for each h E j( a function vZ E L 2 (0, T; Vh ) such that

Ph vZ

-+

WV in L 2 (0, T; F) as h -+ 0.

Proof. The proof is essentially the same as that of Proposition 1.3.1. The result is obvious if v is a step function; since the step functions are dense in L2 (0, T; V), the result follows in the general case by an argument of double passage to the limit as in Proposition 1.3.1. Lemma 5.11. The dimension of the space is n

Ph Uh

-+

WU in L 2 (0, T; F) (strongly),

= 2; then for Scheme

5.1

(5.117)

361

Discretization of the Navier-Stokes equations

Ch. III, §5

as hand k -+ O. Proof. We consider the expression N

x, = It{ - u(T)1 2 + mL;:1 lu~ - u~

J T

+ 2"

0

Ilu. (t) -

uZ (t) I ~

-

11 2

dr,

with u~ defined as in Lemma 5.10. According to (5.21)(Lemma 5.1),

luf I ~ Const.; hence there exists a sequence h’, k' -+ 0, with

zit. -+ X in £2 (n) weakly.

(5.118)

We temporarily assume that

(X - u(T), v)

= 0, V v EH.

(5.119)

and we prove that

Xh , -+ O. We set where

Xl = luml' + 2"

J T

0

(by Lemma 5.10 and (5.90)),

X~

~

lIu:; (t) 0 dr -+ lu(1')I' + 2"

J.

J T

0

Ilu(l)II' dz

T

= -2(tI,[, u(1') - 4"

J

T

- 4"

0 nu(t) It’dl

«u.(t), uZ(t)). -+ -2Iu(1')I'

362

Ch. III, §S

The evolution Navier-Stokes equations

(by Lemma 5.10, (5.90) and (5.118)-(5.119), we recall that u(T) EH),

and

N

X~ =lu:1 2 + m=l L N

N

L

m=l

By summation of the equalities (5.26) for m = 1, ..., N, we get N

X3 h

= luol2 + 2k h

=

2

lu21 + 2

It is then clear that

X~

L tr". uhm )

m=l

J:

1.

if. ro. u. (t)dl.

T

-+

2

luo 1 + 2

(((t), u(t))dt,

as h. k -+ O.

Hence X

h,-+ luol2 + 2 fT (I(t), u(t))dt -lu(T)1 2 - 2vfT IIu(t)II 2dt,

°

°

(5.122)

and this limit is 0 due to (4.55). By a contradiction argument we show as well that the whole family Xh converges to 0: X h -+ 0, as h. k -+ O.

In particular

and

363

Discretization of the Navier-Stokes equations

Ch. III, §5

J:

r

Ip.u. (t) - WU(I)

+

n~dl

<: c

np.u'(t) -

I1: wU(I)I~dl

Ilu. (I) - u. (I)

R~ dt

1-+ 0

and (5.117) follows. It remains to prove (5.119). By summation of (5.12) for m = 1, ..., N, we get N

('1 - u2• vh ) + kv m~1 «u:. vh », N

N

b h (u: - 1. U: Vh) = k

+ k m~-1

I:

:E

m =1

(1m. vh ).

Taking vh = rhv. v E’Y, we easily pass to the limit and get (x - u o' v) + v

L

«u(I), v»dl +

J:

biutf), u(I), v)dl

T

=

(((I), v)dl. Vv E’Y.

But since we deduce by integration of (3.13) a similar equation with X replaced by u(1), we conclude that

(X - u(1), v)

= 0, V v E or.

which implies (5.119) by density. The proof of Lemma 5.11 is complete.

5.7.5. Proof of Theorems 5.4 and 5.5 (other cases) For Schemes 5.2, 5.3, 5.4 and in the case n = 2, the proof is very simi› lar to the above, using the corresponding a priori estimates. For Scheme 5.2, we introduced the condition (5.96) as a sufficient

364

The evolution Navier-Stokes equations

Ch. III, §6

condition to prove (5.104); more precisely, in this case, the analogue of (5.104) is a consequence of (5.77), (5.107) and (5.96). For Schemes 5.3, 5.4, the stability conditions (5.43), (5.56), (5.57) merely ensure that u h and Phuh remain bounded in the suitable spaces. For the proof of (5.97), the condition (5.98) appears as follows: - For Scheme 5.4 the "natural" expression similar to X h in Lemma 5.11 is N

f.

L

m=1

T

+ 2.

lIu,(t) -

uZ (t) Iii dt;

in order to deduce (5.97) from the fact that Yh -+ 0, it suffices that N

L

m=1

Ium _ u m - 11 2 -+ 0, h

h

and this is a consequence of (5.98). - For Scheme 5.3, we consider the same expression X h , but due to some terms involving bh (see (5.47)), the expression X~ is not as simple as (5.121); (5.98) shows that the supplementary terms of X~ involving bh converge to O. If n = 3, we observe first that the stability conditions are not explicitly mentioned in the statement of Theorem 5.5. Actually it is with the help of these conditions, and the a priori estimates that they imply, that we prove the existence of a subsequence h’, k’ -+ 0, such that (5.99)› (5.101) hold; that u is a solution of Problem 3.1 is proved exactly as before.

§ 6. Discretization of the Navier-Stokes Equations: Application of the General Results. We want to state explicitly all the hypotheses and conclusions of the stability and convergence theorems (Theorems 5.1 to 5.5) for specific approximations of the space V. We recall that the final form and effect› iveness of the Schemes studied in Section 5 are also based on the choice

Discretization of the Navier-Stokes equations

Ch.III, §6

365

of the approximation of V, i.e., the discretization in the space variables. In Section 6.1 we consider the approximation of V by finite differ› ences (approximation APX1). Section 6.2 deals with conforming finite element methods: the approximation of V being one of the approxi› mations (APX2) to (APX4). Section 6.3 deals with non-conforming finite element methods (approximation APX5). Then in Section 6.4 we study some algorithms adapted to the practical resolution of the finite dimensional problems (i.e., practical computation of u’f: for one of Schemes 5.1 to 5.4). 6.1. Finite Differences (APX 1)

The general approximation of V considered at the beginning of Section 5.1 is at present taken to be the specific approximation (APX 1). We will successively check and interpret in this case the hypotheses of Theorems 5.1 to 5.5. 6.1.1. Computation of S(h).

The conditions (5.3) and (5.4) are obviously satisfied; according to Proposition 1.3.3, do = 2Q,

IU h I ~ do lIu h Ilh ,

(6.1)

where Q is the smallest of the widths of.n in the directions x l’ ... , x n ’ The purpose of the next proposition is to verify (5.5) and to give an ex› plicit value of S(h). Proposition 6.1. For the approximation (APX 1)

2( ~

n

S(h)

=

Proof. We have n

2

lIuh 11 h n

~2

L

i.] = 1

=

L

t.i = 1

~ h~

I

)1/2

,

(6.2)

The evolution Navier-Stokes equations

366

IL I

n

:E

~4

h}

i,j= 1

(t, h:)

<4

Ch. Ill, §6

IU/h(xll' dx

Iu, I',

I

and (6.2) follows: 6.1.2. The form bh and Sl (h)

For the approximation (APXl), we choose again the form bh defined in Section 3, chapter II (see (3.27) to (3.29)). bh(u h• v h• w h)

= b~(uh’

v h' w h)

b~(Uh’

= b~(uh’

w h' v h)·

v h' w h)

+ b~(uh’

vh' w h)

(6.3)

(6.5)

It is clear that bh is a trilinear continuous form on Vh X Vh X Vh and that (5.6) holds; in order to get some estimate of the constant d 1 in (5.7) we will prove Lemma 6.1. For n Ib~(Uh’

= 2 or

v h' wh)1

~~Iu """...;2 ~ Ib~(uh’

3

h

11/211u h 1I h1/211vh IIlw 11/ 2 11 w 1I1/ 2 (n = 2) h h h h

33/21uh 11/ 4 lIuh IIf4

ilvh " h IWh 11/ 4 IIw h ":/4(n = 3), (6.6)

v h' wh)1

~V ~

2

Iuh 11/211uhh 1I 1/21vh 11/2 11 vhh IP/2I1whh II (n

~ 33/ 21u h 11/4 11uhh 11 3/4 Ivh 11/411vhh 11 3/411w hh II (n

= 2)

= 3).(67) ,

Proof. This proof is based on Proposition 11.2.1. Due to 11.(2.12) and 11.(2.13),

Discretization of the Navier-Stokes equations

Ch. III. §6

n

~1

n

Ilu ih II i4(n)

~

3y'2

~1

IU’h I ( Z

n

~j= 1

367

8 ’h UZ’h 12

)

1/21

1 j

(6.8)

~

(by Holder’s inequality)

(6.9)

Using Holder’s inequality again we now have

Then, ifn

Ib~(uh’

= 2, we apply v h wh)1

(6.8) to get:

~ ~

IUhI1/2I1uhll~/2I1vhllhlwhl~/2

I whll~/2,

Theevolution Navier-Stokes equations

368

Ib;;(u h, Vh, wh)1

~ ~

IUh 11/2I1uhll~/2Ivh

Ch. Ill, §6

11/2I1vhll~/2I1whllh’

For n = 3, using (6.9) we also obtain the results stated in (6.6) and (6.7). Lemma 6.2. Ibh(u h, vh' wh )! ~d1l1uhllh

IIvh llh IIwhllh, YUh' Vh' Wh E

v" ,

(6.10)

with

d1

= 6y'2Q

if n = 2, d 1

= 63/2-J2

if n = 3.

(6.11)

Proof. An immediate consequence of (6.1), (6.3), (6.6) and (6.7). Proposition 6.2. For the approximation (APX1), the inequality (5.41)

holds with

S1 (h) S1 (h)

= 3v'2 S(h)

if n

= 2.33/2 S3/2(h)

= 2,

if n

= 3,

(6.12)

where S(h) is given by (6.2).

Proof. Using (6.1), (6.6) and (6.7) we can write:

Ib~(uh,uh’

wh)l~

~

IUhI1/2I1uhll~/2IwhI1/2I1whllf2 3

~ v'2 S(h)lu h IlIuhllh IWh I

Ib~(uh’

(n = 2),

uh' wh)1 ~ 3 3/21u h 11/4I1uhll~/4I1vhllh

Ib;;(u h, Uh' wh)1

IWh 11/4I1whll~/4 ~ 33/2S3/2(h)luh IlIuhllh IWh I (n = 3),

~ ~

Iuhllluhllh IIwhllh

3

~ v'2 S(h)lu h IlIuh IIh IWh I

(n

= 2),

Ib;;(u h, Uh' wh)1 ~ 33/2IuhI1/2I1uhll~/2l1whllh

~ 3 3/ 2 S3/2(h)lu h IlIuh IIh IWh I

Then, we easily obtain (6.12).

(n

= 3).

Ch.III, §6

Discretization of the Navier-Stokes equations

369

6.1.3. Application of the stability and convergence theorems Schemes 5.1 and 5.2 are unconditionally stable. The stability condi› tions of Scheme 5.3 are given in Theorem 5.2 and Lemma 5.3:

k(t h7~) i= 1

3

k

(

ifn

E:;;

min

I

) 3/2

~1 ~

E:;;

I

(~2

33 2

d")

’

2:~3

(6.13)

3

~

k (

= 3,

if n = 2,

’22

(6.14)

where d’ and d" are defined in the proof of Lemma 5.3. For Scheme 5.4 the stability conditions given in Theorem 5.3 and 5.4 are

k (

t,

if n = 2, 3

k (

~

1

h!I

)

E:;;

1- 5

~

,k

(

L2

i= 1

1)

hl

v5

E:;; 32 26d

1-8 ( L 1)’" h!"1) 24V; 3

E:;;

k

I

i= 1

-

h~I

if n = 3,

E:;;

s ’ v5

2 83 3ds ’

for some 5, 0<5 < 1, and d s is given by (5.58). When applicable, the convergence Theorems 5.4 and 5.5 give "h -+"

in L 2 (Q) strongly, L (0, T; L 2(n» weak star, OD

2

5ih " h -+ Di " in L (Q) strongly or weakly

(6.15)

(6.16)

(6.17) (6.18)

(depending on whether convergence in L2(0, T; F) is strong or weak). Theorems 5.4 and 5.5 are based on the hypotheses (5.89) to (5.92). We must show that these conditions are met; this technical point will be considered now. 6.1.4. Proof of (5.89)-(5.92) Verification of(5.89). If "h’-+" inL2(0, T; L2(n», weakly,

(6.19)

(l )The second condition in (6.14) is consequence of the first one for h sufficiently small.

370

Theevolution Navier-Stokes equations

Ch. III, §6

= (f/Jo’ . , f/J n ) in L 2(0, T; F) weakly, (6.20) then, by Lemma 5.7, f/J must be equal to WV, where v E L 2(0, T; V). On Ph'U h, -+ f/J

the other hand, by the definition of Ph '

L 2 ( Q ) weakly,

Ph,uh’-+ct>O=~in

(6.21)

and the comparison of (6.19) and (6.21) shows that v = u, as elements of L 2 (Q). Hence ct>(t)

= Wu(t) a.e.,

and U E L 2(0, T; V).

Verification of(5.90). If Ph' v h' -+ W v In -

.

2

L (0, T; F) weakly,

Ph' Wh , -+ W W In

2

L (0, T; F) strongly,

then 2

6ih'Vh'-+ Djv in L (Q) weakly, 6ih,wh'-+ Diw in L 2 (Q) strongly,

and it is clear that n

E -+ (

{

t,

(

D,v.D,

s:

w) dx dt =

«v(f), w(i)))dt.

Verification of(5.91). The proof is similar to the proof of Lemma 3.1, chapter II. Let us assume that uh'-+ u in L 2 ( Q) strongly, -

.

(6.22)

2

(6.23)

Ph' Vh' -+ W v In L (0, T; F) weakly,

"’h’ -+ '" in L" (0, T),

(6.24)

and that w is a fixed element of 1"". It follows from (6.23) that vh'-+ v in L 2 (Q) weakly, 6jh'Vh,-+DjvinL2(Q)weakly,

(6.25)

1 ~i,

;~n.

(6.26)

en, III,

Discretization of the Navier-Stokes equations

§6

371

On the other hand, as observed in the proof of Lemma I1.3. in the norm of LOO (S1),

rh W -+ W

(6.27)

6ih rh w -+ D, w in the norm of L OO (S1), 1 ~ i ~ n.

I: b~

(6.28)

We consider first the form b~ :

1

2

», (I), Vh (I), "’h v», w) dr

i I In

i,j= 1

OT

Uih (6 ih V,"h) VJh W,"h dx

dt.

f: L

It is easy to see from (6.22), (6.24), (6.26) and (6.27) that, for each i

and i

I:I

n

U’h’ (6,.. vjh') "'. wjh' dx

dr

~

u, (D’Vj)"’Wj dxdr,

and hence

f:

Similarly,

b~,(uh,(t),

-} [

Vh' (I), "'.,(I)rh'w) dl

b(u(I), 1/J{t)w, v(t)) dr

=~

~

J:

b(u(I), v(I), 1/J{t)W) dl,

and (5.91) is proved.

Verification 0/(5.92). Let

denote a sequence of functions from ~ into v" with support in [0, T] or any fixed compact subset of ~ and such that {V

h,}

372

The evolution Navier-Stokes equations

f: . .,(t)n~, f., ITI

f:

Ch. III, §6

(6.29)

dt " Const.,

2719.,(T)1 2

dT" Const.,

(0 < 7)·

(6.30)

We must prove that this sequence is relatively compact in L 2 ( Q). Since the Ph are stable, we have (6.31)

UP.’ v.,(I)II; dt "Const.,

and hence, the sequence h’ contains a subsequence (still denoted h’), such that Ph'

vh '

-

-+ W

. 2 v In L (61; F) weakly,

(6.32)

ve : v in L 2(61; L 2(S"Z)) weakly,

(6.33)

v being some element of L2(61; V) which vanishes outside the interval

[0, T] (here we have used (5.89)). It suffices to prove that the con› vergence (6.33) holds in L2(61; L2(S"Z)) strongly, which amounts proving that one of the following expressions converges to zero: lh’

f

+ OO

= _

00

IVh’(t) - v(t)1 2 dt

= (by

the Parseval equality)

(6.34) Following the proof of Theorem 2.2, we write lh’

=

J

I'TI<;M

I~h,(r)

- V'(r)1

2

dr

Discretization of the Navier-Stokes equations

Ch. III, §6

~J

l~h•(T)

-

~(T)12

373

dr

ITI<;M

~

(by (6.1), (6.29), and (6.30))

For a given e > 0, we choose M such that C -1-+-M--'l r-’Y ~

2" .

Hence (6.35) where

J h, =

J

l~h•(T)

-

~(T)12

dr

(6.36)

ITI<;M

and the convergence of I h , to zero will be proved if we show that J h , -+ 0,

as h’, k’

-+

O.

(6.37)

This will follow from Lebesgue’s Theorem as we now show. For every T, ~h (T) is equal to

or (6.38)

374

Ch. III, §6

The evolution Navier-Stokes equations

where X is a smooth function with compact support, equal to 1 on [0, T] (so that vh ' = XVh" Y h'). Then 1Vh'(T)1 E;;; 1Vh,IL2(iR;L2(n» lIe-2i1l'tTxIIL2(iR) E;;; Const.

l~h,(T)

- ~(T)12

E;;;

2(Cr + 1~(T)12),

V T,

= C1

(6.39) (6.40)

and, according to (6.33) and (6.38): 9.,(T) -+ 9(T)

J__

=

+00

v(I)x(I) .-’’’’’’’dr,

in L 2(O) weakly,

(6.41)

for every T. We also have, due to (6.38),

This last estimate and the discrete compactness theorem (Theorem 11.2.2 and Remark 11.2.4) show that the convergence result (6.41) holds also in the norm of L 2(O): l~h,(T)

- ~(T)12

~ 0, Y T.

An application of Lebesgue’s Theorem gives (6.37) as a consequence of (6.40) and the preceding convergence results. 6.2. Conforming Finite Elements (APX2) (APX3) (APX4)

The general approximation of V considered at the beginning of Section 5.1 is now taken as one of the approximations (APX2) (APX3) (APX4). We will check and interpret the hypotheses of Theorems 5.1 to 5.5. For all these methods,

Vi,

CHfi(O),

lIuh llh

= Iluhll, YUh

E Vh ,

(6.42)

and Ph is the identity, V h. 6.2.1. Computation of S(h) The conditions (5.3) and (5.4) are obviously satisfied; according to the Poincare Inequality (see Chapter I (l.9))

Discretization of the Navier-Stokes equations

Ch. III, §6

IU h I ~ dO lIu h Ilh ’

do = 2Q,

375

(6.43)

where Q is the smallest of the widths of n in the directions xI' ..., x n ’ For (5.5) we have

Proposition 6.3. For the approximations (APX2) to (APX4),

S(h)=~

p’(h)’

where cq is a constant depending on the degree q of the elements

(6.44) (1).

Proof. The space J.h is a space of polynomials of degree less than or equal to q = 2, 3 or 4 for the approximations (APX2), (APX3), or (APX4). The stability constant S(h) is a bound of the square root of the supremum

(6.45)

SuP

uh E vh

This supremum is related to the supremum of

J

[grad cP(x)1 2 dx

y

(6.46)

among all functions cP which are polynomial of degree less than or equal to q, and all //E!liz. Actually, let J.l q be a bound for this supremum; we then write for each i = I, ... , n, and such // E 5h

(1) p’(h)"is defined in Chapter 1, (4.19).

376

Ch. III, §6

The evolution Navier-Stokes equations

where uh belongs to v" and q has the appropriate value corresponding to the actual space v". Then by summation in i and 9’: n

lIuh

I ~ = pi~

f

2

lgrad Um (x )1 dx

n

~

n

JJ. q

L:

~I

f

2

1Uih (x )1 dx

n

= JJ. q IUhl2

(6.47)

and we can take (6.48)

S(h) =VJJ. q

In estimating (6.46) we use a linear mapping A,

»<ss, which maps 9’ onto a reference n-simplex

ll:

n

a ~Xi~

1,

~ xi~

r-I

1.

The function t$J (Ax) is a polynomial of degree less than or equal to q on S. We observe that

{J_

[grad

'Ii (Xl12 5

} and

9’

{f_

I'Ii(Xl12 dX} ,

9’

are a semi-norm and a norm on the finite-dimensional space of poly› nomial functions of degree less than or equal to q onll. Hence, there exists a constant liq depending only on q (and9’) such that,

t

lgrad 'Ii (X)I2 5';;

:J’

for all such polynomials

~

FI

f

_ .'/

»,

f/

'Ii (Xl12 5,

1/1. In particular,

~

I -aa t/>(&)1 2 dX liq Xj

for the considered polynomials
f

_

Y

It/> (&)1 2 dX,

(6.49)

Discretization of the Navier-Stokes equations

Ch.III, §6

377

But

i I ~(x)12

k=1

=

aXk

~

± i ~cf>(AX)ArfcI2l

k=1

)1

IIAII 2

j=1 aXj

~

1=1

)1

~

aXj

We infer from this and (6.49) that

La~:

~1

I

(AX)I' dx .. jt. IIAII'

y

~

1

cf>(AX)1 2

i

(1)

J_I~

(AXlI'
y

and coming back to the simplex 9’by a new change of variable we find

J

y lgrad

~(x)I’

dx .. Ii. IIAII'

L’~(X)"

dx.

According to Lemma 4.3, Chapter I, p-:IIAII~1.

c-

p~

For the triangulation considered hence

p'(h);

py

IIAII~

p'(h)

and we can take

_-(PY)2 p’(h") .

J,lq - J,lq

(1) IIAII is the norm of the linear operator A associated with the euclidian norm in ~n.

x = ,A-1 x ,

The evolution Navier-Stokes equations

378

Ch. III, §6

Thus (6.44) holds with cq

=VTIq p7·

(6.50)

6.2.2. The form bh and 8 1 (h)

We take again for the approximations (APX2) to (APX4) the form bh defined in Section 3, Chapter II (see (3.55»: bh (uh' vh' wh)

= ~2

b'(u, v, w) b"(u, v, w)

= b'(uh' vh' wh) + b"(Uh, vh' wh)

±J

i,/=1

(6.51)

u.(D.v/)w. dx

n

(6.52)

"J

=- b'(u, w, v),

(6.53)

The form bh is trilinear continuous on JJ, X JJ, X JJ, and (5.6) holds; an estimate of the constants d 1 and 8 1 (h) will follow from the next lemma. Lemma 6.3. For n = 2 or 3, and for u, v, win Ha(n): 1

Ib'(u, v, w)1 <; y'2lu11/211U1l1/211vlllw11/211w1l1/2

<; lul1/411ull3/411vlllwl1/411wll3/4 Ib"(ii, v, w)1

<;~

lul1/211ull1/21vl1/211vll1/211wll

<; lul1/411ull3/41vl1/411vll3/411wll

(n = 2) (n = 3) (6.54) (n

= 2)

(n

= 3)

(6.55)

Proof. For n = 2 the result is proved in Lemma 3.4, observing that b (u, v, w)

= "21 b (u, v, w)

b" (u, v, w)

=- "21 b(u, w, v).

I

For n = 3, the proof is based on Lemma 3.5: 3

L: J f

Ib'(u, v, w)1 <;.!. 2 i,/=1

n

IUi(Div/)wjl dx

(6.56)

Discretization of the Navier•Stokes equations

Ch. Ill, §6

379

3

~ ~2 1,1=1 .2: lIuillL4 (n ) IIDivIII L2(n)lI wpIL4(n) 3

1/2

~ ~ (i~1I1DiVI l12

(n) )

(~

3

(n»)

lIu ill1 4

1/2

(Ji II 3

1/2

wI Il14 (n) )

Due to (3.67), 3

~

3

lI uill14 (n) ~

2

~

(lluilll'1n) llgrad uil i’~n»

(6.57) Similarly, 3

2: IIwI

1=1

11

2

4

L (n)

~

2/w/ 1/2I1wII3/ 2

(6.58)

and then (6.54) follows from (6.56). For b" we simply observe that

b"(u, v, w)

= -b’(u, w, v),

and apply (6.54). Lemma 6.4. Ibh (uh’ vh’ wh)1 ~ d 1 II u h II II v h II II w h II, V uh’ vh’ wh E Vh , (6.59)

with d1 =

2V2f if n = 2, d 1 = 2 3 /2../Q if n = 3.

(6.60)

Proof. An immediate consequence of (6.43), (6.51), (6.54) and (6.55). Proposition 6.4. For the approximation (APX2) to (APX4), inequality (5.41) holds with

81 (h)

= Vi 8(h) if’n = 2, 81 (h) = 28 3 /2 (h) if n = 3,

(6.61)

Theevolution Navier-Stokes equations

380

Ch. III, §6

where S(h) is given by (6.44).

Proof. Using (6.43), (6.54) and (6.55) we write: , 1 Ibh (Uh' vh' wh)1 E;;; V2 S(h)luh IlIuh IIlwh I (n = 2) Ibj,(Uh' Vh' wh)1 E;;; S3/2(h)luh IlIuh IIlwh I (n = 3) Ibl:(Uh' Vh' wh)1 E;;; ~

S(h)luh Illuh IIlwh 1 (n

Ibl:(Uh' Vh' wh)1 E;;; S3/2(h)luh 1 lIuh IIlwh

I

(n

= 2)

= 3)

6.2.3. Application of the stability and convergence theorems

Schemes 5.1 and 5.2 are unconditionally stable. The stability conditions of Scheme 5.3 are given in Theorem 5.2 and Lemma 5.3:

k

[p'(h)]2E;;;min

k

(d’ d")

d’

4crcq

k

ifn=2,

(6.62)

d"

[p'(h)J3 E;;; 4cJ' [p'(h)]2 E;;;

cf ifn = 3,

(6.63)

where d’ and d" are defined in the proof of Lemma 5.3, and cq is defi› ned in the proof of Proposition 6.3. For Scheme 5.4, the stability conditions given in Theorem 5.3 and Lemma 5.4 are. k [p'(h)]2 E;;;

1-5

4vc~’

k

~

1- 6 k k [p' (h)]2 E;;; 4vcr [p' (h) J3 E;;;

lf

v5

[p'(h)]2 ....

16c~ds

32c~

v6 d

1

s

- 2

n-

(6.64)

,

if n = 3,

(6.65)

for some 6, 0<6 < 1, and d s is given by (5.58). The interpretation of the convergence theorems (Theorem 5.4 and 5.5) is very simple in the present case since F = (n); we have: uh -+ U in L2 ( Q) strongly, uh -+ U in L 2(0, T;

H6(n»

Hb Loa (0, T; L 2 (n» weak

star,

(6.66)

strongly or weakly (depending on whether convergence in L2(0, T; F) is strong or weak). (6.67)

Discretization of the Navier-Stokes equations

Ch. III, §6

381

The hypotheses (5.89) - (5.92) on which the proof of these theorems is based, are very easy to check and we will deal with this point very rapidly. Verification of the hypotheses (5.89) to (5.91). (n), and the operators w The condition (5.89) is evident as F = and Ph are identity operators. The condition (5.90) amounts to saying that if

HA

vh'''''' v in L2(0, T; HA(n)) weakly, wh''''''

win L2(0, T; HA(n)) strongly,

then

[ «V.’ (I), w.,(I)))dl -+ f:«V(t), w(t)))dl; this is obvious. One can prove (5.91) as done several times before in similar situat› ions(l). Finally, in the present cases, the condition (5.92) is contained in Theorem 2.2. 6.3. Non-conforming finite elements (APX5) The general approximation of V considered at the beginning of Section 5.1 is now chosen as the approximation (APX5) (piecewise linear nonconforming finite elements). We will successively check and interpret in this case the hypotheses of Theorems 5.1 to 5.5. 6.3.1. Computation ofS(h) The conditions (5.3) and (5.4) are obviously satisfied; according to Proposition 1.4.13. (6.68) where do = c’ (n, a) is a constant rather difficult to express explicitly and depending on n and a (see (4.179) - a has the same signification as in 1.(4.21)). (1) Recall that WE?,; rhW->W in LOO(Q),Di’hw->Diw inC (Q), 1 0;;; t « n, ash all flmte element methods considered.

->

0, for

382

Ch. III, §6

The evolution Navier-Stokes equations

Proposition 6.S. For the approximation (APX5), S

- co(n) (h) - p'(h) ,

(6.69)

where Co (n) is a constant depending only on n, and p’(h) is defined as in 1.(4.20).

Proof. The proof of Proposition 6.3 is valid. The constant S(h) is a bound of the square root of the supremum

J

~

n Igrad

L

~

u,. (xli’ dx (6.70)

lu,. (x ll’ dx

It can be shown as in (6.48) that S(h)2is bounded by the supremum III of the expressions

f, ~ LI~(xl’dx Igrad

(x ll’ dx

among all linear functions (j), and all Y E.!Jh . Using a linear affine mapping A which maps x-simplex Y of eRn :

:I on a reference

n

o ~xi~

I,

~ s,«

1=1

I,

we see that

-

III

where

= III (

p;

2

p'(h) )

iil is the supremum among all linear functions

1/1 of the ratio

Discretization of the Navier-Stokes equations

Ch. Ill, §6

J.~

Igrad

\ t;

383

~(x)I’dX

111’ (xli’ dX

Hence (6.69) holds with

Co (n)

= v’ii;. P:;

(6.71 )

6.3.2. The form bh and 8 1 (h) For the approximation (APX5) we choose again the form bh defined in Section 3.1, Chapter II (see (3.80) to (3.82)):

b’ (ua, va’ w" l = ~ 1.~1 bi,' (uh' Vh' Wh)

=-

(6.72)

L

",a (D’a vIa lW/a dx,

(6.73) (6.74)

b,,(Uh' Wh' vh)'

X v" X v" and that (5.6) holds. An indication on the value of the constant d 1 in (5.7) will follow from next Lemma:

It is clear that b", b,,' and bh are trilinear continuous forms on

Lemma 6.5. For n

v"

=2

Ib,,(Uh' Vh' wh)1 E;;; C1 (n, a, E)luhI1-e/2I1uhllh+e/2I1vhllh /Wh I1- e/2I1whllh+e/2

(6.75)

Ib,,' (uh' vh' wh)1

:e;;;; c1 (n, a, E)luhI1-e/2I1uhllh+e/2IvhI1-e/2I1vhIl1+e/2I1whllh

for any 0 < E < 1, where For n = 3

c1

en, a) depends only on n, a and E.

bi,(uh' Vh' wh)1 :e;;;; C2(n,

a)luhI1/4I1uhll~/4I1vhllh

IWhI1/4I1whll~/4

b,,'(Uh' Vh' wh)1 E;;; C2(n, a)luh 11/4I1uhll~/4Ivh

11/4I1vhll~/4I1whllh’

(6.76)

Theevolution Navier-Stokes equations

384

Ch.I1I, §6

Proof. The estimates for b,,' are deduced from the estimates for a simple permutation of vh and wh' Using Holder inequality as for Lemma 6.1, it appears that

bi"

by

(6.77) In order to estimate the L 4 -norms of uih and vih' we apply Theorem 11.2.3 and Remark 11.2.6. For n = 3, Schwarz inequality allows us to write

10 ut.

dx <;

(

L

1/2 (

uf. dx )

Lufo

<;; (because of 11.(2.42), n <;; IUihl c(n, p,

)

1/2

dx

= 2, p = 2)

n, a)l/uihl/~.

Hence 3

~

i=1

l/uihl/I4 (0)<;; c3 (n,

a)IUhI1/2I/Uh,,~/2,

(6.78)

and the relation (6.67) for bi, follows from this maioration and (6.77). For n = 2, due to lack of majorations of the type found in Proposition 11.2.1, we proceed differently. By the Schwarz inequality,

L L ut,. dx =

<;;

lu,.I'-·lu,.I"'
(J

n ulh dx

)

1-e/2(

J

n 1UihI2(3+e)/1+edx

<;; (because of II. (2.48))

-c c(n, a, Hence

€)luihI1-el/uih"re.

)

1+e/2

Ch. Ill, §6

385

Discretization of the Navier-Stokes equations

2

~

IIuihll14 (0) ~

1=1

C4

(n, a,

e)IUhll-e lIuhll~+e

(6.79)

and the relation (6.75) for b~ follows. The combination of (6.68) and Lemma 6.5 gives easily (5.7): bh (uh' vh' wh)1 ~ d111uhllh IIvhllh IIwhllh' YUh' Vh' Wh E

v,,;

(6.80)

the constant d 1 depends on nand a. Proposition 6.6. For the approximation (APX5), the inequality (5.41) holds with

SI (h) = cl (n, a, e)S(h)l+e, 0 < e

< I arbitrary,

ifn = 2,

= c2 (n, a) S(h)3/2, if n = 3

SI (h)

(6.81 ) (6.82)

where S(h), cl’ c2 are given in (6.69), (6.75), (6.76). Proof. Using (6.69), (6.75) and (6.76) we can write Ib~ (uh' vh' wh)1 ~ cl

h

Ib (uh' uh' vh)1 ~ cl

for n = 2, and for n

(n, a,

e)S(h)l+e IUhllluhllh IWhl

(n, a,

e)S(h)l+e IUhllluhllh IWhl

= 3,

Ib~ (uh' uh' wh)1 ~ c2 (n, a)S(h)3/2!uh1Iluhllh IWhl uh' wh)1 ~ c2 (n, a)S(h)3/2Iuhllluhllh IWhl.

Ib~’(Uh’

6.3.3. Application of the stability and convergence theorems. Schemes 5.1 and 5.2 are unconditionally stable. The stability condi› tions of Scheme 5.3 are given in Theorem 5.2 and Lemma 5.3

k

----:-----::--:-;----:-~-----=-.,...,...-..,----~

[p' (h)] 2(1+e)

k

d’

Co (n )2(I+e)cl (n, a, e)2'

d"

[p’(h)]2 ~ co(n)2' if n

k

--~-----:---~

p'(h)3

k p'(h)2

= 2,

(6.83)

d’ cO(n)3C2 (n , a)2' d"

~ Co (n)2' if n = 3,

(6.84)

The evolution Navier-Stokes equations

386

Ch. III, §6

where a', d" are some constants defined in the proof of Lemma 5.3 (e arbitrarily fixed, 0 < e < 1). For Scheme 5.4 the stability conditions given in Theorem 5.3 and Lemma 5.4 are ----,-~

k

[p'(h)]2

1-5

k

EO; - -

4vcij’

[p'(h)]2(l+e)

EO;

'Y 5 . 22(l+e)lfn=2,

8ds c l Co k [p'(h)] 2

1-5

~ 4vc5’

(6.85)

(6.86)

for some 5, 0 < 5 < 1, some e, 0 < e < 1, and ds given by (5.58). Ofcourse the stability conditions (6.83)• (6.86) are not completely satisfying since we only have imprecise information on the constants in the right-hand side of these relations. The interpretation of the convergence theorems (Theorem 5.4 and 5.5) is very simple: uh - U in L 2 (Q )

strongly, £0 (0, T; L 2(0)) weak star,

Dih uh - DiU in L2 (Q) strongly or weakly (depending on (6.87) whether convergence in £2(0, T; F) is strong or weak). Theorems 5.4 and 5.5 are based on the hypotheses (5.89) to (5.92). We must show that these conditions are met: the verification of these properties is exactly the same as for finite differences, we just have to repeat the arguments of Section 6.1:4, replacing everywhere 5ih by Dih 6.4. Numerical algorithms. Approximation of the pressure The practical computation of the element ulJ’ of J)" defined by one of the considered schemes, is not easy. The difficulty is connected with the constraint "div u = 0" built into the definition of the space J), and, actually, the situation is exactly the same as in Chapter I for the Stokes problem. In this subsection we will show how the algorithms studied in Section 5 of Chapter I (resolution of Stokes Problem) can be extended

Ch. III, §6

387

Discretization of the Navier-Stokes equations

to the resolution of the problems (5.12) to (5.15). At the same time we will introduce the discrete approximation of the pressure. 6.4.1. Approximation of the pressure

For each type of approximation Vh of V, we have also defined an approximation Wh of Hb(f2) (see Chapter I, Section 3 and 4). Essent› ially the elements uh of Wh are exactly of the same type as the elements uh of J)" but no divergence condition is imposed. Later in this section we will show how the element u’C of J), can be approximated by a sequence of elements of Wh , u’C’ r, r = 1, ... 00. Approximation (APX1) The space Wh is the space of step functions

L

Uh =

~M

0

MEn~

whM,

~M

E dtn •

(6.88)

The discrete pressure is an element of the space X h of step functions of the type 1rh

=

L

o

11M WhM' 11M

MEn1

For uh E Wh , we defined the discrete divergence function of Xh given by Dh uh

=

L

0

MEnt

(6.89)

Edt. Dhuh

as the step

(Dh uh (M)) whM'

(6.90) An element

uh

DhUh

of Wh belongs to

=0

v"

if and only if (6.91)

Exactly as in Subsection 3.3, Chapter I (see 1.(3.72)) we prove that if uJ:' is solution of (5.12) (i.e., Scheme 5.1), there exists some step function 7rf:’ of type (6.89), such that

Theevolution Navier-Stokes equations

388

I

k (UI:’

- ul:’-l, Vh) + v «UI:’ , Vh ))h

Ch. III, §6

+ bh (ul:’-l,

- (11’1:’, Dh Vh) = (fm, Vh)’ V Vh E

UI:’, Vh)

(6.92)

Wh.

The main differences between (6.92) and (5.12) are the appearance of the term - (11’1:’ ’ Dh vh) and the fact that the equation is satisfied for all Vh in Wh, i.e., even ifvh is not in some sense divergence free. For Schemes 5.2 to 5.4, the same results are valid. In these three cases there exists a 11’1:’ of type (6.69) such that, respectively: I k

-(ul:’ - ul:’-l, vh)

v I + - «uI;’ + u7/-1, vh))h + - bh(uI;’- I, uI;’ 2

+ ul:’-l, vh) - (11’1:’, Dh vh)=~’

I

k (ul:’

- ul:’-l, vh)

+ v «ul:’,

- (11’1:’, Dh vh) I

k (m uh -

uhm-l , Vh )

2

vh)’

V vh E Wh•

(6.93) (Scheme 5.2)

vh ))h + bh (ul:’-l, ul:’-1, Vh)

= (fm, vh)’

V vh E Who

(6.94) (Scheme 5.3)

+ u«Uhm-l’Vh ))h + bh (m-l m-l’ Vh ) Uh ’ Uh (6.95) (Scheme 5.4)

Other Approximations. For the other approximations the same result holds; the differences only arise in the definition of Dh , the space Xh to which 7rJ:’ belongs (and of course the definition of Jih and Wh ) . For the approximations (APX2), (APX3) and (APX5), Xh is the space of step functions 1I'h

=

2:

'/E5'h

Tl'/ X/' Tl E lR,

(1)

(6.96)

where XfJ' is the characteristic function of the simplex 9’. The discrete divergence is understood as the step function of type (6.96) defined by Dhuh

Tl


=

(l) T/’j -_ 7Th() x,

= ’jE5j" 2: n.’ /X’ /

f

I div uh dx. meas 9' .'/ x E 'I'.

(6.97)

Ch. III, §6

Discretization of the Navier-Stokes equations

389

A function uh of Wh belongs to fh if and only if the step function o, uh vanishes. With this notation we also get the existence of a 7T;:Z satisfying respectively (6.92), (6.93) or (6.95) (Schemes 5.1 to 5.4), in the case of approximations (APX2), (APX3) or (APX5). A similar result can be proved for the approximation (APX4) but then the characterization of the space Xh (7Tf:’ E Xh ) is technically more complicated. 6.4.2. Uzawa Algorithm

We want to study an adaptation of the Uzawa algorithm for the resolution of the problems (6.92) to (6.95). If the elements of the step m - 1 have been computed, we must compute the unknowns

uJ:’

E

fh

and 7T;:Z EXh•

(6.98)

We will obtain them as the limits of two sequences of elements u;:z,rE Wh and 7Tf:',rEXh , r= 0,1, ...

00.

(6.99)

As in Subsection 5.1 and 5.3 of Chapter 1, we start the algorithm with any 7Thm ,o E X h’ (6.100) When 7T;:z,r is known we define u;:z,r+1 and 7T;:z,r+1 (r;" 0) by (Scheme 5.1 or (6.92)) u/:,·7+1 E Wh and (u;:z, r+1, vh) + kv«u;:Z' r+1, vh ))h + kbh (u;:Z-1, u;:z, r+1, vh) - k (7T;:Z’ 7, Dh Vh) = (u;:Z-1, Vh)

+ k (fm, Vh)’ V Vh E Wh .

(6.101)

7T;:Z’ r+ 1 E Xh and m.r q )+p(D u m.r+1 q )=0 Vq EX ( 7Thm. r+1 _ ... "h’ h h h ’h h h.

(6.102)

The existence and uniqueness of the solution uJ:!' r+1 of (6.101) is a consequence of the Projection Theorem (the problem is linear with respect to u/:,·7+1). The same argument is also valid for (6.102) but actually (6.102) explicitly gives 7T/:" r+ 1 without inverting any matrix. The determination of uf{" r+ 1 amounts to solving a discrete Dirichlet problem (see Chapter I, Section 5.1).

The evolution Navier-Stokes equations

390

Ch. III, §6

For Schemes 5.2 to 5.4 (i.e., (6.93) to (6.95)) we leave (6.102) unchanged and replace (6.101) by, respectively, (Scheme 5.2 or (6.93)) ur' r+l E Wh and v) (uhm, r+l 'h

+ kv _ «fdJ- 1 + um, r+l 2

h

h

V )) 'hh

+ ~b

(u m-1 u m- 1 + um, r+l V ) 2hh'h h 'h

- k(Trhm, ', Dh vh)

= (ur- 1, vh)

+ k (fm, vh)' V vh E

Who

(6.103)

(Scheme 5.3 or (6.94)) ur, r+l E Wh and (urI r+l, vh)

+ kv«ur, r+l, vh))h + kb h(ur- 1,ur-1,vh) - k(1TJ:" r,

D", Vh) = (ur- 1, Vh)

+ k(fm, Vh)'

YVh E Who

(6.104)

(Scheme 5.4 or (6.95)) ur' r+l E Wh and (ur,r+ 1, Vh) + kv«ur- 1, Vh))h - k(Trhm, ',

+ k~,

+ kb h(ur- 1, ur- 1, Vh)

D" Vh) = (ur-1, Vh)

Vh)' YVhE Who

(6.105)

We emphasize that all the elements of step m-I are known, and to compute ur, Trhm, we reiterate on the index r computing ur' r+l, 1TJ:" r+l. Regarding the convergence of this algorithm, the interesting point is that the convergence criteria are almost the same as for the linear problem. Proposition 6.7. If the number p satisfies

o< p < 2v - (1) n

then as r ~

Xh/fi{·

00,

(6.106)

ur' r converges to ur in Wh and Tr'/:' r converges to TrW in

Proof. We only give the proof for Scheme 5.1 «6.101) and (6.102) are associated with (6.92)). We drop the indices m and h and set (1) n

= 2 or

3, the dimension of the space.

Discretization of the Navier-Stokes equations

Ch. III, §6

391

v' = u'/:" - u,/: ,

(6.107)

rr' = rr,/:" - rr'/: .

(6.108)

Subtracting (6.92) from (6.101) we obtain (vr+l, vh) + kv«v'+I, vh))h + kb h (u,/:-I, vr+l, vh)

= k(rr', D h vh)'

YVh E Wh;

and with vh = vr+l we get (recall (5.6)) Iv r+11 2

+ kv II vr+1 l1 h = k (rr',

(6.109)

D h vr+l).

uJ:' = 0 since u'/: E ~, we write (6.102) as (rr,-1 -cn’, qh) = - p(Dh v'+I, qh)' yqh E Xh ; setting qh = rrr+l we obtain Irrr+11 2 _ Irr'I 2 -lrr'1 2 + Irr'+1 _rr'1 2 = -2p(Dh vr+l, rr'+l). Recalling that Dh

(6.110)

We continue as in the proof of Theorem 1.5.1. The right-hand side of (6.110) is equal to - 2p (D h vr+l, rrr+l_ rr') - 2p (Dh vr+l, rr')

and, as we prove below, I(D h vh' rrh)1

~..; ;

Irrh1llvhllh' Yrrh EXh, vh E Who

(6.111)

Admitting this point temporarily, we majorize the right-hand side of (6.110) by - 2p(~

v,H, rr')

+ 2pVn II v r+1 l1 h /rrr+l ... rr'l ~ l) Irrr+l _rr'1 2 +

p2

n T" vr+1Ih + 2p(Dh v'+l, rr').

We now add (6.110) multiplied by k, to the equation (6.109) multi› plied by 2p. Dropping two opposite terms and simplifying we find klrr'+11 2 _ klrr'I 2 + (l-l»)klrrr+l _rr'1 2

+ Iv r+112 + kp(2v _ pn )llv'+1112 ~ O.

(6.112)

8

If p satisfies (6.106) then there exists some 0 < l) pn 2v-- >0. l)

< 1, such

that

392

Ch. Ill. §6

The evolution Navier-Stokes equations

By summation of the inequalities (6.112) for r = 0, ... , S, we get s

kl1T5+ 112+

L {k(1-/))I 1Tr+l _ 1T’ 12 + Iv r+112 }

r=O

+ kp(2v -

s

pn) /)

L

r=O

(6.113)

II v r+1112 E;; kl1T 112

This shows that the series GO

are convergent and hence

v r -+ 0 in Wh , as r -+

(6.114)

00.

The relation (6.113) shows also that the sequence r is bounded. According to the preceding and (6.102), any convergent subsequence extracted from 1Ts must converge to 0 in Xhl tH. Thus the sequence 1Ts converges as a whole to 0 as S -+ 00, in Xhl tH (i.e, in X h up to an additive constant). The proof is completed. 0 It remains to prove (6.111). Lemma 6.6. For the approximation (APX 1) to (APX3) and (APX5), I(Dh vh' 1Th)1

E;;;y; 11Th I IIvh IIh ,

V1Th E Xh , VVh E Who

Proof. For finite differences (APXn, since the function e constant on the blocks 0h (M), M E 01,

(" .D. v.l =

f

n ".(xl (

~

V,. V,. (x) )

By Schwarz’s inequality, we estimate this by

dx.

7Th

(6.115)

is

Ch. III, § 6

Discretization of the Navier-Stokes equations

393

But

and then

For finite elements, we observe again that 'Trh is constant on a simplex !/E3h so that (wh' n" vh) =

L

wh (x) div "" (x)

dx.

Then, by Schwarz’s inequality, 1('Trh'

o, vh)1 E;;; l'Trhlldiv vhl E;;; Vn l'Trh1Ilvhll. 0

6.4.3. Arrow-Hurwicz Algorithm We compute uW’ nf, as limit of two sequences of elements

uw•rE Wh , rrW,rEXh , ,;;;:"0, defmed in a slightly different way from before. There are two positive parameters p and cx. We start the recurrence with any

uW’ 0 E Wh, 'TrW' 0 E Xh •

(6.116)

The evolution Navier-Stokes equations

394

Ch. 1Il, §6

When 1I’W’ r and uW’ r are known, we define 1I’W,7+1 and uW’ 7+1 as the solutions of (Scheme 5.1 or (6.92» uW’ 7+1 E Wh and

k«uW,7+1 -uW’ r, Vh»h + (uW’ r, Vh) + kv«uW’ r, Vh»h 1, uW’ r, Vh) - k(~’ ~ D Vh) + kb h(uW-

=

h

(uW- 1, Vh) + k(jm, Vh)’ VVh E Wh,

(6.117)

1I’W’ 7+1 E Xh and cx(1I’W’ 7+1 -~’

’, qh)

+ p(Dh uW’ 7+1, qh) =0,

Vqh EXh

(6.118)

The existence and uniqueness of uW’ It1 follows from the Projection Theorem, those of 1I’W’ 7+1 too, but in practice it is simpler to observe that (6.118) defines ~’ 7+1 explicitly. For the other schemes (Schemes 5.2 to 5.4, or (6.93) to (6.95), we leave (6.118) unchanged and we respectively replace (6.117) by (Scheme 5.2 or (6.93» uW’ 7+1 E Wh and 1 ku k«uW,7+ -uW", Vh»h + (uW’ r, Vh) + 2" «uW’ r + uW-1, Vh»h

k + '2 bh(uW-1,uW-1+uW’ r, Vh)

= (uW-1, Vh) + k(jm, Vh)’

-

k(1I’W’’. o, Vh)

VVh E Wh

(6.119)

(Scheme 5.3 or (6.94», uW’ 7+1 E Wh and

k«uW’7+1_ uW’ r,

+ (uW’ r, Vh) + kv«uW’ r, Vh»h 1, uW-1, Vh) - k(~’ r, D Vh)

Vh»h

+ kbh(uW-

h

= (uW-1 , Vh) + kif"’, vh)’ VVh E Wh’

(6.120)

(Scheme 5.4 or (6.95», uW’ 7+1 E Wh and

k«uW’ 7+1_ uW’ r, Vh»h + (uW’ r, Vh) + kv«uW- 1, Vh»h 1, uW-1, Vh) - k(1I’W’ r, D Vh) + kb h(uW-

= (uW-1, Vh) + k(fm, Vh)’

h

VVh E Wh’

(6.121)

Ch. III, § 7 Aptroximation of the Navier-Stokes equations by the fractional step method 395

Regarding convergence, we have Proposition 6.8. We assume that p and

2o:v 0
r~

00,

0:

satisfy

(1)

(6.122)

uZ" r converges

to

uZ’ in Wh , and Tr'fi' r converges

to

rrZ’

We omit the details of the proof which are similar to those of Theorem 1.5.2. and Proposition 6.7. 0

§ 7. Approximation of the Navier-Stokes Equations by the Fractional Step Method. This section deals with the approximation of the Navier-Stokes equa› tions by a fractional step method. The fractional step, or splitting-up, method is a method of approxi› mation of evolution equations based on a decomposition of the operators. We first present the idea of the method in the following very simple situation. Let us assume that we are approximating a linear evolution equation

u’+du=O,

O
(7.1)

u(O) = uo,

(7.2)

where u(t) is a finite dimensional vector, u(t) E cRh , and.sl is a square matrix of the order m. With a standard implicit scheme (similar to Scheme 5.1), we define a sequence of vectors u’", m = 0, ..., N as follows (T = kN, k = the mesh size, N is an integer):

uo = uo, um + 1 _ u" k

(1)

(7.3)

+ Jlu m + 1

= 0,

n = 2 or 3, the dimension of the space.

m= 0, ... , N -

1.

(7.4)

396

The evolution Navier-Stokes equations

Ch. 1lI, § 7

A splitting-up method is based on the existence of a decomposition of slas a sum q

j=

L:

i= 1

(7.5)

sli’

Starting again with

uO = uo,

(7.6)

we recursively define a family of elements urn +i/q, m = 0, ... , N - I, i = I, ... , q, by setting

urn +i/q _ urn +(i-l)!q . ----k----+JiUrn+i/q =0, m = 0, .. . ,N - 1.

i= l, ... ,q, (7.7)

When urn is known, urn + 1 can be computed in the case of an ordin› ary implicit scheme (Le., (7.4)) by the inversion of the matrix 1+ kj; in case of the fractional step method (i.e., (7.7)) the computation of urn + 1 necessitates the inversion of the q matrices (I + kjl ), ..., (I + kslq ) ; the algorithm is useful if all these q matrices are simpler to invert than I + ksl. This method can be adapted to the Navier-Stokes equations in ’ many ways corresponding to the many possible decompositions of the operators. We will consider two of them. The first one corresponds to q = 2, and n

Y/lU

= - v~ +

L:

i= 1

u;Diu,

(7.8)

whileJ’2 is an heuristic operator taking into account the term grad p and the condition div u = 0. We will be more precise in the course of the section, but we emphasise now that, in case of the Navier-Stokes equations, the method we will study is an interpretation of the fraction› al step method described in (7.5)-(7.7) and not merely a particular case. For the second decomposition of sI, we have q = n + I, the "operator" sin + 1 being like the preceding operator sl2 , and sl l , .. . ,sln’ being defined by

sliu=-vDlu+uiDiu

i=l,oo.,n.

(7.9)

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 397

We can also associate this method with a discretization in the space variables. It would be ineffectual and tedious to study all the possible combinations. Consequently we will restrict our attention to two characteristic cases: the first decomposition above without any dis› cretization in the space variables (subsection 7.1); then the second de› composition with a discretization by finite differences (subsections 7.2 and 7.3).

7.1. A scheme with two intermediate steps We describe here an approximation of the Navier-Stokes equation by a fractional step method without discretization in the space variables. 7.1.1. Description of the scheme

The dimension of the space is n = 2 or 3 and we want to approximate the solution of Problem 3.1 (or 3.2). For simplicity we assume that f is given in L 2(0, T; H) fEL 2(0, T; H),

(7.10)

and, as before

uo E H.

set

(7.11)

The interval [0, T] is split into N intervals of length k (T = kN). We

t" = ~fmk

f(t) dt,

m = 1, .. . ,N.

(7.12)

(m -1)k

We will define a family of elements of L2 (n), denoted u" +i12 , i = 0, I, m = 0, ... , N - 1. These elements are computed successively in the order of increasing values of the fractional index m + i/2. We start with

uO=uo.

(7.13)

When u’" is known (m ~ 0), we successively define u’" + 1/2 and um + 1 : Um + 1/2 EHb(n) and,

~

(u m +1/2 _ um,V)

+ v((u m +1/ 2, v)) + b(um +1/ 2, u m +1/ 2, v) (7.14)

398

Ch. 1IJ, § 7

The evolution Napier-Stokes equations

u m + 1 EH

(u m + 1 , v)

The form

and,

= (u m + 1 / 2 , v),

Vv EH.

(7.15)

g introduced in Chapter II is the

b(u, v, w)

1

= 2"

skew component of b: (7.16)

{b(u, v, w) ~ b(u, w, v)}.

Since n :E;’; 3 it is clear that the form "b, like b, is a trilinear continuous form on HA(n), and that b"(u, v, v)

= 0, V u, v EHA(n).

(7.17)

The equation (7.14) defining u’" + 1/2 is a nonlinear equation very similar to the stationary Navier-Stokes equation, and the proof of existence of at least one element u’" + 1/2 satisfying (7.14) is carried out exactly as the proof of Theorem 1.2, Chapter II, using the Ga1erkin pro› cedure and (7.16). The relation (7.15) amounts to saying that u’" + 1 is the orthogonal projection of u" + 1/2 on H in L2 (n). Thus, the element um + 1 is well defined by (7.15); we write

u m + 1 =PH u m + 1/2 ,

(7.18)

where PH denotes the orthogonal projector in L 2 (n) on the space H. Due to the characterization of Hand H1 given by Theorem 1.4, Chapter I, the difference u" + 1/2 - u" + 1 is the gradient of some function of HI (n) and it is convenient to denote this function by kp’" + 1:

u" + 1 _ k

um + 1/2

+ grad pm+1

= 0,

pm+1 EH1(n).

(7.19)

The relation (7.15) is equivalent to two conditions, namely (7.19) and (7.20) Remark 7.1. The equation (7.14) which defines u" + 1/2 is essentially a nonlinear Dirichlet problem; writing (7.14) with v E !?j(n), we get 1 k

_ (u m + 1/2 _ um) _ vtsu" + 1/2 n

+

i~l

u'{'+1/2Diu m+ 1/2

+~(divum+1/2)um+1/2

=fm. (7.21)

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 399

Remark 7.2. The relations (7.19) (7.20) defining u’" + 1/2 and p" + I are equivalent to a Neumann problem for p" + 1. (1) Application of the operator "div" to both sides of (7.19) leads to !:&pm + I =

k1 div u’" + 1/2 (since

div um+ 1 = 0),

and application of the operator "Y v (= the normal component on leads to apm+1

-av- =0

onr=an,

(7.22)

an) (7.23)

since "Y" u m + I = v, u m + 1/2 = O. It is interesting to observe that this boundary condition "ap/av = 0 on I'", is not satisfied by the exact pressure; this affects the accuracy of the p’" + I as approximations of p; nevertheless, as will be proved later, this does not affect the convergence of the scheme. This peculiarity does not arise in the other fractional step method of subsection 7.2. 0 Our purpose now is to prove a priori estimates for the u’" + i/2 , and then study the convergence of the scheme. 7.1.2. A priori estimates (I) Lemma 7.1. The elemen ts u" + i/2 remain bounded in the following sense:

m = 0, ... , N - 1,

lum+i/2i2 ~ d 2,

i

= 1,2,

(7.24)

N-I

k

2:

(7.25)

m =0

N-I

2:

m=O

lu m+ 1 _um+1/212 ~d2’

(7.26)

um 12 ~ d 2

(7.27)

N-I

2:

m=O

lum + 1/2

-

(l)When pm + I is known, u m + I is directly given by (7.19).

400

The evolution Navier-Stokes equations

Ch. Ill, §7

where (7.28) Proof. If we write (7.14) with v = u" + 1/2 and take into account (7.17), we get lum+1/212 -Iu m 12 + lum+1/2 _ u" /2

+ 2kvllum+1/2112 = 2k(fm, u m+ 1/2) ~ 2klfm Ilum+ 1/2 1~ 2k do Ifm Ilium +1/211 (1) ~ kvllu m+ 1/211 2

kd 2

+ _0 Ifm 12 . v

(7.29)

Hence lu m+ 1/212 - /um 12 + I urn + 1/2 _ u" 12 + kvllu m+ 111 2

~k~

d2

v

Ifm

l2 .

We are permitted to write (7.15) with v (u m+ 1 _ um+ 1/2, u m+ 1 ) = 0

(7.30)

= u m + 1; this gives

or (731)

We add relations (7.29) and (7.30) for m = 0, ... , N - 1, obtaining N-l

luNI 2 +

~

m=O

z

N-l

+ ku kd 2 +_0 J)

m=O

N

~

m=1

Ifm 12 ~ (by (5.29»

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 401

~

d5JT I/(s)1 2 ds = d 2

Iuol 2 + -;

(7.32)

o

This proves the estimates (7.25) to (7.27). Next we add relation (7.30), for m = 0, ... , r, and relations (7.31) for m = 0, ... , r - 1. Dropping some positive terms, we fmd r-1

2:

m=O

11m 12 ~ d 2

Similarly adding relations (7.30) and (7.31) for m = 0, ... , r, we get after some simplification, r

2:

m=l

11'1 2

~

d2 ;

thus (7.24) is proved.

The approximate functions We introduce the "approximate" functions, u~) , i u~i):

= 1, 2, and uk:

[0, T] ~L2(0),

uV) (t) = u’" +i/2

for mk ~ t < (m + 1)k,

i = 1,2

(7.33)

is a continuous function from [0, T] into L2(0), linear on each interval [mk, (m + 1)k], m = 0, ..., N -1, and Uk (mk) = u’", m = 0, ..., N. (7.34) Uk

Lemma 7.1 implies:

Lemma 7.2. The functions uk' u~) , i = 1, 2, remain bounded in oo L (0, T; L2 (0», as k ~ 0. The functions u~l) remain bounded in L2(0, T;

HAcO».

Lemma 7.3. lu~2)

- u~) (2)

IL 2 (0, T; L2(O» ~ Vk d 2 ,

IUk - Uk IL 2 (0, T; L 2 (0»

~

V4kd2 3 .

(7.35) (7.36)

402

en, 1Il,

The evolution Navier-Stokes equations

§7

Proof. Lemma 7.2 and (7.35) are direct consequences of Lemma 7.1. For (7.36), the computations of Lemma 4.8 give N-1

- k

(2)12

IUk-Uk

L2(OTL2(Sl»--

3

2:

m=O

I

m+1

U

-U

m 12

.

But

Iu" + 1

_

u" I ~ 1u" + 1

_

u’" + 1/21 + Ium + 1/2

-

u" I

lu m + 1 _u m l 2 =;;;;2Iu m + 1 _u m + 1/ 2 12 +2Iu m + 1 / 2 _u m l 2

and therefore, by (7.26)-(7.27) (2) 2

IUk - Uk IL 2(0 , T; L 2 (Sl » ~3

4k

d2 •

7.1.3. A priori estimates (II) In order to apply the compactness tools, we need some estimate of the time derivatives of uk' We extend the function uk by outside the interval [0, T] and denote by Ok the Fourier transform of the extended function.

°

Lemma 7.4. The Fourier transform {;k of Uk satisfies

f

+00

ITI'> II ilk _

(T) II},

dr <;; Const,

(7.37)

00

where the constan t depends on ’Y and the data.

Proof. According to Theorem 1.1.4 and 1.1.6, V =.H n n6(0), and the relations (7.14) and (7.15) hold for any v E V. By summation we get

kI (u m + 1 -

u'", v)

+ v«u m + 1/2. v)) + Scum + 1/2, u" + 1, v)

= (fm. v), V v E V. m = 0, ... , N - I

(7.38)

and this equation is equivalent to d (Uk(t), v) + v«u11) (t), v)) + S(u11) (t), u11) (t), v) dt = (fk(t), v), V t E (0, T), Vv E V. (1)

-

(l)The symbol

A

on b has nothing to do with the symbol

A

(7.39)

denoting the Fourier transform.

Ch. lII. § 7 Approximation of the Navier-Stokes equations by the fractional step method 403

or d

Cit (Uk (t), v) = (gk (t), v>,

V t E (0, T), V v E V,

(7.40)

where ’k(t)=, m ,

m=O, ... ,N-I

mk~t«m+l)k,

(7.41)

and gk (t) E V’is defined by (gk(t), v>

=-

1I«U~l)(t),

v» - t(uJP(t), ull)(t), v)

+ (Ik(t), v), V v E V. It is clear from the properties of

Ilgk (t) IIv' ~

II Ilukl) (t)

t that

II + c lIull)(t) 11 2 + Ilk (t)I,

(7.42)

(7.43)

and due to Lemma 7.2, gk remains bounded in L 2(0, T; V’).

(7.44)

After this, we can repeat exactly the arguments of the proof of Theorem 2.2 and arrive at (7.37). 7.1.4. Convergence of the Scheme The behaviour of the approximate functions uii>, uk’ as k -+ 0, is described by the following theorems. Theorem 7.1. The dimension of the space is n = 2, and I and Uo are given satisfying (7.10)-(7.11); u denotes the unique solution of Problem 3.1, ug>, uk are the approximate functions defined by (7.33) (7.34). As k -+ the following convergence results hold:

°

uk;), Uk’ convergence to u in £2(Q) strongly, L"’(O, T; £2 (n» weak-star (7.45) uk!) converges to u in L 2(0, T; nA(n» strongly.

(7.46)

Theorem 7.2. The dimension of the space is n = 3 and I and Uo are given satisfying (7.10)-(7.11). Then there exists some sequence k’ -+ 0, such that

uk’, convergence to u in £2 (Q) strongly, L’"(0, T; £2 (n» weak-star (7.47)

u~>,

ukP converges to u in £2(0, T; nA(n» weakly, where u is some solution of Problem 3.1.

(7.48)

The evolution Navier-Stokes equations

404

Ch. 1lI, §7

For any other sequence k’ -+- 0, such that the convergence results (7.47)-(7.48) hold, u must be a solution of Problem 3.1.

The proof of these two theorems is the same except for the strong convergence in L2(0, T; HA(f}.)). 7.1.5. Strong convergence in L 2 (Q) Due to Lemma 7.2, there exists a subsequence k’ -+- 0, such that

ulV -+- u(1) in tr (0, T; L2 (f}.)) weak,

weak-star, L2 (0, T; HA(f}.))

T;L 2 ( f}. )) weak-star, uk'-+- u; in ir (0, T; L 2( f}. )) weak-star.

u~,-+-u(2)inLOO(O,

(7.49) (7.50) (7.51)

Due to Lemma 7.3, we have u(l)

=u(2) = u..

(7.52)

The functions u~2) and hence the function u(2) belongs to L" (0, T; H). We infer from this the property: u.(t) EHA(f}.)

«s»

V; a.e.,

which implies

u; EL2(0, T; V) n LOO(O, T; H).

(7.53)

The results of strong convergence in L2 (Q), which are essential in order to pass to the limit, are not obtained merely by the application of a compactness theorem; they require some further arguments which will be developed now. We recall that by definition of u~2). ul2)(t) is an element of H for all t E [0, T] and by virtue of Theorem 1.1.4, L 2 ( f}. ) is the direct sum of H and its orthogonal complement H 1 . Denoting by PH and PHl the orthogonal projection in L2 (f}.) onto Hand H 1 , we have u~l)

= PHull)

+ PHwll) ,

and hence lull)(t) - uF)(t)1 2

= IPH

u~l)(t)

- ul2)(t)1 2

+ IPHl ull)(t)1 2 .

(7.54)

From (7.54) and (7.35), we infer that PH u~l)

- ul2) -+-

°

in L 2(0, T; H) strongly,

(7.55)

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 405

°

in L 2 (0, T; L2(n)) strongly.

PHl u~)

~

PH ul~)

~PH

(7.56)

It follows from Remark 1.1. 6 and Lemma 7.2 that PH uP) is a bounded sequence in L 2(0, T; H1(n) n H) and, therefore, we can choose the previous subsequence k’ so that U.


(7.57)

We now apply Proposition 2.1 in the following way: X o = HI (n) n H, Xl = V', the sequences fUm} and {v m} replaced by the sequences {uk'}' {PH u~9}. These sequences possess the required properties; moreover, H1(n) nHCHe V',

and since the injection of HI (n) n H into H is compact, so is the inject› ion of HI (n) n H = X o into V I = Xl' Proposition 2.1 enables us to assert that the sequence PH Uk9 is relatively compact in L 2(0, T; V’) and therefore PH UkV ~ PH U.

= u;

in L 2 (0, T; V’) strongly.

An application of Lemma 2.1 with Xo =H 1 (n ) n H, X leads to: IPHuP) - u.I L 2(0, T; H) '" e IIPH U~P

+ C(e) IIPHu1~) and since PHU~P IPH u~~)

(7.58)

=H, Xl = V’

- u; "L 2 (0.T;H1 (0) n H)

- u.II L2 (0. T; V’)

is bounded in L 2(0, T; HI (n)); - u.I L 2 (0, T; H) ’"

C e + C(e) II PH u~P

- u.IIL 2 (0. T; V’)’

(7.59)

Taking the upper limit of (7.59) we find -.

k~l.To

(1)

IPHuk' - u.I L2(0, T; H)

'"

C e;

since e is arbitrarily small, this upper limit is zero, and therefore PHu}P ~ u; in L 2(0, T; H), as k ’ ~

°

(7.60)

By comparison of (7.56) and (7.60) we get: u~P

~ U. in L 2(0, T; L2 (n )), (strongly) as k ’ ~ 0.

(7.61)

Finally (7.35) and (7.36) imply that u~~)

~ U. in L2 (0, T; L 2 (n)) (strongly) as k’ ~ 0,

Uk’ ~ U.

in L2 (0, T; L 2 (n)) (strongly) as k’ ~ 0.

(7.62) (7.63)

The evolution Navier-Stokes equations

406

Ch. III, § 7

7.1. 6. Proof of Theorems 7.1 and 7.2 Let 1/1 be any continuously differentiable function on [0, T] with 1/I(n = O. We multiply (7.39) by 1/1(1) and, integrate in t. Integrating the first term by parts, we obtain (uk (0) = uo):

-f:

+

+

J:

f:

(Uk (I),

1/1’(f)V) dr + v

f:

((u !l) (I), v 1/1(1))) dr

SCull) (I), u!l) (f), v 1/i(t)) dr = (uo, 1/1(O)v) (1.(1), v!/I(I» dr,

v» E

V.

Due to (7.49), (7.51) and (7.52), [

f: f:

(u..(f), v 1/1 ’(I» dr -+

J:

((u r (f), v1/l(l))) dr -+

(u,(f), v1/l'(t)) dr,

J:

((u, (I), v1/i(t))) dr.

By (7.16), (7.61), and Lemma 3.2, S(u!')(f), u!Il(!), v1/i(t» dr -+

f:

Stu, (I), u, (f), v1/i(t» dr,

Since u; (I) E V a.e., Lemma 11.1.3 and (7.16) imply that

~(u.

(I), a; (I), v) = b(u. (t), u. (t), v), a.e.

f:

We deduce easily from Lemma 4.9 that

f:

(Mf), v1/i(t» dr -+

Thus we obtain in the limit

(f(f), v1/l(f» dr.

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 407

f:

«u. co. V,p(t))) dt

- [

(u. (t), VVi (t)) dt + V

+

btu; (t), u; (t), V,p(t)) dt = (UO’ V,p(O))

+

f: f:

(7.64)

(f(t), V,p(t)) dt, V V E V.

This equation is the same as (3.43) and from it we conclude, as in Theorem 3.1, that u; is solution of Problem 3. I. The same argument can be repeated for any other convergent sub› sequence of uk' uiP. This completes the proof of Theorem 7.2 (n = 3). If n = 2, there exists only one solution u of Problem 3. I; hence u; = u and the sequences uf>, Uk converge to u as a whole, in the sense of (7.49)-(7.51). It remains to prove the strong convergence in £2(0, T; HA(o.»; this is our goal in next subsection. 7.1.7. Proof of Theorem 7. I (strong convergence) Lemma 7.4. lfn

= 2, uN = rIt -+ u(T) in L2 (o.) weakly.

as k -+ O.

Proof. According to (7.24), Irltl ~ Const, and thus, the subsequence k’ can be chosen so that

uf. -+ X in H weakly (ur

and X E H).

By integration of (7.39) we see that (u.
(of, v) = (uo. v) -

-f: S(u~l)(t), f: +

u~l)(t),

v

f: «u~l)(t), v) dt

(!t(t),v)dt, VvE V.

(7.65)

v)) dt

The evolution Navier-Stokes equations

408

Ch. Ill, §7

It is easy to pass to the limit in this relation with the sequence k'; we find

f:

(x, v) = (Uo, v) - v

+

f:

«u(I), v» dr -

f: ~(U(I),

u(I), v) dr

(f(t), v) dr,

By comparison with the relation (3.13) integrated from 0 to T, it follows that (x. v)

= (u(T), v), V v E V,

and since X E H, u(T)

= X.

Since the limit is independent of the choice of the subsequence k', the convergence result (7.65) is in fact true for the whole sequence k.

Lemma 7.5. If n

=2, ui1) ~

u in £2 (0, T;

HA (0)) strongly.

f: lu~l)(t)

Proof. We consider the expression

x, = luN We write Xk

u(1)12

+ 2v

- u(I)1I 2 dr.

= X(l) + x1 2) + x1 3) , 2

X(l) = lu(1)1 + 2v

f:

xl2) = - 2(uN, u(1) XP)

= luNI 2 + 2v

lu(I)P dr,

f: «u~l)

- 4v

f: lu~I)(t)F

dr.

(I), u(I))) dr,

Ch. 1II, § 7 ApproximJZtion of the Navier•Stokes equations by the fractional step method 409

X~2);

The term X(l) is independent of k; we can easily pass to the limit in by (7.49) and Lemma 7.4,

Xl’) -+ - 2Iu(1)I' -

4.

f:

lu(l)n' dt

=-2X(I).

The weak convergence results already proved are not sufficient to pass to the limit in X£3). But, by summation of the relations (7.29) and (7.31) for m = 0, ..., N - I, we can write N-l

2:

1012 + 2kv

lIum + 1/211 2 +

m=l

N-l

2:

m=O

=

or

I:

N-l

I u 12 + 2k o

Xl’) .;; IUo I' + 2

2:

m=O

(t. (I), u~l)

(1m, u m + 1/2)

(t)) dr,

Passing to the upper limit, there results lim Xp>

k-O

~

luo 12 +

2fT (f(t), u(t)) dt. o

Due to the energy equality of the exact problem (see (4.55)), the right› hand side of the last inequality is equal to X(l). Hence lim Xp> ~ X(l)

k-O

and combining the different results, lim

k-O

x, ~

0.

This shows that Xk -+ 0, as k -+ 0, and the proof is complete.

7.2. A scheme with n + I intermediate steps We describe this scheme in a discrete frame. We consider the approxi-

410

The evolution Navier-Stokes equations

Ch. III, § 7

HA

mation of (n) by fmite differences studied in Chapter I, and the corresponding approximation (APX 1) of V. The following can be par› tially extended to other approximations of the space V but the interest of the Scheme would be considerably lessened since the decomposition method is most powerful in the frame of finite differences methods.

7.2.1. The decomposition of the operators For each h = (hi’ ... , h,,), hi> 0, we have defmed in Chapter I, Sub› section 3.3, an approximation Wh of (n) and a corresponding approximation Vh of V (fmite differences, approximation (APXI». Both spaces are finite dimensional spaces, equipped with either the scalar product (u. v) induced by L 2 (n ), or with the scalar product

HA

n

«uh' vh»h =

2:

i= 1

(6ih u h ' 6ih vh)'

We defme on Wh (and hence on Vh ) , n other scalar products «uh' vh»ih

= (6ih u h ' 6ih vh ) '

1 <:i<:n.

(7.66)

Due to the Discrete Poincare inequality (see Proposition 1.3.3), IUh I <: do lIuhllih, V hE Who do

= 2R,

(7.67)

where Rdenotes now the maximum of the widths of n in the xi direc› tions. This inequality implies that 1I.llih is a norm on Wh , and «."»ih a Hilbert scalar product. We now write the trilinear form bh considered in (6.3)-(6.5) in the form bh(uh' Vhf Wh) =

"

~ i= 1

bih(Uh' Vh' Wh)'

(7.68) (7.69) (7.70)

(7.71)

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 411

All of these forms are obviously defined and trilinear continuous on Wh X Wh X Wh; it is also clear that (7.72)

bih(uh,vh,vh)=O,Vuh’ vhEWh’

7.2.2. The scheme

The data I and Uo satisfy (7.10)-(7.11). We assume that we are given a decomposition of I as n

(7.73) this decomposition can be quite arbitrary and the simplest choice of the Ii would bef1 =f, Ii = 0, i = 2, ..., n. The interval [0, T] is divided into N intervals of length k (T = kN) and we set fm +i/q

=~

r

/;(t) dt, i

= I, ..., n,

(7.74)

mk

where

q

= n + 1.

(7.75)

We will now define a family of elements U;:' +i/q of Wh, m = 0, ... , N - I, i = 1, ..., q. These elements are defined successively in the order of increasing values of the fractional index m + if q. We start with

u2 = the orthogonal projection oluo onto Vh in L2 (n ). This definition makes sense since

Wh C L2 (0.),

(7.76)

and obviously

lu21 ~ Iuol, V h,

(7.77)

When ur is known (m ~ 0), we define the u" +i/q as follows: For

~

s ect e;»,

U;:,+i/ q EWh and

(u m +i/q _ u;:’ +i- l/q, vh)

+b

ih

+ v«u;:’

+i/q, vh »ih

(u m +i- l/q u m +i/q V ) h ’ h ’ h

=(fm+i/q,Vh)’ VVh EWh

(7.78)

The evolution Navier-Stokes equations

412

Ch. III, § 7

For i = n + 1 (= q), uZ’+1 E Vh and,

(uZ’+1,Vh)=(UZ’+n/ q,Vh), VVh EVh .

(7.79)

Relation (7.79) means that uZ’ + 1 is the orthogonal projection of uZ’+n/q on Vh in the space Wh equipped with the scalar product (.,.); uZ’ + 1 is well defined by (7.79). Equation (7.78) is linear in uZ’+i/q; the existence and uniqueness of uZ’+ilq is assured by the Projection Theorem (Theorem 1.2.2). Indeed, the form

{uh’ vh}-+

1

k (uh’ vh) + v«uh’

vh»ih

+ bh(uZ’+i-1/q, uh’ vh)

(7.80)

is bilinear continuous on Wh , and the form vh -+

~

(uZ’ +i- l/q. vh) + (fm +i/q, vh),

is linear and continuous; the coercivity of the form (7.80) is a conse› quence of (7.72).

Remark 7.3. (Interpretation of (7.79». We give an interpretation of (7.79) which will enable us to introduce an approximation of the pressure. We proceed as in Subsection 3.3; the element uZ’ +1 - uZ’ +n/q of Wh is orthogonal to Vh (scalar product (.,.». This amounts to saying that (UZ’ + 1 - uZ’ + n/q, vh)

= 0,

when vh E Whand (characterization of Vh ) n

L: .

1=

1

-

V iVih(M) - 0, V ME n1h .

(7.81)

By a classical result of linear algebra, there exists a family of numbers AM' M E such that

nk

q, (uZ’+1 - uZ’+n/ vh)

=

2

0

Menl

AM {

i~1

V ih Vih(M)}, (7.82)

Ch. III, § 7 Approximation of the Navier-Stoke« equations by the fractional step method 413

If 1I’r; + 1 denotes the step function of X h given by 1I’r;+1

=

2

(7.83)

ME~l h

relation (7.82) can then be interpreted as,(2) 1

k (ur; + 1 -

ur; +n/q , vh) - (1I'r; + I, Dh vh)

= 0,

VVh E WhP)

(7.84)

or

~

u~+n/q(M))

(un+I(M) -

+Vih 1I'r;+1 (M)

= 0,

0

VMED. h1 •

(7.85)

Remark 7.4. (Resolution 01(7.78». Relations (7.78) are equivalent to the relations

~

(ur; +i/q (M) - u'li +i-I/q (M» - v

+~

+i- I/q ur; +i/q) (M)

(l)WhM is the characteristic function of the block 0h(M).

n

(3)DhVh(x) =

.2

1=1

V ih "ih(x).

u'li +i/q (M)

un+i-l/q(M) 5ih u'li+i/q(M)

+ ~ 5ih (u~

(2)Compare to I (3.71)-(3.72).

5;"

Theevolution Navier-Stokes equations

414

Ch. III, §7

The unknowns, when we compute u'/(+i/q , are the components of u'/( + tlq (M); the crux of the fractional step method is that the above sys› tem is actually uncoupled into several subsystems involving only the unknowns u'/( +i/q (M) where the M are all on the same line parallel to the Xi direction. This makes the resolution of (7.78) very easy. Remark 7.S. (Resolution 0[(7.79». As in Remark 7.2, we can inter› pret (7.79) as the discretization of a Neumann problem for the pressure .,rn+l (with a boundary condition different from (7.23»: A1Tm + 1

= _1 div u m +n/q

a1Tm+l

_

k

1

-a;- - t»

u

'

m wnlq

(7.86)

This allows us to solve (7.78) by instead solving a discrete Neumann problem for 1T’/(+I. For this procedure, see A.J. Chorin [3], M. Fortin [1] . An alternate procedure for solving (7.79) would be an iterative algo›

rithm of the types considered in Chapter I, Section 5, and in this Chapter, Subsection 6.3. The situation here is very similar and we omit the details. 7.2.3. Unconditional a priori estimates We will establish two types of a priori estimates: unconditional a priori estimates and a priori estimates which are obtained by assuming that k and h satisfy some conditions similar to a stability condition. This subsection deals with unconditional a priori estimates; the condi› tional ones will be studied in Subsection 7.3.3, after the development of some preliminary tools in Subsection 7.3.1 and 7.3.2.

Lemma 7.6. The elemen ts u’/(+ tlq remain bounded in the following sense:

lu’/(+i/qI2~d2’ N-l

k

2: m=O

m=O, ...,N-l, i=l, ... ,q, lIu'/( +i/q 11 2 ~ d 2, i = I, ..., n(= q - 1)

(7.88)

~ d 2i = ’ 1, ... , q ,

(7.89)

u

N

2:

m=O

(7.87)

lu hm +i/q _ u hm +i- l/ql2

Ch. III, § 7

Approximation of the Navier-Stokes equations

I:

where n

by

the fractional step method 415

(7.90)

Ift(s)I' ds.

Proof. We write (7.78) with Vh = uW+i/q; using (7.72) we find: luW+ i/qI2 -luW+ i- 1/qI2 + luW+i/q - uW+ i- 1/ qI2 + 2kv IluW +i/q 11Th = 2k(fm+i/q, uW+i/q) ~ 2klfm+ i/qlluW+ i/ql ~ (by (7.67» ~ 2kdolfm+i/qllluW+i/qllih

~

kvlluW+i/q 11Th + k d5 If m+ i/qI2.

Finally, we obtain luW+ i/qI2 - luW+ i-

(7.91)

v

1/qI2

+ luW+i/q - uW+ i-l/qI2

+kvlluW+i/qIlTh~kdij

v

If m+ i/q\2,

Writing (7.79) with v = uW + 1 , we get luW+ 11 2 + luW+ n/qI2 + luW+ 1 - uW+ n/qI2 We add all the relations (7.92) and (7.93) for i

N - 1. We find after some simplification

l~i~l

= O.

(7.92)

(7.93)

= 1, ... , n, m = 0, ... ,

N-l

L

luW+i/q _u7:+ i- 1/qI2

m=O

+ kv

q-l

N-l

i= 1

m =0

L

+ k dij

v

L

lIuW+ i/qIlTh

q-l

N-l

i= 1

m =0

L

L

~ lu21 2

Ifm+i/qI2

By (7.77), lu21 ~ luol, and because of (5.29), N-l

k

~0

m

Ifm

+~ql’

f

T

..

0

If,(s)I' ds.

(7.94)

Ch. Ill,

The evolution Navier-Stokes equations

416

~

7

Therefore, the right-hand side of (7.94) is less than or equal to d 2 and (7.88) and (7.89) are proved. For rand j fixed, ~ r ~N - 1, 1 ~j ~ q, we add the relations (7.92) and (7.93) for m = 0, ... , r - 1, i = 1, ... , q and for m = r, 1 ~ i ~j;(l) dropping several positive terms, we find

°

~lu212+kd5

q-1

N-1

i= 1

m =0

~

v

~

fm+i/qI2~d,

2

r=O, ... ,N-l, j=l, ... ,q.

The proof of the lemma is complete. 7.2.4. The Stability Theorem.

We introduce the approximate functions u~). [0, T]

il~i):

-+

i = 1, ... , q, and uh:

Wh ,

ug)(t)=uW+i/q formk:t;;;.t«m+ l)k. i= 1, ... ,q,

(7.95)

uh is a continuous function from [0, T] into Wh , linear on each interval [mk. (m + l)kJ, m = 0, ... , N - 1, and uh(mk) = uW, m = 0, ..., N.

(7.96)

We infer from Lemma 7.6 the following stability theorem.

Theorem 7.3. The functions u~i),

and uh, defined by (7.95), (7.96) are unconditionally £"(0, T; L 2 (o. )) stable (l :t;;;. t « q). The functions l)ih u~i) (l :t;;;.i :t;;;.n) are unconditionally £2(0, T; L 2 (o. )) stable.

Remark 7.6. (i) As a consequence of (7.89), we have - Vd .- 2 , .n. Iuh(;) - uh(;-1) IL2(0.T;L2(O» -..... VK,U2’Z-

(7.97)

(ii) As in Lemma 7.3, (q)

IUh

- uh IL 2 (0, T;L 2(O»

~

V(kd; 2'

(l>Wearesurnmingin m andifor 0 ’" me ilq "’r+;/q.

(7.98)

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 417

Indeed, using the computations of Lemma 4.8, we see that N-l

2:

12 2(O,T;L2 - k IUn(q) -UhL (o ») - "2 m=O

.; ~

(1

:~~

kq

N-l

q

m =0

i=

I m+1 _ Uh

m 12

Uh

luw ..’[q - uW•i-l/ql)’

2: 2:1

:0;;;;-

2

7.3. Convergence of the Scheme

We want to prove the convergence of the last scheme, (7.78) and (7.79). First we must establish some further a priori estimates.

7.3.1. Auxiliary Results. We denote by A ih (l :0;;;; i «; n) the linear operator from defined by (A ih uh’ vh)

= ((uh’ Vh))ih

Wh

into Wh

YUh’ Vh E Who

(7.99)

Also we denote by B ih the bilinear continuous operator from Wh X Wh into Wh defined by (B1h(uh’ vh)’ wh)

= bih(uh’ vh’ Wh), YUh’ Vh’ Wh E Wh

(7.100)

In terms of these operators, relations (7.78) can be written as

~

(uK’ +i/q -

uK’ +i- l/q) + vA ih uK’ +i/q

+B. (um+i-l/q ih h

um+i/q)=tm+i/q h h’

(7.101)

Lemma 7.7. lIuhllih :O;;;;Si(h)luhl, YUh E Wh,

(7.102)

TheevolutionNavter-Stokes equations

418

Ch. III, § 7

where

2

Si(h)=- (l hi

(7.103)

~i~n).

Proof. This is essentially proved in Proposition 6.1.(1)

Lemma 7.8. IAih Uhl ~ Si(h) lI uhllih' YUh E Who

(7.104)

Proof. Due to (7.99) and (7.102), I(A ih Uh' vh)1

= 1«Uh' Vh »ih I ~

lIuhllih IIvhllih

~ Si(h)lIuhllihlvhl, YUh' Vh E Wh,

and (7.1 04) follows.

7.3.2. Estimates for the form bih. Lemma 7.9. If n = 2, Ibih(Uh' Ph' wh)1

~

V3

IUhll/2I1uhll!k2I1vhllih IWhll/2I1whlllh2,

(7.105)

Ibi;,(Uh' Vh' wh)1

~ V3luhll/2I1uhll!h2Ivhll/2I1vhllli2I1whllih'

(7.106)

IBih(Uh' vh)1

~ 2 v'3Si(h)luhll/2I1uh 1I!J21vh 11/211vh IIIP,

(7.107)

where ii, 11 is a permutation of the set {l, 2}.

Proof. We prove (7.105) for i = 1, j = 2. In order to simplify the notation, we drop the indices h and we set u={ul,u2}, v={vl,V2}, w={wl,w2}'

By the definition of bih ’ b 1h (u, v, w)

=

2~

± In

1/= 1

••

Ul (li 1h VI/)WI/ dx,

(1) Consider a IlXed value of I in the proof of Proposition 6.1.

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 419

By the Schwarz inequality

The integrations in xl and x2 are independent and thus

1,

lu,w,I' dx

{f

+ OO _

00

(Sup tl

IWQ(~l’

Due to 11.(2.15), Sup lUI (xl’ ~2)12 (2

For the same reasons,

-c

{r~

(S,~p

x2)1 2) dX2

lutlx[, h)I') dx, }

t

S

420

’!he evolution Navier Stoka equation,

Ch. Ill, §7

and we can write

f

IUlw212 dx< 121uIIIB2hUlllw211Blhw21, tR 2

2

Ibih(U, v. w)1 <..;3

IUlll/2162hUlll/216lh V2 IIw211/2

2~1

16 1hW211/2. Finally, using the Schwarz inequality again, we estimate the right-hand side of the last relation by

..;3IUlI1/2IB2hUlI1/2 (

(– 2=1

2

2~1

IW21161hW21)<

16 lhv212

)1/2

.

y3Iul /2162hUlI1/21Ivl hlwl/2I1wl ~2

< y3 lu1 1/2 lIulI!~

IIvlllh Iw1 1/2 Ilwl l~

,

and (7.105) follows. In order to establish (7.106) we simply observe that bi'"(uh' Vhf Wh) = bih (Uh' Vhf wh)

and apply (7.105). For (7.107) we observe that Ibth(uh' Vhf Wh)1

<; 2..;3 St(h)luhll/2I1uhW2Ivhll/2 ~Vh IIJL2Iwhl, YUh' Vhf Wh E Who

Lemma 7.10. If n = 3. Ibih(uh' Vhf Wh)1 < 33/2Iuhll/41Iuhlll/4I1vhllth IWhll/4l1whll1'4

(7.108)

Ibi',,(Uh' Vhf Wh)l< 33/2Iuhll/41Iuhlll/4Ivhll/4I1vhlll/4I1whllth

(7.109)

IBth(Uh. vh)1 < 33/2Iuhll/41Iuhll~4

{S3/4(h)Rvh ll th

+ St(h)lvhll/4I1vhlll/4} .

(7.110)

Ch, III, § 7 Approximation of the Navier-Stokes equations by thefractional step method 421

Proof. The proof of (7.108) and (7.109) is the same as the proof of Lemma 6.1; (7.110) is an easy consequence of (7.108) and (7.109). 7.3.3. Conditional a priori estimates. Lemma 7.11. Weassume that n = 2 and that k and h satisfy k S(h)2 E;;; M,

(7.111 )

where M is fixed, and arbitrarily large. Then, we have N-l

k

L:

lIuJ:! +i/311~

m=O

E;;; Const., i

= 1,2, 3,

(7.112)

with a constant depending on M and the data. Proof. We start with the form (7.101) of the scheme. Here q = 3 and, for i = 2, (7.101) becomes uJ:! +1/3 = uJ:!+2/3 + kv A UJ:! +2/3 + kB (uJ:! + 1/3, uJ:! +2/3) 2h

2h

- kfJ:!+2/3,

and, taking the norms 11-1I 2h of each side,

IluJ:! + 1/311 2h E;;; lIuJ:! +2/311 2h + ku IIA 2huJ:! +2/311 2h + kilB 2h (u hm+1/3 ’ u hm + 2/ 3 ) 112h + kllfhm+ 2/3 11 h : By virtue of (7.102), (7.104), and (7.107), the right-hand side of this in› equality is bounded by lIu;:,+1/3112h + kVS2(h)2I1u;:,+2/3Ihh

+ 2ky3 S2(h)2Iu;:’+1/311/2I1u;:’+1/3111~lur+2/311/2. lIur+2/311¥~

+ kS2(h)lf m + 2/ 31

By the Schwarz inequality,

lIur + 1/311~h

+ 4k 2 v2 S2 (h)4 IluJ:! + 2/311~h + 48 k 2 S2(h)4IuJ:! + 1/31I1 uJ:! +1/311 1h lur+2/31I1ur+2/3112h + 4 k 2 S2(h)2If m+ 2/312. E;;;

411u;:’ + 2/311~h

Because of the previous estimates of Lemma 7.6 and (7.111), the sum from m = 0 to N - 1 of the right side of the last inequality is

422

Theevolution Navier-Stokefl eqUlZtIonfl

Ch. III, § 7

bounded by a constant time k-1 and (7.112) is proved for i = 1. In order to prove the same result for i = 2, write (7.101) for i = 2 (q = 3) as

u’C + 2/3 =u’C + 1/3 - kv A2h u’C + 2/3 - k B 2h (u’C + 1/3, u’C + 2/3) - k I’C + 2/3 ,

take the 11•ll l h norm of each side, and proceed as before. To prove (7.112) when i = 3, we write (7.101) for i = 1 (q = 3) as

u’C =u’C + 1/3 - kv A 1h u’C + 1/3 - k 1’C+1/3,

k B Ih (u’C ’ u’C + 1/3)

and then take successively the norms 1I•ll l h and II• 11 2h of each side, and proceed essentially as before. Lemma 7.12. Weassume that n k S(h)11/4

<. M,

= 3 and that k and h satisfy (7.113)

where M is fixed and arbitrarily large. Then we have N-l

k

~

m=O

lIu’C+1/311~<’Const.,

i=1,2,3,4,

(7.114)

with a constant depending only on M and the data.

Proof. The same as the proof of Lemma 7.11. The estimates (7.107) of Bih are replaced by the cruder estimates (7.110) and for this reason (7.111) is replaced by the stronger relation (7.113). From these lemmas we infer a conditional stability theorem. Theorem 7.4. Assuming that k and h satisfy the stability condition (7.111) (if n = 2) or (7.113) (if n = 3), all of the [unctions 5jhU~/\

are L

2(0,

5jhUh’

i = 1, ..., n + 1, j = 1, ..., n,

T; L 2 (0 » stable.

7.3.4. The convergence theorems. Theorem 7.5. Assuming that the dimension is n = 2 and that k and h re-

Ch. III, § 7 Approximation of the Navier-Stokes equations by the fractional step method 423

main tied by (7.111) the following convergence results hold as k and h go to zero:

u1i), uh’ converge to U in L 2 (Q) strongly, L"" (0, T; L 2 (n» weak-star, t = 1, 2, 3, (7.115) fJ jh u1 , fJ jh uh’ converge to Dju in L2 (Q ) weakly, i 1, 2, 3, j 1, 2,

=

=

fJ ih u10 converge to Diu in L 2 (Q) strongly, i

(7.116)

= 1, 2,

(7.117)

where u is the unique solution of Problem 3.1 corresponding to the data f, uo, in (7.10), (7.11).

Theorem 7.6. Assuming that the dimension is n sequence h’, k’ converging to zero (1), such that

= 3, there

exists a

u19, uh’, converge to u in L 2 (Q) strongly, L"" (0, T; L2 (Sl» weak-star,

(7.118)

fJjh' u1P, 8jh’ uh" converge to Diu in L2 (Q ) weakly.

i = 1, ... , 4, j Problem 3.1.

= 1, 2, 3, where

u is some solution of

(7.119)

The principle of the proof of these theorems is very similar to those of Theorems 7.1, 7.2, 5.4 and 5.5 and we will describe only the main lines of the proof. 7.3.5. Proof of convergence. Lemma 7.13. Under the conditions (7.111) (if n = 2) or (7.113) (if n = 3),

J

2 2 +- """" ITI 'Y 1Uh (T)1

dr ~ Const., for some 0< ’Y <

~

,

where Uh is the Fourier transform in t of the function uh extended by outside the interval [0, T]; the constant depends on ’Y, M and the data.

Proof. We add (7.79) and the relations (7.78) for i the result (1)h’ and k’ satisfying (7.113).

°

= 1, ... , n; this gives

424

Theevolution Navier-Stokes eqUlltion.

Ch. III, § 7

n

!.- (ur + 1 -

ur, Vh)

k

+

L

i= 1

«ur +i/q ,

v« ))ih

n

+

L

i= 1

b (u m + i - 1/q u m + i/q V ) ih

h

’

h

’ h

n

=

L

i= 1

(fm+i/q, Vh)’ VVh E Vh,

which is equivalent (see (7.95)-(7.96)) to n

d -(uh(t) vh)+

dt

’

’" c:

i= 1

«U(t)(t),Vh))ih h

n

+

L

i= 1

bih(ug-1)(t),ug)(t),Vh)

= (fih(t),Vh)’ VtE(O,T), VVh EVh.

(7.120)

Using the previous a priori estimates, we then repeat the proof of Lemma 5.6. Proof of Theorems 7.S and 7.6. By virtue of Theorem 7.3, there exists a sequence h’, k’ -+ 0, such that u~~)

-+ u(i)

in L OO (0, T; L 2 (n)) weak star, i

uh’ -+u. in LOO(O, T; L2(n))

weak-star.

°

= I, ... , q,

(7.121) (7.122)

L 2(Q)

strongly, Due to Remark 7.6, uhi,) - uhi,l) converges to in i = 2, ..., q, and uh’ - u~q) also converges to zero; hence all the limits are the same: u(1)

=... =u(q) ="..

(7.123)

Our goal is to show that". is a solution of Problem 3.1. According to Theorem 7.4, the sequence h’, k’ -+ 0, can be chosen in such a way that the following convergence results also hold:

8th’ "~9,

81h , "h’ -+ DIu., in L 2 (Q) weakly.

(7.124)

By Lemma 7.13 and the property (5.92), which was proved in Sub-

Ch. III, § 7 Approximation of the Navier-Stoke: equations by the fractional step method 425

section 6.1.3 for finite differences, uh’-+u.

inL2 (Q) strongly.

(7.125)

Using again Remark 7.6, we find that u~~)

-+ u;

in L2(Q) strongly. i

= 1, ... , q.

(7.126)

The convergence results (7.121), (7.122) and (7.124) to (7.126) allow us to pass to the limit in (7.120), as we did in the proofs of Theorems 5.4,5.5,7.1 and 7.2. We find that u; is a solution of Problem 3.1. If n = 2, the solution of Problem 3.1 is unique; thus u. = u, and the above convergence results hold for the whole sequence k, h -+ O. The proof of the strong convergence (7.117) (n = 2) follows that of Lemmas 5.11 and 7.5, showing that the expression Xh

= Irt,i + 2.

f:

u(1)1 2

#,

2

lD,u(t) - 6ihug)(t)1 dt

converges to zero, as h, k -+ O. Remark 7.7. The scheme (7.78)-(7.79) is the analogue, discretized in the space variables, of the following scheme: } (u m +i/q _ u m +i-lfq) _

vDl

u m +i/q

n

2:

+

j= 1

+~ u m+ 1 EH

U’!’ +i-ljq (D.u m +i/q) I

I

(divum+i-l/q)um+i/q

and (u m+

1 , v)

=f m + i/q , i= 1, ... , n (7.127)

= (u m+ n/q, r),

Vv EH.

Equations (7.127) are associated with appropriate boundary condi› tions: u’" + ;/q vanishes on some part of r which depends on i; more precisely (see Temam [1]):

u" +i/q cos(v,

Xi)

= 0 on r.

The elements u’" +i/q satisfy a priori estimates similar to those proved in Lemma 7.6 for the u,!: «ilq : however, due to a lack of a priori estimates

Theevolutton Navier-Stoke, eqUlltion,

426

Ch. Ill, §8

similar to those established in Lemmas 7.11 and 7.12, we are not able to prove the convergence of this semi-discretized scheme. In the frame of full discretization in both space and time variables, the difficulty is overcome by requiring that k and h satisfy some stability conditions [(7.111), (7.113)]. These stability conditions allow us to obtain enough additional a priori estimates to pass to the limit.

§ 8. Approximation of the Navier-Stokes equations by the Artificial Compressibility Method

In this section we study the numerical approximation of the Navier› Stokes equations by the Artificial Compressibility Method. This is another method to overcome the computational difficulties connected with the constraint "div u = 0". We introduce a family of perturbed systems (depending on a positive parameter e) which approximates in the limit the Navier-Stokes equations and which do not contain this constraint. One of the most common perturbed systems is essentially the equations of a slightly compressible medium with an artificial state equation P

=Po + ep,

He> 0 small",

(8.1)

where P is the density, p the pressure, and Po a constant which repre› sents a first approximation of the density.(l) Linearizing the equations of motion with respect to e, we obtain as a first approximation, the equations n

"’" -au - vAu + L.. at i=l ap . e at + div u = O.

uiDiu + grad p = f,

(8.2) (8.3)

The equations (8.2)-(8.3) are the perturbed equations we shall study. They are easier to approximate than the original Navier-Stokes equa› tions as the constraint "div u = 0" has been replaced by the evolution (l)In all of the preceding sections we always took Po = 1 for simplification. If Po ". I, we arrive at the same result by dividing the equations of motion by Po'

Ch. III, § 8

427

Solution by the artificial compressibility method

equation (8.3). The problems are now the following: - existence and uniqueness of solutions of the perturbed equations (8.2)-(8.3) (associated with appropriate initial and boundary conditions) - does the solution u e’ Pe’ of (8.2)-(8.3) approximate the solution u, p, of Navier-Stokes equations? - discretization of the perturbed problem; convergence of the dis› crete approximations to the solution of the Navier-Stokes equations themselves. In Subsection 8.1 we state in a more precise way a boundary value problem associated with (8.2)-8.3) and give existence and uniqueness theorems for these equations (e > 0 fixed). The situation is essentially the same as for the Navier-Stokes equations: existence and uniqueness of weak solutions in the two-dimensional case, and existence of weak solutions in the three dimensional case(l). In subsection 8.2 we show how the solutions of the perturbed problems converge to the solutions of the Navier-Stokes equations as e -+ O. Subsection 8.3 deals with numerical approximation of the perturbed equations and many numeri› cal methods are available for their approximation. We do not intend to give a systematic study of the different methods, but choose instead to study in detail the approximation of the perturbed equations by the fractional step method. We obtain an implicit scheme, unconditionally stable in some spaces. Finally, we study the convergence of the discrete approximation to the solution of the Navier-Stokes equations as e, h and k tend to zero. 8.1. Study of the perturbed problems 8.1.1. Description ofthe problem The dimension of the space is n = 2 or 3 and assume that Uo is given as in Problem 3.1,

.n is bounded.

Uo E H,

and, for simplicity.j’ is assumed to be in

We (8.4)

L 2(0,

T; H)

fEL 2(0, T; H).

(8.5)

For any given e > 0, we consider the following initial boundary value problem: , u n e }, a vector function from Q = .n X (0, 1) To find u e = {ul e ’ n into IR and Pe’ a scalar function from Q into IR, such that: 0) We do not study existence and behaviour of strong solutions.

Theevolution Navier-Stokes equations

428

aU e -at

Ch. Ill, §8

n "

f-1= 1

vtlu e +

+ grad Pe e ape + div u,

at

=f

Uie

1 . Diu e + - (div 2

Ue)U e

(8.6)

in Q,

= 0 in Q,

(8.7)

= 0, x E ao, t E (0, T),

Ue

= Uo

Ue

=Po

Pe

at t at t

(8.8)

= 0,

(8.9)

= O.

(8.10)

The function Po, which is not given with Problem 3.1, is arbitrarily chosen (but independent of E), Po E L 2(O).

(8.11)

Equation (8.6) contains the term 1/2(div ue)ue which does not appear in either (8.2) or the exact problem (see 3.5)). This is a stabilization term, already used several times in previous sections, which corresponds to the substitution of the form b for the form b (1). Let us assume that ue and Pe are classical solutions of (8.6) - (8. 0), say u, Ere 2 «2), Pe E fll «2). Then if v E!’1(O) and q E ’@(O), multiply› ing (8.6) by v, (8.7) by q, and integrating over 0, we easily obtain the relations d dt

(U e’

A

v) + v «U e, v)) + b (Ue,

d

Ue’

v) + (grad Pe , v)

= (f, v)

.

e - (Pe’ q) + (div u e’ q) = O. dt These relations are still valid, by a continuity argument, for any v in

HA(O) and any q in L 2(0).

This remark leads to a first formulation of the problem:

Problem 8.1. For e > 0 fixed and f, 110, Po given, satisfying (8.4), (8.5), and (8.11), to find ue and Pe such that

ue E L 2 (0 , T; H6(0)),

"* =

s, EL2(0, T; L 2 (0 )),

(8.12)

(1) We recall that b(u, u, u) 0 if div u"* O.For this reason we introduced the form $ :$ 1’, w) if div u 0, and b(u, v, II) = 0, Yu, II. This allows us to obtain perturbed

(u, v, w) = b(u,

problems which admit solutions for all time.

429

Solution by the artificial compressibility method

Ch. III, §8

d

A

dt (Ue, v) + V «U e, v» + b (Ue’ Ue’ v) + (grad Pe, v)

= (f, -

d

dt

(8.13)

v), V v EHA(Q),

(Pe’ q) + (div u., q)

= 0,

V q E L 2 (Q ),

(8.14) (8.15)

Ue(O) ="0, Pe(O)=Po’

Remark 8.1. If u, and Pe only satisfy (8.12) the conditions (8.15) do not necessarily make sense. We will show, as for the exact Navier› Stokes equations, that if ue’ P e satisfy (8.12) - (8.14), then ue and p, are continuous (into large enough spaces) so that (8.15) makes sense. 0 1

A

A

Foru, v EHo(Q), we defmeB(u, v) EH-1(Q) andB(u) EH-1(Q)

bu setting A

'JJ(u, v), w) A

B (u)

A

A

= b (u, v, w), V u, v, w EHA(Q)

= B (u, u).

(8.16) (8.17)

Lemma 8.1. If u belongs to L2 (0, T; belongs to L1 (0, T; H-1 (Q».

HA (Q», the function t 1-+ B(u (1», A

Proof. We already observed that the form b given by A

b(u, v, w)

1

= 2" {b(u,

(8.18)

v, w) - b(u, w, v)}

is, like b, trilinear continuous on

HA (Q); thus

Ib(u, v, w)] .s;;;; d 1l1ullllvllllwll, V u, v, wE A

H6 (Q)

IIB(u, v)II H - , (0) .s;;;; d l l1 u ll llvll, 2

A

IIB(u)I1H-'(n).s;;;;d1I1ull .

The lemma follows from (8.19).

(8.19)

D

Now if ue satisfies (8.12) and (8.13) then, according to (1.6) and (1.8), one can write (8.13) as

d

-
dt

=
A

B (u e ), v),

V v E V.

It is clear that AU e E L 2(0, T; H- 1(Q», B(u e ) ELI (0, T; H-1 (Q», so

430

Theevolution Navier•Stokes equettom

that t + v liu e 1.1,

-

B(u e ) is in L 1 (0, T; H- 1 (0»

U: E L1 (0, T; H-1 (0» ,

U"

=t + vb.ue -

A

B(u e ) •

Ch.III, §8

and thus due to Lemma

}

(8.20)

Similarly, (8.12), (8.14) and Lemma 1.1 imply that

P; E L2 (0, T; H-1 (0»,

ap at

e .ss. + div U

= o. e

}

(8.21)

An alternative formulation of Problem 8.1 is now the following:

°

Problem 8.2. For e > fixed, f, uo, Po given satisfying (8.4), (8.5) and (8.11), to find u e and Pe’

», E L 2(0, T; HMO», u:E L 1 (0, T; H- 1(0»,

(8.22)

P e EL 2(0, T; L 2 (0 » , P; EL 2(0, T; H- 1(0»,

(8.23)

,

Ue -

vSu, + B(ue ) + grad Pe =t, A

(8.24)

ep; + div ue = 0,

(8.25)

». (0) = uo,

(8.26)

We showed that any solution of Problem 8.1 is a solution of Problem 8.2; the converse is very easy to check (using Lemma 1.1) and thus these problems are equivalent. Our goal now is to study the existence and uniqueness of solutions of these problems for e > fixed; then we will see how the solutions of these problems approximate those of Problems 3.1 and 3.2.

°

8.1.2. Existence of solutions of the perturbed problems

Theorem 8.1. For e > 0 fixed, for f, Uo, Po, given satisfying (8.4), (8.5) and (8.11), there exists at least one solution {u e , Pe} ofProblems 8.1 and 8.2. Moreover,

», E L (0, T; L 2», Pe E L (0, T; L2(0», fIO

fIO

(8.27)

and u e (resp. Pe) is weakly continuous from [0, T] into L 2(0) (resp. L 2 (0 » .

Ch. III,

~

Solution by the artificial compres9ibility method

8

431

The existence is established in the next subsection; the weak continu› ity follows from Lemma 8.1 and the weak continuity of u, and Pe with values in B- 1 (n) and B- 1 (n) has already been proved.

8.1.3. Proof of Theorem 8.1. (i) In order to apply the Galerkin procedure, we consider a basis of HA(n) constituted of elements Wi of !l)(n), and a basis of £2 (0) constituted of elements 'i of!l)(O). For each m, we define an approximate solution u em’ P em, of Problem 8.1 by m u em (t) =

and

~

1= 1

,

(u em (t), wk)

m gim (t) wi> P em (t) =

~ ~jm

1= 1

+ v «u em (t), wk)) + "b (uem (t),

+ (grad

Po« (t), wk)’

(8.28)

(t)’i’

u em (t), wk)

= (f(t), wk)’ k = 1, ... , m, (8.29) (8.30)

Moreover, this differential system is required to satisfy the initial conditions Uem

(0)

= uOm’ P em (0) = POm ’

(8.31 )

where uOm (or POm) is the orthogonal projection ofuo (or Po) onto the space spanned by wI' ... , w n (or ' l ' ... "m)’ in £2(0) (resp. L 2 (n )) . The equations (8.29) and (8.30) form a non-linear differential system for the functions glm’ ... , gmm’ ~lm’ ... , ~mm’ As for Theorem 3.1, we have the existence of a solution defined at least on some interval [0, tm ) , a < tm ~ T, and the following a priori estimates show that in fact tm = T. (ii) If we multiply (8.29) by gkm (t), multiply (8.30) by ~Qm (t), and then add all these equations for k :: 1, "’, m, Q :: 1, ... , m, there results ,

(Uem ’ u em) + v lIuem II

+

2

+ "b (uem’ u em’ u em) + (grad P em, u em)

(P;m, P em) + (div u em’ P em)

= (f,

u em)•

Due to (8.18), "b(uem, u em’ u em) = 0, and since u em vanishes on (grad Pem, u em) + (P em, div u em) = O.

ao,

432

Theevolution Navier-Stoke. equatiom

Ch.llI, §8

Thus there remains

d

2

2

2_

{Iueml + elpeml } + 2vlIuemII - 2(/, uem)•

dt

(8.32)

The right side is bounded by d2

21/1 IUemI E;;; 2doIlIlIuemII < vlluem11 2 + ~ 111 2 v so that d dt

2

2

2

d6

2

{Iueml +elPeml }+vlluemll <-III. v

(8.33)

Integration of (8.33) form 0 to s shows that

<: Iuol’ + ’1P01’ +

~~

f:I/(I)I’ dt, 0 < s < t", .

Hence t m = T, and Sup

Sf

[0, T]

{lu em(s)12 + eIPem(s)12 }
d 3 = luol2 + IP ol

2

2JT I/(t)/2 dt.

+ dvo

0

(1)

(8.34)

(8.35)

Next, we integrate (8.33) from 0 to T:

J:

IU.m(TlI’+ .IP.m (TlI’+.

lIu.m(1)11’ dt <: Iuo m I’ + ’lPom I’

(1) We are interested in smaI1 values of Ii thus we 88IIIJI1Cl I bounded from above, say I

< 1.

Solution by theartificial compresribility" method

Ch.III, §8

433

Thus (8.36) (iii) In order to pass to the limit in the nonlinear term we need an estimate of the fractional derivative in time of uem. Setting tP m (t)

,. = f(t) + v6.uem -B(uem),

it follows from (8.36) and (8.19) that IItP m (t)IIH- l

(.0) :E;;;

1!(t)1

+ v lI uem (t)1I + d1 IIuem (t)1I2

(8.37)

a- 1 (0)).

(8.38)

and therefore tP m remains in a bounded set of £1 (0, T;

The relations (8.29) and (8.30) can be written (u;m (t), wk)

+ (grad

Pem (t), wk) = (tP m (t), wk),

k = I, ... , m,

e(p;m (t), ’11) + (div uem (t), ’11) = 0, 2 = I, ... , m.

°

As done several times before, we extend all functions by outside the interval [0, T] and consider the Fourier transform of the different equations. The following relations then hold on lR: d __ __ -(u em’ wk) + (grad Pem, wk) dt

’):’

= (’Pm’ wk) + (uOm’ wk)c5(O)

- (u em (T), wk)c5(T)

e

~dt

CPem’ ’lZ) + (divuem, ’lZ)

= e(po m, ’lZ) c5(0)

- e (Pem (T), ’lZ)

c5(T).

After taking Fourier transforms, there results

,.

,.

2i1TT(Uem (T), wk) + (grad P em (T), wk)

,.

+ (uo m, wk)

,.

,.

= (tP m (T), wk)

- (u em (T), wk) exp (- 2i1TTT),

2i1TTe(Pem (T), ’11) + (div u em (T), ’11) = e(POm’ ’lZ) - e(Pem (T), ’lZ) exp(-2i1TTT).

The evolution Navier-Stokes equations

434

Ch. III, §8

’" (1’) (gkm '" = We multiply the first of the last two equations by gkm ,.,."

"A.

Fourier transform of gkm)’ and the second by ~km (1’) (~iim = Founer transform oft'2m), and then add these relations for k = I, ... , m, Q = I, ... , m, obtaining 2i1l'1' {Iu’"em (1’)1 2 + elP’"em (1’)1 2 } + (grad P’"em (1’), u’"em (1'»

+ (div u'"em (1’), P"em (1’»

=(lP" m (1’), u"em(1'»

+ (uo m, u'"em (1’» + e(POm’ '" P em (1'» - {(u em (1’), u em (1’» + e(Pem (1’), P em(1’»} exp (-2i1l'T1').

(8.39)

The term

" u'"em) + (div uem’ P'"em) (grad Pem» vanishes. With this simplification, we deduce from (8.39) that 2 211’11’1 {Iu’"em (1’)1 2 + e IP" em (1’)1 } <;

’" (1'»1 l(lP’"m (1’), U em (1’) 1 + eIpo IIP" (1')1

+ IUOm Ilu" em m em " + luem(1’)lluem (1’)1 + e IPem (1’)IIP'"em (1’)1

Due to the previous estimates (8.34) and (8.36), 211'11'1 IU em (1’)1 <;

’"

2

"lP’"m (1')"0-

lI~m (T)I H~

r~ lI~m

1

,..

(0) "u em (1')"

+ 2..;;J; I;m (1’)1 + 2..;;J; elPem (1')1.

But (n) ..

. f:

(t)IIH-•(n) dt" (by (8.37»

2

{If(l)1 + v lIu,m (1)11 + doIlu,m(1)11 ) dt

<; (by (8.38» <; Const. ,

EIP,m (T)I .. E

f.,

IP,m WI dt = E[IP,m (1)1 dt

<; (by (8.34» <;...;e

v’d;

T<; Const. ;

Ch. Ill, §8

Solution by the artificial compressibility method

435

finally we have " 2 21TITII"em(T)1 ~cll1"em(T)1I " +c2’ As in the proof of Theorem 2.2, this inequality implies that

f~

ITI'TI;;.m (T)I'dT <;; Const., for some ’Y,

O<’Y<~’

(8.40)

(8.41)

(iv) We want to pass to the limit as m ~ 00 in (8.29) - (8.31) using the estimates (8.34), (8.36) and (8.41). We recall that at the present time e > is fixed, and we are only concerned with a passage to the limit as m ~ 00. There exists a sequence m’~ 00, such that

°

"em’ -r u,

in L2 (0, T;

HA (n»

weakly,

(8.42)

L (0, T; L2 (n» weak-star, and (due to (8.41) and Theorem 2.1), L2 (0, T; L 2 (n» strongly; OO

e. in L

Pem’ ~

OO

(0, T; L2 (n» weak-star.

(8.43)

Let 1/1 be a continuously differentiable scalar function on [0, T] , with 1/1 (n = 0. We multiply (8.29) (resp. (8.30» by 1/1 (t), integrate over [0, T], and then integrate the first term by parts:

-f: f:

(u.m (I), .... 11<’(1)) dr +

+

=

v«u. m(I), .... 11« t))) dr

{b (u. m (I), u.m(I), w.II<(I») + (grad Po«

-f:

f:

f:

(t)' wk1/l(t»} dt

=(uom , .... )11«0)+

([(I), ....W)) dr,

J:

e(P.m (I), r,lI<’ (t))dl +

(POm’ 'lI)1/I(0), I ~ k, Q ~

(8.44)

(eli. u.m(I), r,II<(I» dr

m.

(8.45)

436

Theevolution NllVier-Stoke, eqUlltion,

Ch.III, §8

It is easy to pass to the limit in (8.44) and (8.45) with the sequence m’ and we find

-f:(U,(t), 14\011/(1)) +. f:«u, (I), 14\01/1(1»)) dl

f:

+

{b (u, (I), u,(I), 14\01/1(0)) + (grad 1’,(1),14\01/1(1))}dl

J:(f(t), J:

= (uo, 14\0)1/1(0) +

-·J:

14\01/l(/) dl,

(P,(I), ,,1/1’(1)) dl +

(diY

(8.46)

..,(t), ’,1/1(1)) dr

= e(po, '1l)I/I(O), 1 <; k, ~ <; m.

(8.47)

The relations (8.46) and (8.47) imply that {u" p,} are solutions of Problem 8.1 : the proof is identical to the interpretation of (3.43). The existence of solutions of Problem 8.1 and 8.2 is thereby proved and the proof of Theorem 8.1 is complete.

Remark 8.2. It is useful, in view of passing to the limit e ~ 0, to establish some a priori estimates, independent of e, satisfied by

u" p,.

Due to (8.34), (8.36), (8.42) and the lower semi-continuity of the norm for the weak topology: lu,IL-(o, T; L 1 (0» <; m~

_ IU,m,I L- (0, T; L '

lIu,IIL, (0, T; B~

_ lIu,m’ IIL, (0, T,’ B~

(0)

<;

m~

(0»

<; "V’d;,

(0)

ft.

(8.48) (8.49)

Due to (8.34) and (8.43)

Vi Ip,IL-(O,T;LI(O» <; m~_ve

IP,m,IL-(O, T;L'(O»
(8.50)

Solution by the artiftcilll compreBBibility method

Ch. III, §8

437

Similarly, by inspection of the proof of (8.40) and (8.41), we observe that the constants appearing in these relations are independent of e (and of course m). Thus, by (8.42), we obtain

f

+OO

ITI2")' 1~,(T)12

dt";;;

lim m'-+

_00

f+oo ITI2")’1~,m’(1)12

00

_00

with a constant independent of e, and for

°<

'Y

dr <; Const; (8.51) I

< ’4 .

8.1.4. Uniqueness of Solutions of the perturbed problems. Theorem 8.2. Weassume that n = 2 and otherwise the assumptions are the same as those of Theorem 8.1. There exists a unique solution {u e , Pel ofProblems 8.1 and 8.2 which belongs to LOO(O, T; L2 (.0» X LOO (0, T; L2 (.0», and v, is a continuous function from [0, T] into L2 (.0). The theorem follows from several lemmas, some of them giving add› itional regularity properties of ue Lemma 8.2. If n U

= 2, U E L2(0, T; HA (.0» n LOO (0, T; L2(.0»,

then

E L4 (0, T; L4 (.0»,

BU EL4/3(0, T; L4/3(,Q».

(8.52) (8.53)

Proof. Property (8.52) follows from Lemma 3.3 which gives the estimate IIu(t)II14(n)";;; clu(t)1 2 IIu(t)II 2 , a.e, in t.

(8.54)

For (8.53) we observe first that

b(u, v, w) = I,Fl .~

+~

J

uleDi Vj)Wj dx

n

z In{ 2

2

j=1

(divu)vjWjdx.

(8.55)

The evolution Navier-Stoke, equation,

438

Ch.I1I, §8

The relation (8.55) is obvious if u, v, w E~(O); it is still valid by con› tinuity for u, v, wE JCA(O). In fact, due to the Holder inequality, 2

It (u, v, w)] E;;; i,j=1 ~ +~

lIuiIlL4(n)IDivh2(si)lwh4(n)

2

~

[div uIL2(si)lvjIL4(n)lwjIL4(n)

and therefore

" u, w)1 E;;; clluII L4(n) lIullllwIlL4(n) E;;; (because of (8.54» Ib(u, E;;;clull/2I1uIl3/2I1wIlL4(o), V uEH5(0), VwEL 4(0)

Since the space L 4/3(0) is the dual of L4(0), it follows that IIBuIlL4/'(n) E;;; clul l / 2I1uIl3 /2 , V u EHA(O)

(8.57)

and (8.53) is easily deduced from this estimate.

Lemma 8.3. If ue is a solution ofProblem 8.1 belonging to L- (0, T; L2 (0», and n = 2, then: u e EL2(0, T; HA(O)

n L4(0, T; L4(0»

u~ E L 2 (0, T; H-l (0» + L 4/3 (0, T; L 4/3 (0».

(8.58) (8.59)

Proof. This is merely a consequence of Lemma 8.2; (8.58) is equivalent to (8.52) and for (8.59) we use (8.24):

u; =f+ vtsu, - grad P« -

A-

BUfi'

The first three terms on the right are in L 2 (0, T; B- 1 (0»; the last one belongs to L4/3(0, T; L 4/3(0» (by (8.53».

Lemma 8.4. For any function u, satisfying (8.58) (8.59), , )_ d 2 2 (ue(t), ue(t) - dt lue(t)1 ,

(8.60)

in the distribution sense on (0, T). Moreover, Ufi is almost everywhere equal to a continuous function from [0, T] into L2(0).

439

Solution by the artificial compressibility method

Ch. III, § 8

Proof. The same as the proof of Lemma 1.2, observing that the spaces in (8.58) and (8.59) are in duality. Proof of Theorem 8.2. It remains to prove the uniqueness. For this we drop the indices e and denote by {uI’ PI} , {U2' P2} two solutions of Problem 8.1, and then set u

= ul -

u2' P

=PI -

P2'

By subtracting the relations (8.24) (resp. (8.25» satisfied by ul and u2 (resp. PI and P2), we see that u' - vb.u

'" - BUI '" + grad P = BU2

ep’« div u = O.

(8.61)

Taking the scalar product of (8.60) with u(t) and (8.61) with pet) and adding these equations, we find (u'(t), u(t»

+ e(p'(t), p(t» + v lIu(t)1I 2

= - '"b(u(t), u2 (t), u(t»;

(8.62)

the term (grad P, u) + (P, div u)

has disappeared, and we have used the property of b’":

b'"(v, w, w)

=0

V v, w.

Thanks to Lemma 8.4 we can write (8.62) as -d {lu(t)1 2 + elp(t)12} + 2v lIu(t) 11 2 = - 2b’"(u(t), u2 (t), u(t). dt

The inequalities established in the proof of Lemma 8.2 enable us to estimate the right side of this equation by c3Iu(t)llIu(t)lIl1u2(t)1I +c4Iu(t)III/2I1u(t)1I3/2Iu2(t)II/2I1u2(t)1I

~ vllu(t)1I2 + Cs lIu2(t)1I2Iu(t)1 2 + vllu(t)1I 2 + c6 1u 2(t)1 2I1 u2 (t)1I2Iu (t)1 2 .

Thus (8.63)

Theevolution Navier•Stokes eqUlltions

440

Ch. III, §8

where a is a scalar function in Ll (0,1’); more precisely,

a (t) = (cs + c61u2 (t)1 2) lIu2 (t)1I 2.

By Gronvall’s Lemma and since u(o)

= 0,

p(o)

= 0,

(8.63) implies that lu(t)12 + elp(t)1 2 = 0,

°

~ t ~ T,

and the uniqueness is proved. 8.2. Convergence of the Perturbed Problems to the Navier-Stokes

Equations

= 2, as e -+ 0, the solutions {u e, Pel ofProblem 8.1 converge to the solution U ofProblem 3.1 and the associated pressure p in the following sense: u -+ U in L 2(0, T; Hi (.Q» strongly,

Theorem 8.3. If n

e

i: (0, T; L2(.Q» weak-star,

grad Pe -+ grad pin H- l (Q).

(8.64) (8.65)

Theorem 8.4. Ifn = 3, there exists a sequence ue’ , Pe’, of soluttons of Problem 8.1, such that

ue ’ -+ u in L2 (0, T; L2 (.Q» strongly, L2 (0, T;

HA (.Q»

(8.66)

weakly, tr (0, T; L 2 (.Q» weak-star,

grad Pe’ -+ grad pin H-l (Q) weakly,

(8.67)

where U is some solution ofProblem 3.1 and p denotes the associated pressure. For any other sequence Ue" Pe" such that (8.66) - (8.67) hold, U must be some solution ofProblem 3.1 and P the corresponding pressure. Proof of Theorems 8.3 and 8.4. We pointed out in Remark 8.2 that the solutions ue ’ Pe constructed in the proof of Theorem 8.1 satisfy a priori estimates independent of e. By virtue of these estimates, there exists a sequence e'-+ 0, such that

441

Solution by the artificial compressibility method

Ch. llI, § 8

~

UE,

p

U.

in Loo (0, T; L2 (0)) weak-star,

P E’ ~ X in

L2(O, T; Hb(O)) weakly

(8.68)

tr (0, T; L 2(O)) weak-star.

(8.69)

Due to (8.68), (8.51), and the compactness theorem (Theorem 2.1), we also have UE’~

U.

in L 2 (0, T; L2(O)) strongly.

(8.70)

We can pass to the limit in (8.14), for the sequence e’;

ve’-

d

d

dt(pe’,q)~dt(x,q)

in the distribution sense; hence

,d e - (p, dt

q)~O

E’

in the same sense, and (8.14) gives in the limit the equation (div

U., q)

= 0,

which shows that div U. U.

Vq E L 2(0),

=

°

and hence

EL 2 (0, T; V) n LOO(O, T; H).

(8.71)

Let v be an element ofr; equation (8.13) then gives d

- (uE , v) dt

~

+ v«u E , v)) + b (u E , uE ’

v)

= (f, v)

(8.72)

since (grad PE, v)

= (pE’ div v) = 0.

°

If I/J is a continuously differentiable scalar function on [0, T] with I/J(n = we can multiply (8.72) by I/J(t) integrate in t, and then inte›

grate by parts, obtaining

-f

T 0 (u,(I),

"if"(I»

L T

dr

+v

«u, (I), ""’(I») dr

f; T

+

o

(u,(I), u,(I),

""’(I»

dr

(8.73)

Theevolution Navier-Stokes equations

442

f

Ch. III, §8

T

=(.... v),p(O) + 0«(I), v,p(l»dl, V V E V.

We can pass to the limit in (8.73) using the weak convergence results (8.68) and the strong convergence (8.69). We obtain

-f

T (u. (I), v,p'(I» dl

f f

T

+v

o

((u(t). v,p(I») dr 0

T

+

b(u(I),u(I). v,p(t))dl

o

=(uO’v),p(O) +

J T

0 (((I). v,p(lll dr,

VvE’f:

(8.74)

The relation (8.74) is the same as (3.43) and we conclude as for (3.43) that U. is a solution of Problem 3.1. If n = 2, the solution u of Problem 3.1 is unique and the whole sequ› ence u, converges to u in the sense (8.68) - (8.69). The strong con› vergence in £2(0, T; HA(Q» is obtained with a technique already used and which we skim over: we first prove that (ue(T) - u(T), v) -+ 0, Vv E V,

and this, along with the previous weak convergence results, suffices to prove that the expression

J

T

X, = IU,(Tl - u(TlI' + elp,(TlI' + 2v

0 lIu,(I)

- u(l)II' dz

converges to zero, as e -+ 0. It remains to prove (8.65) and (8.67). For this we write (8.24) as grad Pe
The convergence results for ue show that the right-hand side converges to

!-u’+v!::.u -Bu

" =Bu) (Bu

Solution by the artificial compressibility method

Ch. Ill, §8

443

in B- 1 (Q), and by comparison with (3.129), this is exactly grad p. Hence grad Pf -+ grad p in B-1 (Q); the convergence holding for the whole sequence e, in B- 1 (Q) strongly if n = 2, and for a subsequence e’ in B-1 (Q) weakly if n = 3.

8.3. Approximation of the Perturbed Problems. Our goal is now to approximate the perturbed problems, Problems 8.1 and 8.2. Among the many available methods, we will study the approximation by a fractional step method, with a discretization in the space variables by finite differences. The study will be similar in several respects with the study of the fractional step scheme of section 7.2.

8.3.1. Description of the scheme. The discretization of (.0) and V is the discretization (APX1) and all the notations of Subsection 7.2.1 are maintained. For the approxi› mation of the pressure we will need an approximation of the space L 2 (.n) ; we take simply the space Xh which has already appeared a few times: X h is the space of step functions of type

BA

.L:

1I'h =

~M

1 MEf1. h

WhM'

~M

E

cR

(~M

= 1I'h(M»,

(8.78)

where whM is the characteristic function of the block ah (M) centered at M, whose edges are parallel to the Xi axes and of length hi’ i = 1, ’" , n. The space Xh is equipped with the scalar product induced by L 2 (n ),

f

("h’"h)~

"h(X)"h(x)dx

n

= (hI

.... hn )

.L:.9.1

MEHh

1I'h (M)

1 ’~

(M).

(8.79)

Moreover, the data Po- "0’ and f are given satisfying (8.11), (8.4), and (8.5) and we choose some decomposition of f n

(8.80)

444

Theevolution Navier-Stokes equations

Ch. Ill, §8

as in (7.73) this decomposition is quite arbitrary and the simplest choice could be It = f.li = 0, i = 2, ... , n, The interval [0, T] is divided into N intervals of length k (T = kN) and we set .

fm+lln

1 J(m+l)k

=k

fi(t) dt, i

= 1, ... , n.

(8.81)

mk

The fractional step scheme to be studied involves n intermediate steps, where n is the dimension of the space (n = 2 or 3). We define a family of pairs {u’l( +i/n, 7T’I( +i/n } of Wh X Xh , where m = 0, ..., N - 1 and i = 1, ..., n. These elements are defined success› ively in the order of increasing values of the fractional index m + iln. We start with

u2 = the orthogonal projection o[uo onto Vh in L2 (n), 7T2 = the orthogonal projection o[Po on X h in L 2 (n ).

(8.82) 2(n)) These definitions make sense (Vh C L2(n), X h C L and it is worth

observing that

lu21 ~ luol, 17T21 ~ IPol, V h.

(8.83)

When u’l( +i-I/n, 7T’I( +i-I/n are known, we define u’l( +i/n, 7T’I( +i/n (m = 0, ... , N - 1, i = 1, ... , n) by means of the following conditions: u’l( +i/n E

Wh , 7T’I( +i/n E Xh and,

~(um+i/nh

k

_um+i-I/n V )+v«u m+ i/n v)). h

h

h’ h

Ih

+ b.Ih (u hm +i-l/n u hm +i/n Vh ) _ (7Thm +i/n ’ vv ih.

= (fm +i/n.

~ (7Thm +i/n

k

V 7Th E Xh .

vh)’ V v« E

_ 1rm +i- lin h

Wh ,

7T ') + (V.th

’h

U’!’ +i/n Ih

7T ') =

’h

°

V.

Ih

)

(8.84)

’

(8.85)

There are no conditions on the discrete divergence of the computed elements u’l( since they are allowed to belong to Wh and not only to Vh' The equations (8.84)-(8.85) form a linear variational equation for the pair {u’l( + i/n, 1T'{( + i/n} E Wh X Xh ; the coercivity of the bilinear

-«.

445

Solution by the artificial compressibility method

Ch. Ill, §8

form

+ ek ( 1rh’ 1rh, ) + bih (Uhm +i-l/n ’ Uh’ Uh, ) - ( 1rh’ V ih U;h) (Vih Uih; 1rh),

is ensured by (7.72). The existence ofu7(+i/n, 1r7(+i/n is a consequence of the Projection Theorem.

Remark 8.3. As in Remark 7.3, we can interpret the equations (8.84)› (8.85) in the following way:

~

(u7(+i/n(M) - u7(+i-l/n(M» - v

~rh

u7(+i/n(M)

+ ~ u’!h+i-l/n(M)

~ih

+} ~ih(u’!h+i-l/n

u7(+i/n) (M) +Vih 1r7(+i/n(M)

=17( +i/n (M),

u7(+i/n(M)

V ME s’!h’

(8.86)

~ (1r7(+i/n(M) -1r7(+i-l/n(M» + Vih u’!h+i/n(M)

= 0,

VMEn~,

(8.87)

where .,

17(+l,n (M)

= - - 1- -

hI" .., hh

1

Im+i/n(x)dx,

VMEn~.

(8.88)

G1i(M)

The unknowns when computing u$ +i/n , 1r7( +i/n are the n components of u7( +i/n (M) and the numbers 1r7( + in. Here again (see Remark 7.3) the main point of the fractional step method, is that the above equations are actually uncoupled into several much smaller subsystems which only involve the unknowns u7(+i/n (M), 1r7( +i/n (M), corresponding only to nodes M on the same line parallel to the xi direction. This makes the resolution of (8.86) and (8.87) very easy.

446

Theevolution Nfl'Iier-Stokes eqUiltion,

Ch. III, §8

8.3.2. Unconditional a priori estimates The stability of the Scheme will be established through two types of a priori estimates, as in Section 7.2: unconditional a priori estimates, leading to unconditional stability results, and conditional a priori esti› mates leading to conditional stability theorems, and also used in the proof of convergence. This subsection deals with unconditional a priori estimates. Lemma 8.5. The elements u7: +i/n remain bounded in the following sense: i

= I, ..., n,

(8.89)

N-l

k

L I l u hm + i/ n l l ’~h

m=O

:t::.~ d4 v

, .l

= I ,..., n,

(8.90)

N-l

L:

(8.91)

m=O

N-l

e

L:

m=O

l/nI2:t::.d m+i/n _lI’m+ih ...... 4’

11I’h

where

f:

Proof. We write (8.84) with

lu7: +i/n 12

_

Ph

1'=

I , ... , n.

(8.92)

If,(tll' dt.

= u7:+ i/n; due to (7.72) we find:

1u7: +i- lin 12 + Iu7: +i/n _ u7: +i- lIn 12

+ 2kvllu7: +i/nlll h - 2k(1I’f: +i/n , Dh u7: +i/n)

= 2k(fm+i/n, u7:+ i/n) ~

2kIF+ i/nll u7:+ i/nl

~ (by (7.67» ~ 2k do IF +i/nlllu7:+i/nllih <; kvnu7:+i/nnlh + kd~

V

If m+i/nI2.

(8.93)

Solution by the artificial compressibility method

Cho III, §8

447

After simplification, there remains

luW+ i/nI2 -luW+ i- 1/nI2 + luW+i/n _uW+ i-l/nI2 + kvlluW+i/nllth - 2k(7rW+ i/n , "V ih u’ft,+i/n) kd2

~ -...Q V

Moreover, taking 7rh

11m +i/n /2.

= 7rW +i/n

(8.94)

in (8.85), we get the relation

el7rW+ i/nI2 - el7rW+ i - 1/nI2 + el7rW+ i/n _7rW+ i- 1/nI2

=

Tfl +i/n) + 2k(7rhm +i/n ’ "V , h uih 0

°

(8.95)

.

Next we add all the relations (8.94) and (8.95) for i = 1, ..., n. m = 0, ... , N - 1. After some simplification, and after dropping some positive terms, we find:

1tt,;1 + el~12 2

+

n

N-l

i'" 1

m=O

2:

L:

+el7rW+ i/n _7rW+ i- 1/ nI2} +kv

n

N-l

i= 1

m=O

2:

L:

N-l

L:

(8.96)

m=O

As done in several other places, we bound the right-hand side of this inequality by d

2

4

= lu 0 12 + Ip0 12 + do V

m fT L: 1/(t)1 2 dt . i= 1

o

(see (8.83) and (5.29) and recall that e ~ 1). With the second member bounded by d 4 , relation (8.96) implies (8.90) and (8.91). For rand j fixed, ~ r ~ N - 1, 1 ~ j ~ n, we add relations (8.94) and (8.95) for m = 0,. 00’ r - 1, i = 1, ... , q, and for m = r, i = 1, ... , j;(l)

°

(l)The summation concerns those indices m, i; such that 0 .;; ni + tln

.;; r + ;/n.

448

The evolution Navier-Stokes equations

Ch. III, §8

dropping several positive terms and simplifying, we obtain

luh+ i/q I 2 + eI1l’h+ i/QI 2 ~ lu21 2 + el7l'21 2 2

kd +_0 V

2

i.m

0<. m+i/n <.r+i/n 2

kd +_0 v

n

N-1

;=1

m=O

2 2

Ifm+i/nI2~d4’

r=O, ...,N-I,

j= I, .. ..n,

Thus (8.89) is established and the proof of Lemma 8.5 is complete.

The approximate functions. We consider now the following approximate functions associated with the u’I( +i/n, 11’’1( +i/ n:

uhl): [0, T] -+ Wh ,

uhi)(t) = u’l( +i/n , for mk ~ t < (m + l)k, i = I, ... , n.

(8.97)

uh: [0, T] -+ Wh; uh is a continuous function linear on each interval [mk, (m + I)k], m = 0, .. . ,N- I; and (8.98) uh(mk) =u’l(, m = 0, ..., N. The results of Lemma 8.5 can be interpreted as a stability result: Theorem 8.5. The functions uhi) and uh’ defined by (8.97) are uncondi› tionally Loo (0, T; L2(n)) stable (l ~.i~ n). The functions 6ih uhi) and 6nh uh’ (l ~ i ~ n) are unconditionally L2 (0, T; L2(n)) stable. Remark 8.4. (i) As a consequence of (8.91), we have

lu~i)

- u~i-l)IL2(0,

T; L2 (n»

~

vikd4, i = 2, ... , n,

(ii) As in Lemma 7.3 and Remark 7.6,

N-1

2

m=O ~›

N-1 k 3 m=O

2

(8.99)

449

Solution by the artificial compressibility method

Ch. III, §8

~›

N-I

n

L: ~lL:

kn

3

luW+ i/n _uW+i-IlnI2

m=O

~kn

3

d . 4

(8.100)

Remark 8.5. Let 1f~i), defined by

1fh

1fhi)(t) = 1fW+ i / n

denote the function from [0, T] into X h

for mk ~ t

< (m + l)k,

i = 1, ... , n,

1fh is continuous on [0, T], linear on each interval [mk, (m + l)k], and 1fh (mk) 1fW .

=

(8.101) (8.102)

Then (8.92) amounts to saying that

Vi

11f~i)

- 1f~i-

1)I L 2(0, T; L2(,n» ~ yfkd;"

i

= 2, ..., n

(8.103)

On the other hand, we can prove as for (8.100) that

v. r:e

(n) 11fh - 1fh IL 2(0, T; L2(n» ~

J

knd4

(8.104)

-3- .

8.3.3. Conditional a priori estimates The following estimates are obtained by the methods of Lemmas 7.11 and 7.12, using in particular the lemmas in Sections 7.3.1 and 7.3.2. It is convenient to consider also the linear operators, denoted Dih , continuous from Xh into Wh and such that (D ih 1fh' vh)

=(1fh ' ’Vih vih)'

V1fh EXh , VVh E Who

(8.105)

Using the operators A ih , D ih • Bih , we reformulate (8.84) as

~

(uW +i/n - uW+i-l/n)

+ vA ih uW +i/n

+ B.th (u hm + i-lIn ' U hm + i/n) + D.tb 7f'h -111 + i/n m = 0, ..., N - 1, i = 1, ..., n.

= fmh + i/n ' (8.106)

Lemma 8.6.

(8.107) (l)Essentially Dih 7Th is the vector function whose only non zero component is its i th components which is equal to Vih 7Th.

450

The evolution Navier-Stokes equations

Ch. Ill, §8

Proof. Due to (8.101), I(Dih ’Trh’ vh)1

= 1(’Trh’ V ih Vih)1 ~

= l’Trhll6ih

l’TrhllVih Vihl

Vihl ~ l’Trh1llvhllih ~ 8 i(h)I’Trh ll Vh l,

V Vh E Wh , and (8.107) is proved. Lemma 8.7. We assume that n

= 2 and

that k and h satisfy

(8.108)

k 8(h)2 ~ M,

where M is fixed, and arbitrarily large. Then we have, N-l

k

L:

m=O

Ilur+i/211~

~Const.,

i= 1,2,

(8.109)

where the constant depends only on M and the data.

Proof. We write (8.106) with i

= 2 (and n = 2):

ur +1 =ur +1/2 - kv A 2h ur +1 - k B 2h (ur +1/2, ur +1) +kfm+l - kD2h ....m+l "h h’

Taking the norm 11-11 Ih of each side, using (7.102), (7.1 04), (7.107), and (8.107), we estimate the right hand side as follows: lIur+ 111 1h ~ lIur+l/2111h + kvllA 2h ur+ 111 1h + kIlB 2h(Ur+ 1/2, ur+ 1) 1I 1h + kllD 2h 1TW+ 111 1h + kllfr + I11 1h ~ Ilur +111 2h + kv 8 1(h) 8 2(h) lIur +111 2h + k 8 1(h) 8 2(h)lur +1/211/2I1ur +1/211¥1IuW +11 1/2• •lIuW+111¥1 + k 8 1(h) 8 2(h)I1TW +11 + k 8 1(h)lfr +11.

2V3

By the Schwarz inequality

lIur+1l1h ~5(l

+k 2 v 2 81(h)2)lluW+ll1~h

+ 60 k2 8 1(h) 8 2 (h)2Iur +1/2111uW +1/211 1h luW +lllluW +1112h

+5 k 2 8 1(h)2 8 2(h)2 l'Trr +11 2 +5 k 2 8 1(h)2Ifr+ 11.

Ch. III, §8

451

Solution by the artificial compressibility method

Due to the previous estimates of Lemma 8.5 and (8.108) the sum from 1 of the right-hand side of this inequality is bounded by k-1 times a constant and (8.109) is proved for i = 2. In order to prove that

o to N -

N-l

k

2:

m=O

we write (8.106) with i

ofur+ 1/2

= 1, and estimate

appropriately the norm IH 2h

Lemma 8.8. Weassume that n = 3 and that k and h satisfy k S(h)11/4 -----==--- ~ M,

(8.110)

V

where M is fixed and arbitrarily large. Then we have N-l

k

2:

m=O

"ur+i/n, ~

~

Const, t = 1,2,3,

(8.111)

with a constant depending only on M and the data. Proof. The same as the proof of Lemma 8.7 (and Lemmas 7.1 1, 7.12), the estimates (7.107) of B i h being replaced by the estimates (7.110). We infer from these lemmas a new stability result: Theorem 8.6. Assuming that k, hand e satisfy the stability condition (8.108) (if n = 2) or (8.110) (if n = 3), all the functions

5j h

U{h),

1 ~ i, j ~ n,

are L2 (0, T; L 2 (n)) stable. 8.3.4. The Convergence Theorems. Theorem 8.7. Weassume that the dimension is n remain connected by (8.108). Then, if k, hand e go to zero and k --+0,

e

= 2 and

that k, h, e.

(8.112)

452

Ch. III, §8

The evolution Navier-Stokes equations

the following convergence results hold: u~i), "h’ converge to" in L2 (Q) strongly, weak-star, i = 1,2,

tr (0, T; L2 (0))

(8.113)

8jh "~i), 8jh "h’ converge to Dj " in L2 (Q) weakly, 1 <. t: j <. 2,

(8.114)

8ih "~i)

(8.115)

converge to Di " in L 2 (Q) strongly, i = 1, 2,

where" is the unique solution of Problem 3.1 corresponding to the data I, "0’ in (8.4) and (8.5). Theorem 8.8. We assume that the dimension is n = 3. There exists a sequence h’, v, e’, converging to zero (1) such that

u~~), "h" converge to" in L2 (Q) strongly, L (0, T; L2 (0)) weak-star (8.116) OO

8jh " "~9, 1 <. i, j

8jh , "h" converge to Dj " in L2 (Q) weakly, 3,

<.

(8.117)

where u is some solution of Problem 3.1. For any other sequence h’, r. e’ ~ 0, with k'[e' ~ 0.

(8.118)

and such that the convergence results (8.116) and (8.117) hold, u must be a solution of Problem 3.1. The main lines of the proof are given in Subsections 8.3.5, and 8.3.6. 8.3.5. Proof of Convergence Lemma 8.9. Under the conditions (8.109) (ifn = 2) or (8.110) (ifn = 3),

and if k/e remain bounded,

f

+ OO

__ ITI"lu.(T)I' dT';; Const., for some 0 < 'Y <~,

(8.119)

°

where {;h is the Fourier transform in t of the function "h extended by outside the interval [0, T], the constant depending on "/’ M, the bound on kle, and the data. (l}e’.

v. r, satisfying

(8.106), and k’/e’ -> O.

453

Solution by the artificial compressibility method

Ch. III, §8

Proof. We add (separately) the relations (8.84) and (8.85) for i = 1, ... , n; this gives after an easy calculation n

~

(u m+ 1 _ u m v ) h h » h

k

2:

+

i= 1

«u m+ i/n, v»• h h th

n

+

2:

b ih (Ur/ + i - 1/n , ur/+i/n, vh) - (wW+ 1,

i= 1 n

= i=2:1 (fm+i/n ’ h V )+ ’VVh E

k (m+1 1rh

o,

vh)

n-1

2:

i= 1

(1rm+1_1rm+i/n V. v.) h h ’ ih Ih ,

Wh ,

(8.120)

m ’ ) (D m+1 ’) - 1rh , 1rh + h uh ’ 1rh n-1

=

2:

i= 1

m +i/n ’ ) ( "v ih Uihm + 1 - "v ih Uih ’ 1rh

(8.121 )

V1r}, EXh . We reformulate these equations as: n

dtd

(uh (t), Vh)

;-1

+ "" n

+

2:

i= 1

bih (Uhi-1)(t), Uhi)(t), Vh) n

h

- (1r n)(t),

o, Vh) = i=2:1

(fih (t), vh)

+(1)h(t),Vh)’ VVh EW h , VtE(O, T),

~

dt where

(1rh (t), 1r},)

+ (D h

uhn) (t), 1rh)

(8.122)

= (0 h (t), 1rh)’

V 1r}, E

x..

V t E (0, T),

(and

0h)

are two step functions from [0, T] into Wh (and Xh )

1>h

(8.123)

454

The evolution Navier-Stokes equations

Ch. III, §8

defined by n-l

rph(t)

= i=L1

mk <, t

+1 D .".m+ i/n) (Dih .".m "h ih "h ,

< (m + l)k n-l

(Jh (t)

=

L:

i= 1

'Vih

(8.124)

un

+1 -

un

~h

+i/n,

mk <, t

< (m + l)k,

m = 0, .. . ,N - 1.

(8.125)

The derivation of (8.119) is based on the same principles as those of similar results (see Lemma 5.6 and the proof of Theorem 8.1, point (iii)) except for the treatment of the terms corresponding to rph and (J h ’ which we must estimate suitably. We observe that for t E (mk, (m + l)k), n-l

L:

+1 ""m + i/n ( .".m "h - "h ’

i= 1

(

.n =1

L..

I ~h

2) Vih 1

ih

ih

1/2

"

<’c /Iv /I 1

'V v )

h h

(–

i=2

l7r m + i /n_7r m + (i- l) /n I2\ 1/2

h

h

)

Thus, denoting by 1I-II.h the dual norm of 1I-lI h on Wh ,

mk <, t

< (m + l)k

so that (8.126)

455

Solution by the artificial compressibility method

Ch. III, § 8

Similarly, if t E (mk, (m + l)k), n-l

IOh(t)1

~

L

i= 1

" I

m + 1 - "v ih Uih m + iln 1,

v ih Uih

n

L

i= 2

f:

and thus (using (8.91), (8.108) and (8.110))

10.U)I' dt" Const.

(8.127)

The estimates (8.126) and (8.127) suffice in order to bound the terms corresponding to ¢h and 0h in (8.122) and (8.123). Proof of Theorems 8.7 and 8.8. (ii) In virtue of Theorem 8.5, there exists a sequence h’, k’ -+ 0, such that uh~)

-+ u(i) in

Lea (0, T; L2 (n)) weak-star, 1, ’" , n

uh’ -+ u; in Lea (0, T; L2 (n)) weak-star.

(8.128) (8.129)

Due to Remark 8.4, Uhi ) - uK-l) converges to 0 in L 2 (Q) strongly (2 ~ i ~ n) and uh’ - Uh~) does too. Therefore, all the limits are equal: u(1)

= ... = u(n) = u. ,

(8.130)

and we would like to prove that u; is a solution of Problem 3.1. According to Theorem 8.6, the sequence h’, k’ -+ 0, can be chosen in such a way that we also have

5jh , Uh9, 5jh , uh’ -+ Di u; in L 2 (Q) weakly.

(8.131)

We can also choose the sequence so that k'[e' -+ O. Then, by virtue of Lemma 8.9 and the property (5.92).(1) uh’ -+ u; in L2 (Q) strongly.

(8.132)

Again using Remark 8.4, we get

UhP -+ u;

in L2 (Q) strongly, 1 ~ i ~ n.

(l)See the proof in Subsection 6.1. 3 for finite differences

(8.133)

Theevolution Navier-Stokes equations

456

Ch. III, §8

= 0 by passing to the limit in (8.85). Let a be a function in.@(n) and let 7Th be defined by

(ii) Let us show now that div u.

1l'h (M) = a(M),

VM E

f],h.

It is clear that

1l'h ... a in roo (n), as h-« O.

(8.134)

Let I/; denote a continuously differentiable scalar function on [0, T] with I/;(T) = O. If we multiply (8.85) by I/;m = I/;(mk), and then add these relations for m = 0, ..., N - 1, i = 1, ..., m, we get N

e

L

m=l

+k

=

n

N-l

i= 1

m =0

L

L (v.th u1!'ih + i/n

.I,m

,’I’

7T’h )

7Th 1/;(0)).

(8.135)

It is easy to pass to the limit in (8.135) using the estimates (8.89) on the 7T7(, and the above weak convergence. The limit relation is

f:

(8.136)

(div u; (I), o 1/1(1)) dl = 0,

and since a E.@(n) and I/; are arbitrary, (8.136) is equivalent to div u;

=0

so that (8.137) (iii) It remains to pass to the limit in (8.122). Let v be an element of 1/ and let us write (8.122) with vh = rh v where rh = the restriction operator of the approximation (APX 1) (D h rhv = 0). Let I/; be as before and multiply (8.122) by I/;(t), integrate in t, and then integrate

by parts; we find T

-

o (Uh(I), 1/1’(1) vh dr +

J

n

~,

f.

T

v

«U10(1), Vh 1/I(t)))'h dl

n

+ ~

i= 1

= +

f: (u~’ f: (f,. V.

Solution by the artificial compressibility method

Ch. III, § 8

~1

I:

b"

-

(I),

(~.

(I),

v.

1) (I),

u~1)

457

(I), 1/;(1) V. ) dr

1/;(1)) dl + (u2,

V.

)1/;(0)

(8.138)

1/;(1)) dz.

Passing to the limit in (8.138) for all the terms except the last (and for the sequence h’, k', e’) is standard. The last term in the right-hand member converges to zero because of estimate (8.126) on ifJh' and since we assumed that k’[e’ -+ O. This is the only point of the proof where we need this hypothesis on the ratio k]e. In the limit we get an equation similar to (3.43) (with" replaced by u; ), from which we infer that u, is a solution of Problem 3.1 (as done in Theorem 3.1). If n = 2, the solution of Problem 3.1 is unique: thus a,
= Iuf{ + 2v

I:

"(1)12 + elnfl 2

,t

converge to 0, as h, k, e

-+

ID,u(l) - 5,.

O.

u~i)(I)1

dl

COMMENTS AND BIBLIOGRAPHY

Chapter I Section I contains a preliminary study of the basic spaces V and H; the trace theorem is proved by the methods of J. L. Lions-E. Magenes, see ref. [I]. The characterization of H 1 given here is based on a theorem of De Rham of the current theory. A more elementary proof is given in O. A. Ladyzhenskaya [1] for n = 3. A simplified version of O. A. Ladyzhenskaya's proof valid for all dimensions, was given in R. Temam [9]. Remark 1.9 gives another way for avoiding De Rham's theorem. We have not given any systematic study nor review concerning the Sobolev spaces. We restrict ourselves to recalling properties of theses spaces when needed (Sections 1.1 of Chapters I and II in particular). As mentioned in the text, the reader is referred for proofs and further material to J. L. Lions [1] , J. L. Lions and E. Magenes [1], J. Necas [1] , L. Sobolev [I], and others. The variational formulation of Stokes equations was first introduced (in the general frame of the non-linear case) by J. Leray [1], [2], [3], for the study of weak or turbulent solutions of the Navier-Stokes equations. The existence of a solution of the Stokes variational problem is easily obtained by the classical Projection Theorem, whose proof is recalled for the sake of completeness. The study of the non-variational Stokes problem, and the regularity of solutions is based on the paper of Cattabriga [I] (if n = 3) and on the paper of Agmon, Douglis and Nirenberg [I] on elliptic systems (any dimension); these results are recalled without proofs. The concept of approximation of a normed space and of a variational problem was studied in particular by J. P. Aubin [I] and J. Cea [I]; the presentation followed here is that of R. Temam [8]. The discrete Poincare Inequality (Section 3.3) and the approximation of V by finite differences are in J. Cea [I]. The approximation of V by conforming finite elements was first studied and used by M. Fortin [2] ; our description of the approximations (APX 2), (APX 3) (conforming finite elements), follows essentially M. Fortin [2]. In this reference one can also find many results of computations using this type of discretization. The idea of using the bulb function is due to P. A. Raviart; the presentation of the approximation (APX 2') given here, is new. The approximation (APX 4) has been studied and used by J. P. Thomasset [1]. The material related to the nonconforming finite elements for the approxi-

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459

mation of divergence free vector functions is due to Crouzeix, R. Glowinsky, P. A. Raviart, and the author. Other aspects of the subject (non-conforming finite elements of higher degree and more refined error estimates) can be found in Crouzeix and P. A. Raviart [I] ; for numerical experiments, see F. Thomasset [2] and also P. Lailly [I] in the case of the axisymmetric three dimensional flow. For other applications of finite elements in fluid mechanics, see Oden, Zienkiewicz, Gallagher, Taylor, [1], and the proceedings of a conference held in Italy, June 1976 (to appear). Concerning the general theory of finite elements, let us mention the synthesis works of I. Babushka and A. K. Aziz [1], P. G. Ciarlet [1], P. A. Raviart [2], G. Strang and G. Fix [1], and the proceedings edited by A. K. Aziz [1]. For more references on finite elements (in general situations) the reader is referred to the bibliography of these works. The description of finite element methods given here is almost completely self-contained; we only assume a few specific results whose proofs would necessitate an introduction of tools quite remote from our scope. After discretization of the Stokes problem, we have to solve a finite dimensional linear problem where the unknown is an element uh of a finite dimensional space Vh • There are then two possibilities: (a) either this space Vh possesses a natural and simple basis, such that the problem is reduced to a linear system with a sparse matrix for the components of uh in this basis; in this case we solve the problem by resolution of this linear system; (b) or, if not, the finite-dimensional problem is not so simple to solve (ill-conditioned or non-sparse matrix), even if it possesses a unique solution. In this case, appropriate algorithms must be introduced in order to solve these problems; this is the purpose of Section 5. The algorithms described in Section 5 were introduced in the frame of optimization theory and economics in Arrow, Hurwicz and Uzawa [I] ; the application of these procedures to problems of hydrodynamics is studied in J. Cea, R. Glowinski and J. C. Nedelec [1], M. Fortin [2], M. Fortin, R. Peyret, and R. Temam [I]. See in D. Begis [1] , M. Fortin [2] , an experimental investigation of the optimal choice of the parameter p (or p and a); a theoretical resolution of this problem in a very particular case is given in Crouzeix [2]. The approximation of incompressible fluids by slightly compressible fluids, as in Section 6, has been studied by J. L. Lions [4) and R. Temam [2). The full asymptotic development of a, given here is new and due to M. C. Pelissier; for further investigations of this point

460

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the reader is referred to M. C. Pelissier [I]. Cf. also new related developments in R. S. Falk [I] and M. Bercovier [I]. Chapter II Section I develops a few standard results concerning the existence and uniqueness of solutions of the nonlinear stationary Navier-Stokes equations. We follow essentially O. A. Ladyzhenskaya [I] and J. L. Lions [2]. A more complete discussion of the regularity of solutions and of the theory of hydrodynamical potentials can be found in O. A. Ladyzhenskaya [I] ; for regularity, see also H. Fujita [I]. The stationary Navier-Stokes equations in an unbounded domain have been studied by R. Finn [I] -[5], R. Finn and D. R. Smith [I], [2].

Some recent theoretical results concerning the stationary NavierStokes equations will be given in C. Foias and R. Temam [2], R. Temam [11]. Section 2 gives discrete Sobolev inequalities and compactness theorems, whose proofs are very technical. The principle of the proofs in the case of [mite-differences parallels the corresponding proofs in the continuous case (see, for instance, J. L. Lions [I], J. L. Lions-E. Magenes [I]). The proof of discrete Sobolev inequalities has not been published before, the proof of the discrete compactness theorem can be found in P. A. Raviart [I]. For conforming finite elements the proofs are much simpler: in particular, for the discrete compactness theorem, the problem is reduced by a simple device to the continuous case. For non-conforming finite elements the proof of the Sobolev inequality is based on specific techniques of non-conforming finite element theory. The discrete compactness theorem is proved by comparison between conforming and non-conforming elements; these results are new. The discussion of the discretization of the stationary Navier-Stokes equations follows the principles developed in Chapter I. The general convergence theorem is similar to that of Chapter I and the same types of discretization of V are considered; the differences lie in the lack of uniqueness of solutions of the exact problem. The numerical algorithms of Section 3.3 have been introduced and tested in Fortin, Peyret and Temam [I]. The non-uniqueness of stationary solutions of the Navier-Stokes and related equations has been investigated in recent years. The main results in this direction are due to P. H. Rabinowitz [2] and W. Velte [I], [2]. In [2] Rabinowitz establishes the non-uniqueness of solutions of the

Commentsand Bibliography - Chapter III

461

convection problem by explicitly constructing two different solutions (the first is the trivial case when the fluid is at rest, the second is constructed by an iterative procedure). The works of W. Velte are based on topological methods, the bifurcation theory and the topological degree theory; the problem considered in [1] is the convection problem as in Rabinowitz [2]. In [2], W. Velte proves the non-uniqueness of solutions of the Taylor problem and the situation is very similar to the problem for which existence is proved in Section 1, although not identical. Section 4 follows closely this presentation. For other applications of bifurcation theory see in particular, J. B. Keller and S. Antman [I], 1. Nirenberg [1], P. H. Rabinowitz [4] and volume 3, number 2 of the Rocky Mountain J. of Math. (1973). Chapter III The existence and uniqueness results for the linearized Navier-Stokes equations (Section I) are a special case of general results of existence and uniqueness of solutions of linear variational equations (see for instance, J. 1. Lions-E. Magenes [I], vol. 2). For completeness, we have given an elementary proof of some technical results which are usually established as easy consequences of deeper results {i.e. Lemma L I which is more natural in the frame of vector valued distribution theory (1. Schwartz [2]) or Lemma L2 which can be proved by interpolation methods (J. 1. Lions-E. Magenes [I])}. Theorem 2.1 is one of the standard compactness theorems used in the theory of nonlinear evolution equations. Other compactness theorems are proved and used in J. 1. Lions [2] . The existence and uniqueness results related to the non-linear Navier-Stokes equations and given in Sections 3 and 4 are now classical and prolong the early works of J. Leray [I], [2], [3] ; see E. Hopf [ 1] , [2], O. A. Ladyzhenskaya [I], J. 1. Lions [2], [3], J. 1. Lions and G. Prodi [I], and J. Serrin [3]. Further results on the regularity of solutions and the study of the existence of classically differentiable solutions of the Navier-Stokes equations can be found in the second edition of O. A. Ladyzhenskaya [I]. For the analyticity of the solutions see C. Foias and G. Prodi [I], H. Fujita and K. Masuda [I], C. Kahane [l], K. Masuda tn, and J. Serrin [3]. Let us mention also two recent and completely different approaches to the existence and uniqueness theory that we did not treat here. The first one is that of E. B. Fabes, B. F. Jones, and N. M. Riviere [I] based on singular integral operator methods and giving existence and uniqueness results in IF spaces. The other one is the method of D. G. Ebin

462

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and J. Marsden [I] connecting the Navier-Stokes initial value problem with the geodesics of a Riemann manifold and thus using the methods of global analysis. The material of Section 5 containing a discussion of the stability and convergence of simple discretization schemes for the Navier-Stokes equations is essentially new; a similar study for different equations or different schemes was presented in R. Temam [2], [3], [4]. Stability and convergence of some unconditionally stable one step schemes are given in O. A. Ladyzhenskaya [5] ; for fractional step schemes see A. J. Chorin [2], O. A. Ladyzhenskaya and V. I. Rivkind [I]. In all these references except in A. J. Chorin [2] the convergence is proved, as here, by obtaining appropriate a priori estimates of the approximated solutions and the utilization of a compactness theorem; in [2], A. J. Chorin assumes the existence of a very smooth solution and compares the approximated and exact solutions. Section 7.1 is essentially an introduction to Section 7.2. The fractional step scheme described in Section 7.2 (the Projection Method) was independently introduced by A. J. Chorin [I] , [2], [3] and the author R. Temam [3]; A. J. Chorin considers a slightly different form of the scheme, without the stabilizing term Y2 (div u) u (i.e. without replacing b by 75). Applications and other aspects of this scheme are developed in particular in C. K. Chu and G. Johansson [I], C. K. Chu, K. W. Morton and K. V. Roberts [I] , M. Fortin, R. Peyret and R. Temam [I] , M. Fortin [I], M. Fortin and R. Temam [I], G. Marshall [I] , [2] and C. S. Peskin [I], [2]. This scheme is an interpretation of the fractional step method introduced and studied by G. I. Marchuk [I] and N. N. Yanenko [I] (see Section 8). The approximation of the Navier-vStokes equations by the equations of slightly compressible fluids (subsection 8.1) was introduced by N. N. Yanenko [I] who considers slightly more complicated perturbed equations. The introduction of these perturbations enabled N. N. Yanenko to use the fractional step method which is studied in Subsection 8.2. Let us point out that the schemes of Section 7 are fractional step schemes not needing the consideration of perturbed equations. The proof of convergence of the fractional step scheme which is given here is due to R. Temam [3], [4] and follows the method introduced in R. Temam [I]. For other aspects of the Fractional Step Method, see G. I. Marchuk [I], N. N. Yanenko [I], [2] and their bibliographies; see also R. Temam [I], [6], [7]. Other types of perturbed problems, whose purpose is to overcome the

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463

difficulties of the constraint "div u = 0" (but not to apply fractional step methods) are studied in J. L. Lions [4] and R. Temam [2]. For the alternating direction methods see O. A. Ladyzhenskaya and V. I. Rivkind [l]. The material of Sections 5 to 8 is only a very small part of a considerable amount of work on the approximation of fluid mechanic equations; up-to-date results and very useful references can be found in the proceedings edited by O. M. Belotserkovskij [1], M. Holt [2], H. Cabannes and R. Temam [I], and Richtmyer [1]. See also K. Bryan [1] , C. K. Chu and H. O. Kreiss [I], and the list of references compiled by the Los Alamos Scientific Laboratory. Many other problems can be handled by the methods used here. For the Navier-Stokes equations properly speaking one can consider different boundary conditions (see looss [1]), or periodic solutions (G. Prouse [I], [2]), variational inequalities (Lions [2]); Stochastic Navier-Stokes equations are studied in Bensoussan-Temam [I], Foias [I] and Vishik [1]. Optimal problems for systems governed by the NavierStokes equations appear in Cuvelier [1]. The difficulties encountered in the mathematical theory of the NavierStokes equations lead several authors to reconsider the fluid mechanic hypotheses leading to these equations and to propose new models with a better mathematical behaviour; see Kaniel [I], Ladyzhenskaya [1]. Similar models involving other equations (most often the NavierStokes equations coupled with other equations) are: the convection equations whose treatment is almost identical to the treatment of the Navier-Stokes equations, several fluid models, pollution (Marshall [I]) or blood models (Peskin [1]), and oceanography models (having the appearance of a concentration equation). More elaborated are the magnetohydrodynamic equations and the Bingham equations (see Duvaut-Lions [I], [2]) which are an example of non-Newtonian fluids. The mathematical theory of the Euler equations has not been developed here. For a treatment based on analytical methods, cf. C. Bardos [1], T. Kato [I], [2], J L. Lions [2], R. Temam [10], [12], Yudowitch [I]. Some results related to the behaviour of the Navier-Stokes equations as v-+>O are given in Lions [2], Yudowitch [1.]. A similar problem for a model equation related to the Burgers equation is completely studied in C. M. Brauner, P. Penel and R. Temam [I], P. Penel [I] ; cf. also C. Bardos, U. Frish, P. Penel and P. L. Sulem in R. Temam [12].

REFERENCES S. AGMON, A. DOUGLIS, L. NIRENBERG [1) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12, 1959, p.623-727. [2) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math., 17, 1964, p.35-92. B. ALDER and S. FERBACH [1) Editors, Methods in Computational Physics, Academic Press, New York, 1965. A. A. AMSDEN See F. H. Marlow and A. A. Amsden. A. A. AMSDEN and C. W. HIRT [I] YAQUI: An arbitrary Lagrangian-Eulerian Computer Program for Fluid Flow at all Speeds, Los Alamos Scientific Laboratory, Report LA-5100, March 1973. [2] A simple scheme for generating General Curvilinear frids, J. Compo Phys, to appear. S. ANTMAN See J. B. Keller and S. Antman. A.ARAKAWA [1] Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow (part 1) J. Compo Physics, 1, 1966, p.119-143. D.G.ARONSON [1] Regularity property of flows through porous media, SIAM J. Appl. Math., 17, 1969,

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Regularity properties of flows through porous media: a counterexample, SIAM J. AppL Math., 19, 1970, p, 299-307. Regularity properties of flows through porous media: the interface, Arch. Rat. Mech. Anal., 37, 1970, p.1-10.

K. ARROW, L. HURWICZ, H. UZAWA [1] Studies in nonlinear programming, Stanford University Press, 1968.

J. P. AUBIN [1]

Approximation of Elliptic Boundary Value Problems, Wiley Interscience, New York, 1972.

A.K. AZIZ [1] Editor, Proceedings of the Symposium on Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, University of Maryland, Baltimore County (June 1972), Academic Press, 1972.

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L BABUSHKA and A. K. AZIZ [1] Lectures on Mathematical Foundations of the Finite Element Method, Technical Report B.N. 748, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, 1972. C. BARDOS [I] Existence et unicite de la solution de l'equation d'Euler en dimension deux, J. Math. Anal., Appl., 40, 1972, p.769-790. C. BARDOSand L. TARTAR [1] Sur l'unicite retrograde des equations paraboliques et quelques questions voisines, Arch. Rat. Mech. Anal, SO, p.l0-25. D. BEGIS [1] Analyse numerique de l'ecoulement d'un fluide de Bingham, These de 3eme cycle, Universite de Paris, 1972. G. M. BELOTSERKOVSKII [1] Editor, Proceedings of the First International Conference on Numerical Methods in Fluid Dynamics, Novossibirsk, August, 1969; Publication of the Academy of Science of U.S.S.R., 1970. A. BENSOUSSAN and R. TEMAM [1] Equations Stochastiques du type Navier-Stokes, J. Funct. Anal., 13, 1973, p.195-222. M. BERCOVIER [I] Regularisation duale et problemes variationnels mixtes, These, 1976. BORSENBERGER , [1] These de 3eme cycle, Universite de Paris-Sud, Ii paraitre.

J. P. BOUJOT, J. L. SOULE and R. TEMAM [I] Traitement Numerique d'un Probleme de Magnetohydrodynamique, in M. Holt [2]. C. M. BRAUNER, P. PENEL and R. TEMAM [1] Sur une equation d'evolution non lineaire liee Ii la theorie de la turbulence, C.R.A.S., Paris 279, Serie A, 1974, p.65-68 et 115-118, and a developed paper to appear in Annali Scuola Norm. Sup. di Pisa. K.BRYAN [I] A numerical investigation of a nonlinear model of a wind driven ocean, J. Atmos. Sci. 20, 1963, p.594-606. [2] A numerical method for the study of circulation of the world ocean, J. of Compo Phys.,4, 1969, p.347-376. [3] Editor, Proceedings of a Conference on Oceanography, held in 1972. BUIANTON [1] Fractional derivatives of solutions of nonlinear evolution equations in Banach spaces. To appear.

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IL CABANNESand R. TEMAM [1] Editors, Proceedings of the Third International Conference on Numerical Methods in Fluid Dynamics, Paris, Iuly 1972, Lecture Notes in Physics, 18 and 19, SpringerVerlag, 1973.

1.. CATIABRIGA [1] Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova, 31, 1961, p.308-340. I.CEA [1] [2]

Approximation variationelle des problemes aux limites, Ann. Inst. Fourier, 14, 1964, p.345-444. Optimisation, Theorie et Algorithme Dunod, Paris, 1971.

J.CEA,R.GLOWINSKIandJ.C.NEDELEC [1] Minimisation de fonctionelles non differentiables, Conference on Applications of Numerical Analysis (Dundee, March 1971). Lecture Notes in Mathematics No.128 Berlin, Heidelberg, New York, Springer Verlag 1971. A.I. CHORIN [1] A numerical method for solving incompressible viscous flow problems, I. of Compo Phys., 2,1967, p.12-26. [2] Numerical solution of the Navier-Stokes equations, Math. Comp., 23,1968, p.341354. [3] Numerical solution of incompressible flow problems, Studies in Num, Anal, 2, 1968, p.64-71. [4] Computational aspects of the turbulence problem, in M. Holt [2].

C. K. CHU and G. JOHANSSON [1] Numerical studies of the heat conduction with highly anisotropic tensor conductivity II. To appear. G. K. CHU and H. O. KREISS [1] Computational Fluid Dynamics, a boole, in preparation. C. K. CHU, K. W. MORTON and K. V. ROBERTS [1] Numerical studies of the heat conduction equation with highly anisotropic tensor conductivity; in Cabannes-Temam [1]. P. G. CIARLET [1] Numerical analysis of the finite element method, Presses de I'Universite de Montreal, 1975, and a book tobe published by North Holland. P. G. CIARLET and P. A. RAVIART [1] General Lagrange and Hermite interpolation in Rn with applications to finite elements, Arch. Rational Mech. Anal., 46 (No.3), 1972, p.177-199. [2] Interpolation theory over curved elements, with applications to imite element methods, Computer Methods in Applied Mechanics and Engineering, 1, 1972, p.217249. [3] The Combined Effect of Curved Boundaries and Numerical Integration of Isoparametric Finite Element Methods, in I. Babushka [1]. [4] Theory of Finite Element Methods, a book, to appear.

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P. G. CIARLET and G. WAGSHALL [1] Multipoint Taylor formulas and applications to the finite element method, Nurn. Math., 17, 1971, p.84-100.

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Application de la mecanique des milieux continus Ii la Biomecanique, Cours de 3eme cycle, Universite de Paris VI, 1972.

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Resolution Numerique des Equations de Stokes et Navier-Stokes Stationnaires, Seminaire d'Analyse Numerique, Universite de Paris VI, 1971-72. Thesis, Universite de Paris VI, 1975.

M. CROUZEIX and P. A. RAVIART [1] Conforming and non conforming Finite Element Methods for Solving the Stationary Stokes Equations, to appear. CUVELIER [1] Thesis, University of Delft, 1976. S. C. R. DENNIS [1] The numerical solution of the vorticity transport equation, Report CERN, Geneva, 1972.

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Les espaces du type de BEPPO LEVI, Ann. Inst, Fourier,S, 1954, p.305-370.

A. DOUGLIS See S. Agmon, A. Doughs and L Nirenberg. G. DUVAUT and J. L LIONS [I] Les inequations en Mecanique et en Physique, Dunod, Paris, 1971, English translation, to appear. [2] Inequations en therrnoelasticite et magnetohydrodynamique, Arch. Rat. Mech, AnaL,46, 1972, p.241-279. D. G. EBIN and J. MARSDEN [1] Groups of diffeomorphisms and the motion of an incomprehensible fluid, Annals of Math., 92, 1970, p.l02-163. [2] Convergence of Navier-Stokes fluid flow as viscosity goes to zero. Initial boundary value problem. To appear. S. FERNBACH See B. Alder and S. Fernbach.

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On the exterior stationary problem for the Navier-Stokes equations and associated perturbation problems, Arch. RationalMech. AnaL, 19, 1965, p.363-406. On the steady state solution of the Navier-Stokes equations III, Acta Math., lOS, 1961, p.197-244. Estimates at inimity for stationary solutions of the Navier-Stokes equations, BulL Math. Soc. Sci Math. Phys. R. P. Roumanie, 3 (51),1959, p.387-418.

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On steady state solutions of the Navier-Stokes partial differential equations, Arch. RationalMech. Anal, 3,1959, p.381-396. Stationary solutions of the Navier-Stokes equations, Proe. Symp., Appl Math., 17, 1965, p.121-153.

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R. S. FALK [1] An Analysis of the penalty method and extrapolation for the Stationary Stokes equations. Advances in Computer Methods for P.D.E's, Proceedings of A.I.C.A. Symposium. R. Vichnevelsky editor.

G. J. FIX See G. Strang and G. J. Fix. C.FOIAS [1] Essais dans l'etude des solutions des equations de Navier-Stokes dans l'espaceL'unicite et la presque periodicite des solutions petites, Rend. Sem, Mat. Un. Padova XXXII, 1962, p.261-294. [2] Solutions statistiques des equations d'evolution non lineaires, Lecture at the C.I.M.E. on non-linear problems, Varenna 1970, Ed. Cremoneze Publishers. [3] Statistical study of Navier-Stokes equations I and II, Rend. Sem, Math. Un. Padova, 48,1973, p.219-348 and 49,1973, p.9-123. [4] Cours au College de France, 1974. C. FOIAS and G. PROm [1] Sur Ie comportement global des solutions non stationnaires des equations de Navier-Stokes en dimension 2, Rend. Sem, Mat. Padova XXXIX, 1967, p.I-34. C. FOIAS and R. TEMAM [1] On the stationary statistical solutions of the Navier-Stokes equations and Turbulence, Publie. Math. d'Orsay, 1975. [2] Structure of the set of stationary solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., to appear. [3] Remarques sur les equations de Navier-Stokes stationnaires et les phenomenes successifs de bifurcation, Annali Scuola Norm. Sup. di Pisa, volume dedicated to J. Leray, to appear. M.FORTIN [1] Approximation d'un operateur de projection et application un schema de resolution numerique des equations de Navier-Stokes, These de 3eme cycle, Universite de Paris XI, 1970. [2] Calcul numerique des ecouIements des fluides de Bingham et des fluides newtoniens incompressibles par la methode des e16ments finis. These, Universite de Paris, 1972.

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Resolution numeriques des equations de Navier-Stokes pour un fluide incompressible, J. de Mecanique, 10, No.3, 1971, p.3S7-390, and an announcement of results in M. Holt [2]. J. E. FROMM [1] The time dependent flow of an incompressible viscous fluid, Methods of Compo Phys. 3, Academic Press, New York, 1964. [2] A method for computing nonsteady incompressible viscous fluid flows, Los Alamos Scient. Lab. Report LA 2910, Los Alamos, 1963. [3] Numerical solutions of the nonlinear equations for heater fluid layer. Phys. of Fluids, 8, 1965, p.17S7. [4] Numerical solution of the Navier--Stokes equations at high Reynolds numbers and the problem of discretization of convective derivatives. Lectures in J. J. Smolderen [1].

A. V. FOURSIKOV See M. I. Vishik and A. V. Foursikov. H. FUJITA [1] On the existence and regularity of the steady-state solutions of the Navier-Stokes Equations, J. Fac, Sci. Univ. Tokyo, Section I, 9, 1961, p.S9-102. H. FUJITA and T. KATO [1] On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal, 16, 1964, p.269-315. See also T. Kato and H. Fujita. H. FUJITA and K. MASUDA

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D. GREENSPAN and D. SCHULTZ [1] Fast finite-difference solution of Biharmonic problems, Comm. of the A.C.M. H. P. GREENSPAN [1] The theory of rotating fluids, Cambridge University Press, 1969. See also M. Israeli and H. P. Greenspan.

F. H. HARLOW and A. A. AMSDEN [1] A Numerical Fluid Dynamics Calculation Method for all flow speeds, J. Compo Phys., 8, 1971, p. 197. [2] "Fluid Dynamics: A LASL Monograph", Los Alamos Scientific Laboratory report No. LA-4700, 1971. F. H. HARLOW and J. E. WELCH [1] Numerical calculation of time dependent viscous incompressible flow of fluid with a free surface, Phys, Fluids, 8, 1965, p.2182-2189. J. E. HIRSH [I] The finite element method applied to ocean circulation problems. To appear.

c. W. HIRT

See A. A. Amsden and C. W. Hirt.

M. HOLT [I] Editor, Basic Developments in Fluid Dynamics, VoL I, Academic Press, New York, 1965. [2] Editor, Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics, Berkeley, Sept. 1970, Lecture Notes in Physics, 8, Springer-Verlag, 1971. [3] La Resolution Numerique de Quelques Problemes de Dynamique des Fluides, Lecture Notes No. 25, Universite of Paris-Sud, Orsay, France, 1972. [4] Numerical methods in fluid dynamics, Springer-Verlag, to appear. E. HOPF [1] Uber die Aufangswertaufgabe fiir die hydrodynarnischen Grundgliechungen, Math. Nachr.,4 (1951), p. 213-231. [2] On nonlinear partial differential equations, Lecture Series of the Symposium on Partial Differential Equations, Berkeley, 1955, Ed. The Univ. of Kansas (1957), p.I-29. L. HURWICZ See K. Arrow, L. Hurwicz, and H. Uzawa.

G.100SS [1] Bifurcation of a T-periodic flow towards an nT-periodic flow and their non-linear stabilities, Archiwum Mechaniki Stojowanej, 26, 5, 1974, p.795-804.

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M. ISRAELI and H. P. GREENSPAN [1] Nonlinear motions of a confined rotating fluid, Report, Department of Meteorology, M.I.T. To appear. V. I. IUDOVICH [1] Secondary flows and fluid instability between rotating cylinders PMM 30, n? =4, 1966,p.688-698 [2] On the origin of convection, PMM, 30, n? = 6, 1966, p. 1000-1005. G. JOHANSSON See C. K. Chu and G. Johansson. B. F. JONES See E. B. Fabes, B. F. Jones, and N. Riviere. C.KAHANE [1] On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. RationaiMech. AnaL, 33, 1969, p.386-405. S. KANIEL [1] On the initial value problem for an incompressible fluid with non-linear viscosity, J. Math, Mech., 19, 1969-70, p.681-707. S. KANIEL and M. SHINBROT [1] Smoothness of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. AnaL, 24, 1967, p.302-324. [2] A reproductive property of the Navier-Stokes equations, Arch. Rational Mech. AnaL, 24, 1967, p.363-369. See also Shinbrot and S. Kaniel. S.KARLIN [1] Total positivity and applications V.I. Stanford University Press, Stanf., Cal 1967. T.KATO [1] On classical solutions of two dimensional non stationary Euler equation, Arch. Rational Mech. Anal., 25,1967, p.188-200. [2] Nonstationary flows of viscous and ideal fluids in R3, J. Functional Analysis, 9, 1972, p.296-30S. T. KATO and H. FUJITA [I] On the stationary Navier-Stokes system, Rend. Sem, Univ. Padova, 32,1962, p.243-260.

J. B. KELLER and S. ANTMAN [I] Editors, Bifurcation Theory and Nonlinear Eigenvalue Problems, Benjamin, New York,1969. R. B. KELLOG and J. E. OSBORN [1] A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal., 21,1976, p.341-397. K. KIRCHGASSNER [1] Die Instabilitiit der Stromung zwischen zwei rotierenden Zylindern gegeniiber Taylor-Werbeln fiir beliebige Spaltbreiten. Z.A.M.P. 12, 1961, p.14-30.

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M. A. KRASNOSELSKII [1 J Topological Methods in the theory of non linear integral equations, Pergamon Press, New York, 1964. M. G. KREIN and M. A. RUTMAN [1 J Linear operators leaving invariant a cone in a Banach space, Trans. A.M.S., Series 1, 10, 1962, p.199-32S. H.O.KREISS See C. K. Chu and H. O. Kreiss. A. KRZWICKI and O. A. LADYZHENSKAYA [IJ A grid method for the Navier-Stokes equations, Soviet Physica Dokl., 11, 1966, p.212-216. O.A.LADYZHENSKAYA [1 J The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, English translation, Second Edition, 1969. [2J Sur de nouvelles equations dans 1a dynarnique des fluides visqueux et leur resolution globale-Troudi Math. Inst. Stekloff, CII, 1967, p.8S -104 (in Russian). [3J Sur des modifications des equations de Navier-Stokes pour de grands gradients de vitesses, Sem, Inst, Stekloff, Leningrad, 7,1968, p.126-154 (in Russian). [4 J Sur 1a solution globale unique du probleme de Cauchy tridimensionne pour les equations de Navier-Stokes Ii symetrie axiale, Sem, Inst, Stekloff, Leningrad, 7, 1968, p.155-177 (in Russian). [SJ On convergent finite differences schemes for initial boundary value problems for Navier-Stokes equations, Fluid Dyn. Trans., 5,1969, p.125-134. See also A. Krzywicki and O. A. Ladyzhenskaya. O. A. LADYZHENSKAYA and V. I. RIVKIND [1 J On the alternating direction method for the computation of a viscous incompressible fluid flow in cylindrical coordinates, Izv. Akad. Nauk, 35, 1971, p.259-268. P. LAILLY [IJ These, Universite de Paris Sud, Orsay, 1976. P.LASCAUX [1 J Application de 1a methode des elements finis en hydrodynamique bidimensionnelle utilisant les variables de Lagrange, Technical Report, C.E.A. Limeil, 1972. P. LAX and B. WENDROFF [1] Difference Schemes for Hyperbolic Equations with high order of accuracy, Comm. Pure AppL Math., XVII, No.3, 1964, p.381-398. J. LERAY [1] Etude de diverses equations integrales nonlineaires et de quelques problemes que pose l'hydrodynamique, J. Math. Pures Appl., 12, 1933, p.I-82. [2] Essai sur les mouvements plans d'un liquide visqueux que limitent des parois, J. Math. Pures et Appl., 13, 1934, p.331-418. [3J Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63, 1934, p.193-248.

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J. LERAY and J. SCHAUDER , [1 J Topologie et Equations fonctionnelles, Ann. Sci. Ec. Norm. Sup. 3eme Serle, 51, 1934, p.45-78.

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G.MORETTI [I] Lectures in J. J. Smolderen [I]. K. W.MORTON See C. K. Chu, K. W. Morton, and K. V. Roberts; R. D. Richtmyer and K. W. Morton.

J. NECAS [1] Les methodes directes en theorie des equations elliptiques, Masson, Paris, 1967. [2] Equations aux derivees partielles, Presses de I'Universite de Montreal, 1965. L. NIRENBERG [1] Topics in non linear functional analysis, Lecture Notes, Courant Inst of Math. Sc., 1973-74.

See also S. Agmon, A. Douglis, and L. Nirenberg. W.F.NOH [1] A time dependant two spaces Dimensional Coupled Eulerian - Lagrange Code, in B. Alder and S. Ferbach [1]. J. T. ODEN [1] Finite elements of nonlinear continua, McGraw-Hill, New York, 1972.

ODEN, ZIENKIEWICZ,GALLAGHER, TAYLOR [1] Finite element methods in flow problems, UAMPress, Huntsville, Alabama, 1974.

J. E. OSBORN [1] Regularity of the Stokes problem in a polygonal domain, in Numerical Solution of P.D.E. (III), B. Hubbard editor, Academic Press, 1976. See also R. B. Kellog and J. E. Osborn. M. C. PELISSIER [1] Resolution numerique de quelques probIemes raides en meeanique des milieux faiblement compressibles, Calcolo, 12, 1975, p.275-314.

P.PENEL [I] Sur une equation d'evolution nonltneaire Universite de Paris Sud, 1975. See also C. M. Brauner, P. Penel and R. Temam.

liee Ii.]a theone de la turbulence, These,

C. S. PESKIN [I) Flow patterns around heart valves, J. ofComp. Phys., 10, 1972, p.252-271. R.PEYRET See M. Fortin, R. Peyret and R. Temam. PIRRONNEAU [1] Sur les probIemes d'optimisation de structure en mecanique des fluides, Theses, Universite de Paris, 1976.

References

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G. PROm [1] Un teorema di unicita per le equazioni di Navier-Stokes, Annali di Mat., 48, 1959, p.173-182. [2] Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso bidimensionale, Rend. Sem, Mat. Padova, 30, 1960, p.I-15. See also J. L. Lions and G. Prodi; C. Foias and G. Prodi, G.PROUSE [1] Soluzioni periodische dell'equazione di Navier-Stokes, Rend. Acad. Naz. Lincei, XXXV, 1963, p.443-447. [2] Soluzioni quazi-periodiche dell'equazione di Navier-Stokes in due dimensioni, Rend. Sem, Mat. Padova, 33, 1963. P. H. RABINOWITZ [1] Periodic solutions of nonlinear hyperbolic partial differential equations (I) and (II), Comm. Pure AppL Math., 20, 1967, p.145-205, and 22,1969, p.25-39. [2] Existence and nonuniqueness of rectangular solutions of the Benard problem, Arch. Rational Mech. AnaL, 29,1968, p.32-57. [3] Some aspects of nonlinear eigenvalue problems, M.R.C. Technical Summary Report No. 1193, University of Wisconsin, 1972. [4] 'Iheorie du degre topologique et application Ii des problemes aux limites non lineaires, Lecture Notes of a Course at the University of Paris VI and XI, 1973. [5] A priori bounds for some bifurcation problems in Fluid Dynamics, Arch. Rat. Mech. Anal, 49, 1973, p.270-285. M. A. RAUPP

[1]

Galerkin methods for two-dimensional unsteady flows of an ideal incompressible fluid, Thesis, University of Chicago, 1971.

P. A. RAVIART [1]

[2]

Sur l'approximation de certaines equations d'evolutlon lineaires et non lineaires, J. Math. Pures AppL, 46,1967, p.ll-l07 and 109-183. Methode des Elements Finis, Cours de 3eme cycle, Universite de Paris VI, 1972.

See also P. Ciarlet and P. A. Raviart; M. Crouzeix and P. A. Raviart. G.deRHAM [1] Varietes Differentiables, Hermann, Paris, 1960. R. D. RICHTMYER

[1]

Editor, Proceedings of the Fourth International Conference on Numerical Methods in Fluid Dynamics, Boulder, June 1974, Lecture Notes in Physics, 35, SpringerVerlag, 1975.

R. D. RICHTMYER and K. W. MORTON

[1]

M. RIESZ

[1]

Difference Methods for Initial Value Problems, Wiley-Interscience, New York, 1957. Sur les ensembles compacts de fonctions sornmables, Acta Sci. Math. Szeged, 6, 1933, p.136-142.

N. RIVIERE See E. B. Fabes, V. F. Jones and N. Riviere.

References

476

V. I. RIVKIND See O. A. Ladyzhenskaya and V. I. Rivkind.

K.. V. ROBERTS See C. K.. Chu, K. W. Morton, and K. V. Roberts. M. A. RUTMAN See M. G. Krein and M. A. Rutman. E. SANCHEZ-PALENCIA

[1] [2]

Existence des solutions de certains problemes aux Iimites en magnetohydrodynamique, J. Mecanique, 7,1968, p.405-426. Quelques resultats d'existence et d'unicite pour les ecculements magnetohydrodynamiques non stationnaires, J. Mecanique, 8, 1969, p.509-541.

J.SAmER [1] The initial boundary value problems for the Navier-Stokes equations in regions with moving boundaries, Thesis, University of Minnesota, 1963. N. SAUER See H. Fujita and N. Sauer. J. P. SAUSSAIS [1] Etude d'un probleme de diffusion non lineaire lie Ii la physique des plasmas, These de 3~me cycle, Universite de Paris-Sud, 1972. J. SCHAUDER See J. Leray and J. Sehauder. D. SCHULTZ See D. Greenspan and D. Schultz. L. SCHWARTZ

[1] [2]

Theorie des Distributions, Hermann, Paris, 1957. Theorie des distributions Ii valeurs vectorielles, I, II, Ann. Inst. Fourier, 7, 1957, p.1-139 and 8,1958, p.1-209.

J. SERRIN [1] A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. RationalMech. Anal, 3, 1959, p.120-122. [2] Mathematical Principles of ClassicalFluid Dynamics, Encyclopedia of Physics, voL 13, 1, Springer-Verlag, 1959. [3] The initial value problem for the Navier-Stokes equations, in "Non-linear Problems", R. E. Langer editor, University of Wisconsin Press, 1963, p.69-98. [4] On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Rational Mech. AnaL, 9, 1962, p.187-195. M. SHINBROTand S. KANIEL [1] The initial value problem for the Navier-Stokes equations, Arch. Rational Mech. AnaL, 21, 1966, p.270-285.

References

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D. R. SMITH See R. Finn and D. R. Smith.

J. J. SMOLDEREN [1] Numerical Methods in Fluid Dynamics, AGARD Lecture Series, No. 48, 1972. S. L. SOBOLEV [1] Applications de l'analyse fonctionnelle aux equations de la physique mathematique, Leningrad, 1950, in Russian, and English translation by A.M.S., 19. G. STAMPACCHIA [1] Equations elliptiques du second ordre a coefficients discontinus, Pressesde l'Universite de Montreal, 1966. See also Magenesand G. Stampacchia. G. STRANGand G. J. FIX [1] An Analysis of the Finite Elements Method, Prentice Hall, Inc, Englewood Cliffs, 1973. P.SZEPTYCKl [1] The equations of Euler and Navier-Stokes on compact Riemannian manifolds, Technical Report, University of Kansas, 1972. L. TARTAR

See C. Bardos and L. Tartar.

R. TEMAM [1] Sur la stailite et la convergence de la methode des pas fractionnaires, Ann. Mat. Pura Appl, LXXIV, 1968, p.191-380. [2] Une methode d'approximation de la solution des equations de Navier-Stokes, Bull Soc. Math. France, 98, 1968, p.115-152. [3] Sur l'approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires (I), Arch. Rational Mech. Anal, 32 (No.2), 1969, p.135-153. [4] Sur l'approximation des equations de Navier-Stokes par la methode des pas fractionnaires (II), Arch. Rational Mech. Anal, 33 (No.5), 1969, p.377-385. [5] Approximation of Navier-StokesEquations, AGARD Lecture Series, No. 48,1970. [6] Sur la resolution exacte et approchee d'un probleme hyperbolique non lineaire de T. Carleman, Arch. Rational Mech. Anal, 35 (No.5), 1969, p.351-362. [7] Methodes de Decomposition en Analyse Numerique, Proceedings of the International Conference of Mathematicians, Nice, 1970, Gauthier-Villars, Paris, 1971. [8] Numerical Analysis, Reidel Publishing Company, Holland, 1973 (in English), and P.U.F. Paris, 1969 (in French). [9] On the theory and numerical analysis of the Navier-Stokes equations, Lecture Note No.9, Department of Mathematics, University of Maryland, 1973. [10] On the Euler equations of incompressible perfect fluids, J. Funct. Anal, 20, 1975, p.32-43. [11 J Une propriete generique de l'ensemble des solutions stationnaires ou periodiques des equations de Navier-Stokes, Actes du Symposium Franco-Japonais, Tokyo, Sept. 1976, to appear.

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Turbulence and Navier-Stokes equations, Lecture Notes in Math. vol. 565, Springer Verlag, 1976.

See also C. M. Brauner, P. Penel and R. Temam, A. Bensoussan and R. Temam, C. Foias and R. Ternam, M. Fortin and R. Temam; M. Fortin, R. Peyret and R. Temam; J. P. Boujot, J. L. Soule, and R. Temam; H. Cabannes and R. Temam, F. THOMASSET Etude d'une methode d'elements finis de degre 5. Application aux problemes de [1] plaques et d'ecoulement de Fluides, These de 3eme cycle. Universite de Paris-Sud, 1974. [2] Methodes d'elements finis non conformes en hydrodynamique. Rapport IRIA Laboria, paraitre.

a

R. TREMOLIERES See R. Glowinski, J. L. Lions, and R. Tremolieres.

H. UZAWA See K. Arrow, L. Hurwicz, and H. Uzawa. R. S. VARGA [1] Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 1962. W.VELTE [1] Stabilitiitsverhalten und Verzweigung Stationiirer Losungen der Navier-Stokesschen Gleichungen, Arch. Rational Mech. Anal., 16, 1964, p. 97-125. [2] Stabilitiits und Verzweigung stationiirer Ldsungen der Navier-Stokesschen Gleichungen beim Taylor problem, Arch. Rational Mech. Anal, 22, 1966, p.I-14. M. I. VISHIK and A. V. FOURSIKOV [1] L'equation de Hopf,les solutions statistiques,les moments correspondants aux systemes des equations paraboliques quasilineaires, J. Math. Pures AppL, to appear. C. WAGSHALL See P. G. Ciarlet and C. WagshalL J.E. WELCHS See F. H. Harlow and J. E. Welchs. B. WENDROFF See P. Lax and B. Wendroff. WILKINS [1] Calculation of elastic plastic flow, in Meth. ofComp. Phys., 3, Academic Press, New York,1964. H.WITTING [1] Uber den Emflu Pder Stromlinienkriimmung auf die Stabilitiit larninarer Stromungen, Arch. Rat. Mech. AnaL, 2 (1958), p.243-283.

References

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N. N. YANENKO [1] Fractional Step Methods, English translation, Springer-Verlag, 1971. [2] In the proceedings of the International Conference of Mathematicians, Nice, 1970, Gauthier-Villars, Paris, 1971. V. I. YUDOVICH [1] Nonstationary flow of a perfect nonviscousfluid, (in Russian), J. of Num. Math. and Math. Phys., 3, 1963. [2] Periodic solutions of viscousincompressiblefluids, (in Russian), DokL Akad. Nauk, 130,1960, p.1214-1217. [3] A two dimensional problem of unsteady flow of an ideal incompressible fluid across a given domain, Amer, Math. Soc. Translation, 57,1966, p. 277-304. A. ZENISEK

[1]

Polynomial interpolations on the triangle, Numer. Math., 15, 1970, p.283-296.

M.ZLAMAL [I] On the imite element method, Numer. Math., 12, 1968, p.394-409. [2] A finite element procedure of the second order of accuracy, Numer. Math., 14, 1970, p.394-402. See also I. Babushka and M. ZlamaL

APPENDIX by F. Thomasset Implementation of non-conforming linear finite elements (Approximation APX5 - Two dimensional case)

O. Test Problems Two test problems are considered. Problem 1: Flow in a cavity

n is the square (0,

I) X (0, I) of the plane.

u is given on an, U = 0 except on the side y = I where u = {U, O}

(U to be specified).

Problem 2: Flow between nonconcentric rotating cyltndars

n is the annular domain of the plane, Oxy, which is limited by the two circles C1

x 2 + y2

C2

(x - 1)2

-

25

+ y2

=0 -

4=0

C1 and C2 are rotating with algebraic angular velocities, 01 and be specified. In both cases, no body forces are applied.

02

to

Appendix

481

1. The triangulation The triangulation ffh is defmed by numbering the vertices and the triangles. An algorithm of automatic triangulation provide us the following entries

i

X( .)

Y(. )

XCi)

Y(i)

coordinates of the i th vertice.

Nu(., .)

Nu(l,j)

Nu(2 ,j)

Nu(3,j)

number of the vertices of the jth triangle.

It is necessary to be able to recognize in a simple way the boundary vertices: an economical way of doing it is to provide them with numbers > N, where N is the total number of interior nodes. (i) There are two different types (2) of automatic triangulation algorithms (l>rn the case of a domain with several boundaries (Le, multi-connected domain), we are led to introduce another set of integers in order to distinguish the points belonging to different boundaries. (2)we are concerned with the treatment of general domain shapes. Of course, for a specific geometry which must be considered many times, one may find a triangulation more economical than that given by a general program.

Appendix

482

- the method of "successive subdivision" leading to regular nets - the method of "contraction of the domain" by Alan George [1] , which is convenient for a domain with a curved boundary.

The method ofsuccessive subdivision The domain is initially covered with a coarse triangulation; then at each step, every triangle is divided into k 2 smaller triangles, similar to the initial triangle.

k • 2

k • 3

The procedure can be reiterated until we reach the appropriate number of triangles.

Example (k =2).

Initial triangulation

N • 1 , Nl • 5 , M• 4

lit subdivision

N • 5 , Nl • 13, M· 16

Appendix

483

For a given triangulation, let N be the number of interior vertices, N 1 the total number of vertices, M the number of triangles. Let N', Ni, M', denote the corresponding parameters after the subdivision into k 2 triangles has been performed. Then observing that 2C+ CB = 3M,

CB =N l

-

N,

(C and CB denoting the number of interior and boundary edges), we find Ml

=k2

XM

Nl =(k-l)(k-2) M+(k-l)(3M+N-Nd +N

2 N~

2

=N' + k(Nl - N).

After n subdivisions, we note that

= k"

Mn ;: k 2n MO ,

~

=2N n + C~ MO =2N 0 + C~

+ 2(T - 1) + 2(T - 1)

M"

~

where T is the number of wholes in the domain. The parameters of the triangulation are then

N"

=} «k 1

2n

-

NJ. =2: «k 2n -

1)M° - (I{' -

I)C~)

+ NO

I)Mo + (k n + I) cj) + NO

(MO =M, ... ).

We emphasize that the triangles obtained by subdivision are similar to the triangles of the initial triangulation: therefore the angles do not become smaller than those of the initial triangulation. The method of successive subdivision can be adapted to domains with a simple curved boundary: we just replace the new vertices appearing on the boundary edges by their projection on the boundary of n.

484

Appendix

Example (k =2)

Initial Triangulation

Method of Contraction of the domain The principle of the method is to cover 0 with triangles as similar as possible to equilateral triangles (for which the numbers aT are minimal and thus also, the bound of the error). An initial discretisation of an is given, constituted by a closed connected sequence of segments. If n is multi-connected, we must introduce artificial boundaries in 0, and these boundaries will be removed after the triangulation is done. Given the boundary polygon, the algorithm can construct a triangle in two different ways: - by introducing a new node in the interior (notching) - by joining two consecutive boundary vertices (trimming). Then we start again with a new domain constituted with the initial domain less the last triangle, and so on until the domain is reduced to one triangle.

i)

ii)

-iii)

(i), (iv): notching, (ii), (iii): trimming.

iv)

Appendix

485

Examples. Figures 1-4 show two triangulations of the two considered geometries.

Fig.!.

Fig. 2.

486

Appendix

Fig. 3.

Fig. 4.

Appendix

487

2. The nodes

The nodal variables of a function l/J are the values of l/J at the midedges which will be called the nodes. Given the entries X, Y, Nu, we form the corresponding entries for the nodes

i

X .) m(

Y .) m(

Xm(i)

Y (i) m

coordinates of the i th node

(the boundary nodes are numbered with numbers> Nm ,Nm = the number of interior nodes).

Num(l,i)

Nu

m(2.i)

Nu m(3.i)

(this amount to numbering the edges).

numbers of the nodes of the jth triangle

Appendix

488

It will appear later that we need to know rapidly the numbers of the nodes which are neighbors of a given nodes (i.e. those belonging to a same triangle: there are at most 4 neighbors).

i

For this reason we will compute from NU m the entry of the neighbors: TABV(., .)

I

TABV(l,i)

TABV(2,i)

TABV(3. i)

TABV(4.i)

Numbers of the nodes which are neighbors of the i th node

3. Computation of the basis function on a given triangle

Let there be given a triangle Twith verticesM I • M']" M 3 , mid-edges Pl. P'],. P 3 , and let VI. V'],. V3 denote the corresponding basis functions (Vi(Pj )

= fJij)'

"

We introduce a reference triangle T and the linear mapping trans-

489

Appendix

forming T'"into T

'" M='Y(M)=

1)

1») M'" + (G(3, 1»)

(G(1, G(2, G(1,2) G(2,2)

G(3,2)

1

=A - -A - -A- -A - - -A - -A - -" - -A (M 1 -M3 ) 1 (M2 -M3 h

-

(M 1 -M 3)2 (M2 -M 3 ) 1

h)(-(M -M hh '" 2 -M "3 (M

M 1 -M3)1 -(M 2 -M3 ( X (M 1 -M 3 h (M 2 -M 3 h = (M 1

-

(M 1

-

M 3 ) 1 (M 2 M 3 h (M 2

-

-

M3 ) 1 M3 h

)

•

1

3

C

" is a 2 X 2 matrix independent of T. where C

1») =M

G(3, ( G(3,2)

1)

1

1») . M .

_ ( G(1, G(2, G(1,2) G(2,2)

1

We easily deduce 'Y- 1 from v and:

M=

-I M 'Y ( )

= (G1(1,

G1

1) (2, 1) ) • M G1(1,2) G1(2,2)

+(

G1(3, 1»)

G1(3,2)

490

Appendix

We then possess all what we need to construct the basis functions. Indeed, let At denote the barycentric coordinates in T with respect to the Mj(At(Mj ) fJ ij ). We have

=

V1

= A1 + A2 - A3

V2

= A2 + A3 - A1

V3=A3+ A1- A2'

On the other hand

A1 (M)

'" (M) " =X2

"" " " A3(M) =X2(M) A2 (M) = I -

X 1(M) -

" x"2(.M)

from which we infer that

I area T = - det G 2

i

IT

vi dM=

~ area(T)

v,v/dM =

~

6,/ area (T).

It is interesting to notice that the basis functions are orthogonal in

Appendix

L 2 (T). The gradient of the

491

v;'s can be computed by using the matrix

G1 •

aUj

" a(X 1o'Y)

L

aXk

k=1,2

aXj

aXi =

aXk

=

L

GI(j, k)

k=1,2

a(Xi;'Y) aXk

aX 2 aX 2 aX 3 .• • • • -a = Gl(" 2), -a = - Gl(l, I) - GI(j, 2), -a = GI(j, ~

~

aVl

1)

~

.

-=-2GI(j,I) aXj

aV2

-

aXj

.

=- 2 GI(j, 2)

aV3

-a = 2 GI(j, 1) + 2 GI(j, 2). Xj

Numerical integration of the right hand side

J[

T f(M) vi(M)

dM ~

area (T) . 3 j(Pi), l = 1,2,3.

4. Solution of Stokes' Problem 4.1. The basis functions

A basis of Wh (the approximation of H~(O»

wi(i= 1, ... ,Nm

is constituted with the

):

- wi is linear on each triangle - wi

= a on aOh

- Wi =

a at each node (= at each mid-edge), except i th where wi(i) = I.

node,

The restriction of Wj to a triangle T is one of the three functions Vj defined in Section 3. The support of wi is the union of the two triangles containing i.

492

Appendix

A basis of Wh (approximation of {(Wj,

0), (0,

Wj),

i

nb(n»

is provided by the

= 1, .. .,Nm } .

With N m =number of interior nodes. The discrete homogeneous Stokes problem leads to the solutions of Nm

2:

j= 1 Uh

=

x, Wj

Nm

2:

j= 1

Yj

Wj

II

t

+ ""

'ffh (T)

area (T) aWk ay (T)

with div(uh) = 0 on T, V T. The et;k are computed by the following expressions

et;k =

2:

TU,k)

area (T). VwlT). VWk(T) ,

where the summation is over all the triangles T adjacent to the nodes j and k. For the nonhomogeneous problem, we also introduce the Wj associated to the boundary nodes (i > Nm ), and the linear system has the same form as (4.1), provided we respectively add to the right-hand sides the expressions

493

Appendix

L

j>N

(The X j and Yj for j

>N

are known and given by the boundary data).

4.2. Uzawa algorithm Let us now write the discrete Uzawa Algorithm: - we start with an arbitrary 1T2 = {1T2(T), T

Effh}

(for example 1T2(T) = 0, V T, - when 1T~ is known, we compute u~+

1

by

II

II

+

L T

~(T)

aWk_

area (T) -

ay

(T) k - 1, .. ,

(4.2)

and then we compute 1T£+1 by

= 1T~(T)

1T~+l(T)

- P div(u~+l)(T),

VT.

(4.3)

The two components of the velocity in (4.2) are actually uncoupled; we just have to solve a linear system of the type Nm

L

j= 1

Zj

OI.;k I'

=f k •

k

= 1, ... , N m . .

(4.4)

494

Appendix

We proceed by overrelaxation (S.O.R.) (cf. Varga [I

D·

Overrelation Optimal parameter It is easy to verify that the matrix (Oljk) is symmetrical and positive definite. Let w denote the relaxation parameter. We start with an arbitrary vector Z7; when zn k+l

Z; is known, we compute Z;+ 1 by

=(1- w)Z~

!

+

~

_

Oltk Z;

I=k+ 1

~1

(

Olle, k

lYJ

1= 1

- .!.. Ik)

-ik

zn+l j

.

II

The stoping test is, as usual, of the type, - Z~I

Max IZ~+l k

<eret

The optimal value of the relaxation parameter w can be determined by the following algorithm: We apply the Gauss-Seidel method to the solution of the homogeneous system

f

Zj Oltk = 0,

starting with a vector ZO of components Z?

~1

(

z

Nm

+

I=k + 1

j= 1

lYJ

-7k

> O.

Then

zn+l 1

"t> Z; )

(4.6)

We set, for each n : l Pn+ p wn+ 1 p

zn+l

= Min _k_ k

=

znk

'

pn+ 1 g

2

I+VI-p;+l'

zn+l

=Max _k_ k

w n+ 1 g

ZZ

=

(4.7)

2

1+v'I-p;+I

(4.8)

The numerical tests performed show that both w;+l and w;+ 1 converge

Appendix

495

Storage of the matrix. A A (i) o

A(I, .)

TABV

A(2, .)

A(3,.)

I

A(4,.)

TABV (1 •• )

TABV (2 •• )

TABV(3•• )

TABV(4•• )

!

I 0

a.

L

a. i.i

a. l

1,

i

a ..

1,1

Z

a.

1.

3

i

i

4

i

1

2

i

3

i

4

I iI' i

2•

i

3•

i

4

• the neighbourhood nodes of

i

Direct methods (Cholesky, Frontal method, ...) may be used to solve (4.2) provided a renumbering of the nodes is done. However it will be uneasy to use these methods in the non-linear case.

4.3. Numerical results The choice of the parameter p in the Uzawa algorithm is done by experiment. We perform several tests with different p, and we note the results for a given number of Uzawa iterations (say, 10, 20, 30, ... ). A good test for the convergence is the quantity c~n)

= Max Idiv(u~ T

)TI.

(2)

Hereafter we give the results for the geometry 2, with v = I, 01 =0, 02 = I (C 1 is at rest and C2 is rotating with a unit angular velocity, no volumic forces, and v = I). The results given in Fig.S, 6, clearly show the existence of an optimal p, Popt =0.96 in this case. (1) p;:1 and p;+1 converge to p(t 1) = the spectral radius of the Gauss-Seidel matrix, and P(B)2

=P(t1),

Wopt =.J 1+

2

I-p(B)

2

n+l)(T) - 1r~(T)1 (2)This number, equal to (lIp) Max 11r1 preuureL T We also consider c'iw~n) =M.ax [IXr 1 I vergence of discrete velocities, I

xli,

measures the convergence of discrete

rr 1 - Yfll

which measures the con-

n=100

n=50

---

n=30

cl.n '

0.7

10-&1

10-5

1O-3~

10-2~

lO-,T

' 0.8

'---

Fig.S.

,

0.9

!

,

I

I

I

,,

0.95

I

Pmax · '1

... p

Theoretical bound:

/

0.7

10-&'

10-5

10- 4

-

~.

n =60

1O-3~=50

-

n=40

n-30

I n.20

10-2r~

10-'1' liw\:"

' 0.8

Fig. 6.

, 0.9

~

~I

1

... p

\()

~

)t'

):.

:g ::s '" I:l.

0\

497

Appendix

4.4. The augmented Lagrangian technique This is a combination of penalty and Uzawa techniques. \\e simply add to I (4.20.g) a penalisation term (l le)ldivh Vh 12 , and this leads to the following set of equations (Fortin-Glowinski): I

+ - (divj, Uh' divh e

v«Uh' Vh»h

=(f, Vh)'

VVh E

Vh) - ('lrh'

Wh

divh Vh) (4.9)

.

We write the Uzawa algorithm for (4.9). Setting as before

Unh + 1 --

I

L

x1I n + l) w.I ,

"" nn+ 1) W.

L..

I

I '

we just have to replace the left-hand side of (4.2) by Nm

II

L

j= 1

Nm II

L

j=1

N 1m

ry

+1

I OItk + e-

L

j= 1

+ ry+1(D hy

Wj.

D hy wk)} •

(4.10)

We see on (4.10) that the equations for the X and Y components are no longer decoupled and then we must solve a linear system twice bigger than in the case lie =0 previously studied. Moreover, although the S.O. R. method is applicable for the solution of (4.10), the parameter w cannot be found any more with the technique shown in Section 4.2; hence, the choice of w is made empirically. For the tests made on Problem 2, the value w = I, 6 has been selected. The number ne of iterations which is necessary to achieve convergence (c~n) ~ 10-5 and {)WC ~ 10- 5 ) decreases as lie increases (starting from 0); it attains a minimum for lie = 8, with the best p = p(e) = 9.2 in this case. The convergence was achieved after only 15 iterations (n, = 15).

498

Appendix

5. Solution of Navier-Stokes Equations We will denote

Wh = the approximation of HI (,Q) Wh = the approximation of HI (,Q) Wh = the approximation of Wh

HA (,Q)

= the approximation of HA(,Q).

X h is the space of step functions 1rh which are constant on each

f/E!Jh and vanish outside ,Q(h). The basis functions vk which span Wh correspond to k

<: N m .

We set

f J

s, =

vi dM

fih

and we recall that

r

Jfih

vkvQdM=Oifk*£.

As previously we introduce the discrete differentiation operators Dhx v, D hy v and the discrete divergence Dhv

=Dhxvx +Dhy vy

.

The notations b h , ah are the same as in Chap. II, (3.78) and (3.80). The discrete problem is stated in Chap. II (3.93). \\e have tried two of the algorithms presented in Chapter II. Algorithm I. (Uzawa, cf. 11.(3.107), 11.(3.108)).

pW)

We start with some E Xh (for instance +1 such that known (m ~ 0), we compute

ur

p1°) =0).

p1

When m ) is

v«ur+ 1, vh ))h + bh(ur+ I, ur+ 1, vh) + ~ (D h uW +1, e

o,

vh)

Appendix

499

((5.1) may be solved by an under-relaxation technique). Then we compute pg' + 1

p,// + 1

=p'l(

- pDh

(5.2)

U'// + 1 •

Algorithm II. (Implicit Arrow-Hurwicz. Cf. 11.(3.122), 11.(3.123».

U'// + 1 is computed according to ((u,// + 1 - U'//, vh »h + p [ v((u,//,

+~(Dh e

vh »h

u,//+l,D h vh)+bh(u'//, u'//+l, vh)

-(P,//,DhVh)-(f,Vh)]=OVVhEWh'

Then p'l( + 1 is defined by

o(p'// + 1

-

p,// ) + P D h

u'// + 1 =0 .

Examples. Fig. 7 shows the streamlines of the flow for Problem I for the best experimental choice of p, ex, e (Algorithm II, v = 10- 2 , U = I, 512 triangles; I/e =0,5,7, 10 has been tested).

Fig. 7.

(5.3)

500

Appendix

Fig. 8 shows the streamlines of the flow for Problem 2 (v =4.10- 2 , ul = 1, u2 =0, 412 triangles, 655 nodes (Fig. 4); the selected values are p = O. 1, lie = 10, a =0.004).

Fig. 8.