O P E R A T O R THEORY AND N U M E R I C A L M E T H O D S
STUDIES IN MATHEMATICS AND ITS APPLICATIONS
VOLUME
30
Editors: Paris PAPANICOLAOU, New York H . F U J I T A , Tokyo H . B . K E L L E R , Pasadena J.L. LIONS,
G.
ELSEVIER A M S T E R D A M - LONDON - NEW YORK - OXFORD - PARIS - S H A N N O N - TOKYO
OPERATOR THEORY AND NUMERICAL METHODS
HIROSHI
FUJITA
The Research Institute of Educational Development Tokai University Tokyo, Japan
NORIKAZU
SAITO
Faculty of Education Toyama University Toyama, Japan and
TAKASHI
SUZUKI
Department of Mathematics, Graduate School of Science Osaka University Toyonaka, Japan
2001 ELSEVIER AMSTERDAM-
LONDON - NEW YORK - OXFORD - PARIS - SHANNON-
TOKYO
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9 2001 Elsevier Science B.V. All rights reserved.
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Preface Recent developments in numerical computations are amazing. A lot of huge projects in applied and theoretical sciences are becoming successful by them, while similar things are happening even in the level of personal computers. Under such a situation, theoretical studies on numerical schemes are fruitful and highly needed. The purpose of the present book is to provide some of them, particularly for schemes to solve partial differential equations. In 1991, we published an article on the finite element method applied to evolutionary problems from Elsevier Publishers (Fujita and Suzuki [148]). This book follows basically that way of description. We study various schemes from the operator theoretical point of view. Many parts are devoted to the finite element method, of which history is described in Oden [306]. We deal with elliptic and then time dependent problems in use of the semigroup theory and so forth. Some other schemes and problems are also discussed, with the later development taken also into account. We are led to believe that any scheme used practically has significant and tight structures mathematically and the converse is also true. Besides some corrections and supplementary descriptions, we have added the following topics: (1) /Y' estimate of the Ritz operator associated with the finite element approximation (2) asymptotic expansions and Richardson's extrapolation for the finite element solution (3) Trotter-Lee's product formula for holomorphic semigroups (4) mixed finite element method (5) Nehari's iterative method for non-stable solutions of elliptic problems (6) finite element approximation of nonlinear semigroups and applications (7) boundary element method for elliptic problems (8) charge simulation method for elliptic problems (9) domain decomposition method for elliptic problems. To make the description consistent, the domain in consideration is mostly supposed to be a convex polygon, and piecewise linear trial functions are adopted unless otherwise stated. The other cases are described in the notes at the end of chapters. The authors thank Professors M. Katsurada, A. Mizutani, H. Okamoto, and T. Tsuchiya for reminding us of some recent developments in the theoretical study and actual computations. Thanks are also due to Ms Y. Ueoka for typesetting the manuscript carefully. Tokyo / Toyama / Osaka April 2001 Hiroshi FUJITA Norikazu SAITO Takashi SuzuKI
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Contents Preface 1
Elliptic Boundary Value Problems and FEM 1.1 Elliptic B o u n d a r y Value P r o b l e m s . . . . . . . 1.2 R i t z - G a l e r k i n M e t h o d . . . . . . . . . . . . . 1.3 F i n i t e E l e m e n t M e t h o d ( F E M ) . . . . . . . . 1.4 Inverse A s s u m p t i o n . . . . . . . . . . . . . . . 1.5 L ~ E s t i m a t e . . . . . . . . . . . . . . . . . . 1.6 L p E s t i m a t e . . . . . . . . . . . . . . . . . . . 1.7
2
v
Asymptotic Expansion
. . . . . .
1 1 7 10 14 19 27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Semigroup Theory and FEM 2.1 E v o l u t i o n a r y P r o b l e m s . . 2.2 S e m i - d i s c r e t i z a t i o n . . . . 2.3 F r a c t i o n a l Powers . . . . . 2.4 F u l l - d i s c r e t i z a t i o n . . . . . 2.5 I n h o m o g e n e o u s E q u a t i o n . 2.6 H i g h e r A c c u r a c y . . . . . 2.7 L ~ E s t i m a t e . . . . . . . 2.8 H y p e r b o l i c E q u a t i o n . . .
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95 108 118 126 131
4
Other Methods in Time Discretization 4.1 R a t i o n a l A p p r o x i m a t i o n of S e m i g r o u p s . . . . . . . . . . . . . . . . . . . . 4.2 M u l t i - s t e p M e t h o d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 P r o d u c t F o r m u l a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
145 145 153 165
5
Other Methods in Space Discretization 5.1 L u m p i n g of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 U p w i n d F i n i t e E l e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . .
171 171 181
3
Evolution 3.1 3.2 3.3 3.4 3.5
Equations
. . . . . . . .
. . . . . .
and FEM
Generation Theories A Priori Estimates . Semi-discretization . Full-discretization . . Alternative Approach
. . . . .
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95
vii
5.3 5.4 5.5
Mixed Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Element Method (BEM) . . . . . . . . . . . . . . . . . . . . . . Charge Simulation M e t h o d (CSM) . . . . . . . . . . . . . . . . . . . . . .
182 190 195
Nonlinear Problems
207
6.1 6.2 6.3
207 211 217
Semilinear Elliptic E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . Semilinear P a r a b o l i c E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . Degenerate Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . .
Domain Decomposition Method 7.1 7.2 7.3 7.4 7.5 7.6
243
Dirichlet to N e u m a n n (DN) Map . . . . . . . . . . . . . . Dirichlet to N e u m a n n (DN) I t e r a t i o n . . . . . . . . . . . . Dirichlet 2 to N e u m a n n 2 ( D D - N N ) I t e r a t i o n . . . . . . . . . . . . . . . . . . Robin to Robin (RR) Iteration . . . . . . . . . . . . . . . . Exterior P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . T h e Stokes S y s t e m . . . . . . . . . . . . . . . . . . . . . .
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243 246 255 257 258 261
Bibliography
275
Index
303
viii
Chapter
Elliptic
1
Boundary
Value
Problems
and
FEM
The present chapter is concerned with the finite element method (FEM) applied to elliptic boundary value problems. For this topic, we have several monographs such as Strang and Fix [359], Ciarlet [83], Raviart and Thomas [326], Johnson [193], Ciarlet and Lions [85], Szab5 and Babuguka [372], and Brenner and Scott [60]. Here, we describe it in the framework of operator theory, picking up approximate operators of the schemes. This way is natural and efficient, particularly in dealing with time dependent problems, because the finite element method is regarded as a discretization of the underlying variational structure.
1.1
Elliptic
Boundary
Value
Problems
To fix the idea, let f~ C R 2 be a convex polygon, and consider the Poisson equation -Au = f
in f~
(1.1)
with Dirichlet condition
u= 0
on Of~.
(1.2)
This boundary value problem has the weak form, described under the following notations. 1~ LP(ft) denotes the set of p-integrable functions for p E [1, oc) and that of essentially bounded functions for p = oc. Unless otherwise stated we assume functions to be real-valued and Banach spaces real. 2 ~ Wm'P(f~) denotes the Sobolev space, the set of measurable functions with their distributional derivatives up to m-th order belonging to LP(ft), where m = 0, 1, 2 , . . . 3 ~ C~(ft) denotes the set of infinitely many times differentiable functions having compact supports contained in f~.
4 ~ Wo'P(f~) denotes the closure of C~(f~) in
wm'p(~).
5 ~ Hm(f~) and Hg(f~) stand for Wm'2(~) and W~'2(f~), respectively. We set Ivlm = I~lm,a - E,<=m IID~ll~=(a) 9
1. Elliptic Boundary Value Problems and FEM 6~ ( , )
and . 4 ( , ) denote the L 2 inner product and the Dirichlet form, respectively:
(f, g ) = ~ f(x)g(x) dx,
A(~,v)= j~ w(~). w(~) d~. 7 ~ The trace operator is denoted by 3' =
" Ion: Hi(f~) ---* gl/2(Of't) 9
It holds that Ker(-),)= Hl(f~). Then, (1.1) with (1.2) is reduced to the problem in V = Hl(f~): find u C V satisfying .4(u, v) -- (f, v)
(1.3)
for any v E V. Supposing that u is smooth, we have by Green's formula that
A(u,v) = s V u . Vv dx = - fn (Au) v dx = (f,v), since v e H~(f~). Equality (1.1) follows from arbitrariness of v. Henceforth, various generic constants are denoted indifferently by C. If it depends on some parameters, say c~, g , . . . , we may write it as Ca,0,.... Poincar6's inequality is indicated as (v E V)
(1.4)
so that "4(, ) provides an inner product to tile Hilbert space V = H~(f~). Let V' be tile dual space of V. Then, regarding f C L2(ft) as an element of V' through
(f , v)v, v = (f , v) for v E V, we can verify the unique solvability of (1.3) from Riesz' representation theorem. Because f~ is convex, regularity of the weak solution follows; 'u E V belongs to H2(ft) if f E L2(f~). Therefore, the above calculation is justified and u becomes a strong solution. We proceed to the case of V = Hl(f~) in (1.3). Assuming that u is smooth, we obtain similarly t h a t
A(u, v ) =
V u . Vv dx =
-:~nv d S -
(Au)v dz = (.f, v).
Here, 0 / 0 n denotes the differentiation along the outer unit normal vector n = (nl, n2) on 0f~ and dS denotes the surface element of 0f~. Taking v E H~(f~), we have
~o O'u 9 ~ ~-~nv
d8=0
1.1. Elliptic Boundary Value Problems so that equality (1.1) follows from arbitrariness of v E H~(f~). This implies
a-~n v dS = O for any v E Hl(f~) and hence
Ou --0 On
on O~
follows. In other words, Neumann problem -Au
Ou On
=
f
inf~
(1.5)
=
0
on Of~
(1.6)
is reduced to variational problem (1.3) for V = HI(f~). Since Poinca%'s inequality (1.4) dose not hold for V = Hl(f~), this time . 4 ( , ) does not provide an inner product to Hl(f~). Actually, problem (1.5) with (1.6) is not uniquely solvable. For instance, any constant function u = c E R satisfies (1.5) with (1.6) for f = 0. Above considerations are generalized in the following way. Let aij(x) = aOi(X), bj(z) and c(x) be real-valued smooth functions on f2 and suppose that the uniform ellipticity 2
E
i,j=l
giJ(X)~i~J ~ (~1 I~1 ~
(1.7)
holds for ~ = (~1, ~2) E R 2 and x E ~ with a constant (~1) 0. Given
s
D) = -
~'~ 0-~~
0
2
a'i(x)-~x j + E
i,j=l
0
by(x)-~x j + c(x),
j=l
we take the boundary value problem
12(x, D)u u
= f
in f~
=
onOf~.
0
(~.s) (1.9)
Then it is reduced to problem (1.3), where V = H~(f/) and
r
V) ~-
i,j=l aij(X)~xj-~Xi + j=l bj(x)~VOxj + c(x)?.l,V
Similarly, if a smooth function a(~) on Of/is given and 8
2
i,j=l
a
dx.
1. Elliptic Boundary Value Problems and FEM denotes the outer co-normal differentiation associated with s (1.8) with
Ou - - + cru = 0 One:
b o u n d a r y value problem
on Of~
(1.10)
is reduced to variational problem (1.3) with V = H~(ft) and
,A(u, v)
=
aij(x)-~xj--~x i i,j=l
+
bj(x)--~xjV
+
c(x)uv
dx
j=l
+ / o n au . v dS
(1.11)
for u, v E V. Those bilinear forms are b o u n d e d in the sense t h a t
I,A(~, ~)1 _< c II~llv Ilvllv
(1.12)
(u, v ~ V).
Given e > 0, the trace operator -), = "Ion admits a constant C~ > 0 satisfying
(1.13)
IIG~llg~(oa) <-~ IlVllg,(~) + Cr II~llg~(~) for any v E Hl(f~). Therefore, for 5 E (0, 51) and ,k E R, it holds that
.,4(v, ~) -> 5 I~11~v - A II'vl %
(v e v)
(1.14)
by (1.7). Here and henceforth, we set X = L2(f~). In the case t h a t b j ( x ) =_ O,
c(x) :> 0,
and
V = H~(f~),
or
bj(x) -- 0,
c(x) > 0,
a(~) k 0,
and
V = Hl(f~),
we can take A = 0 in (1.14). Then, problem (1.3) is uniquely solvable by Lax-Milgram's theorem. In use of the dual space V' of V, boundedness (1.12) of A( , ) implies the welldefinedness of the b o u n d e d linear operator A : V --+ V' through the relation
A(u, v) = (Au, V)v,,v for u, v E V. On the other hand, identifying X' with X by Riesz' representation theorem provides a triple of Hilbert spaces V C X C V' with continuous and dense inclusions. Let
D(A) = {'u E V [ Au E X}. We shall write the restriction of A : V --, V' to D(A) by the same symbol A. It is regarded as an operator in X , and (1.3) is written as an abstract equation in X if f E X = L2(f~): Au = f
1.1. Elliptic Boundary VMue Problems The elliptic regularity is expressed as
D(A) = H2(f~) n H~(~) for boundary condition (1.9) and
+av=O
D(A) = {v 9 H2(f~)
on0t2
(1.15)
for boundary condition (1.10). Those relations hold if 0t2 is sufficiently smooth for instance. We get a strong solution u 9 H2(t2) of (1.8) with (1.9) or (1.10), whenever A = 0 holds in (1.14). The operator A in X arising in such a way from the bilinear form A( , ) on V x V is called m-sectorial. To describe the meaning of this terminology, let us suppose A = 0 in (1.14) for simplicity, and specify the constant C in (1.12) as C1. We can take natural complex extensions of the Banach space X and the operator A. This means that the functions in X are extended to be complex-valued with the inner product
(f, g)= J(n f(x)9(x) dx and Af = Afl + zAf2 if f = fl + zf2 with fl and f2 being real-valued, where z = v/-Z-1. Denote them by the same symbols X and A. Then, inequality (1.12) keeps to hold, while (1.14) is replaced by
R~ .A(v, v)>__ a Ilvl G
(v c V).
A sector is given as
~0 = {~ ~ c I 0 __ I~rg zl ~ 0}, where 0 9 (0, 7r/2). If cos0 = 5/C1, an elementary calculation gives that the numerical range u(A) of A is included in E0, where
~(.A) _= {.A(~, ~)1 ~ ~ V,
II~llx - 1}.
A fundamental property of the numerical range says a(A) C u(A). The relation u(A) C E0 implies that C \ E0 C p(A), where or(A) and p(A) denotes the spectrum and the resolvent set of A, respectively. More precisely, if we take 01 9 (0, 7r/2) and z 9 C in 01 _<[arg z I _< 7r, then we have
li( z- A)-'tlx,
< -
1 sin (01 - 0). I~1"
In particular, - A generates a holomorphic semigroup are discussed in the next chapter. Inequality (1.16) is proven in the following way. Let
d(z)=dist (z, 0E0) = i n f { I z - ~ l l
(1.16)
{e -tA}t>o in X, of which details
~ 9
1. Elliptic Boundary Value Problems and FEM Then, because z C E01 we have
d(z) >__ Izl sin (01
-
-
0)
(1.17)
and also
(Au, u) lU 2
> dist (z, L,(A)) > dist (z, Eo) = d(z).
This implies
I((zz- A)u, ~)1 ___d(z)ll~ll ~, or
l l ( z / - A)~II >_ d(z)ll~ll
(1.18)
for u r D(A). In particular, Ker ( z I - A*) = Ker ( z I - A) = {0} follows and hence Ran ( z I - A) = X holds by the closed range theorem. Inequality (1.16) now follows from (1.18) and (1.17). The following notation is commonly used in the operator theory: A densely defined closed linear operator A with the domain and the range in a Banach space X is said to be of type (0, M) for 0 E (0,7r) and M _> 1, if C \ E0 C p(A), M
I)(z/- A)-' I _< V~ for z < 0, and
II (zi
M~
- A)-'II
;zi
for 0 + e _< larg z I _< 7r with e > 0, where M~ is a positive constant. In use of this terminology, we can say that any m-sectorial operator is of type (0, M) for some 0 C (0, ~/2). The adjoint form ,4* of .4 is defined by
A*(,., ~) = A(~, ~)
(., ~ ~ v ) .
If ,,4 = `4", we say that it is symmetric. In this case it holds that D(A 112) = V and
a'/~ II~llv < IIA'/=vllx < .~, Ilvllv, _
_
~1/2
where A 1/2 is the square root of A defined through its spectral decomposition.
(1.19)
1.2. Ritz-Galerkin Method
1.2
Ritz-Galerkin
Method
Considerations of the previous section are summarized as follows. Let V C X c V' be a triple of real Hilbert spaces, with the first inclusion continuous and dense so that the second inclusion follows from identifying X' with X by Riesz' representation theorem. Let A( , ) be a bilinear form on V x V and assume that it is bounded:
I,A(u, ~)1 _< c II~,llv II~llv
(u, v E V)
(1.20)
and is strongly coercive, which means that (1.14) hold for A = 0:
A(v v)> a I~11= '
~
V
(v E V)
(1.21)
Given f E V', the problem in consideration is formulated in an abstract manner so as to find u E V satisfying
A(u, v) = ( f , V)v,,v
(1.22)
for any v E V. Then it is uniquely solvable by Lax-Milgram's theorem. Ritz-Galerkin method approximates (1.22) in the following way: Prepare a family of finite dimensional subspaces {Vh}h>0 of V approximating the latter as h ~ 0. Then, we take the problem to find Uh E Vh satisfying
,A(uh, Xh) -- ( f , Xh)v,,v
(1.23)
for any Xh E Vh. Unique solvability of (1.23) follows from the same reasoning as for (1.22). Let Xh be the space Vh equipped with the topology induced from X, and Ph : X ---+Xh the orthogonal projection. The linear operator Ah : X h --+ Xh is defined through
A(uh, Xh) = (A~uh, Xh) for Uh, Xh E Xh. If f E X, problem (1.23) is equivalent to the equation
AhUh = Phf in the finite dimensional space X h. Stability and error estimate of the approximate solution are verified as follows. Let Rh : V --+ Vh be Ritz operator defined through
for ~h E Vh and u E V. Its well-definedness follows from Lax-Milgram's theorem similarly to the unique solvability of (1.23). If u E V is the solution of (1.22), then the approximate solution uh of (1.23) is nothing but Rhu. This gives the relation
R h A - I = Ah~Ph.
(1.24)
1. Elliptic Boundary Value Problems and F E M If .4( , ) is symmetric, the operator Rh " V --+ Vh is nothing but the orthogonal projection with respect to the inner product .4( , ). Therefore, we have
.A(R,,~, R,~u) _ .a(u, u) and
A(R,,~- ~, R,,~- ~) _ A(x,,- u, x , , - u) for any Xh E Vh. Then boundedness (1.20) and strong coerciveness (1.21) o f . 4 ( , ) imply the stability
IIR,,** Iv _ o Ilullv
(1.25)
and the error estimate
IR,~.,,- **llv _< o inf
Xh CVh
IIx,,- **1 v.
(1.26)
Those relations hold even in the general case of `4" 5r .4. In fact, from (1.21) and (1.22) we have ,5 IIR,,ull~
_< ..4(Rh**, R,,~,) = ..4(**, R,,u) _ C I1~,,~11~
9I~ Iv
and 6 IIRhu - ** I =V
<
.A(R,~u - u, R,,.,, - u) = .A(R,,u
<
c IIRh,~- ullv" Ilxh - ullv
-
-
- u, x,, - **)
for any Xh e Vh. Hence (1.25) and (1.26) follow in turn. Operator theoretical features of Ah in Xh are rather similar to those of the continuous version, A in X. If the complex extensions are taken as in the continuous case, Ah in Xh is m-sectorial of type (0, M) for some 0 E (0, rr/2) and M > 1 independent of h. If X = L2(~) and g = Hg(~) or V = H I ( ~ ) , and A ( , ) is associated with the elliptic boundary value problem described in w this uniform structure of a (Ah) is refined in the following way. T h e o r e m 1.1. The spectrum a(Ah) of Ah is contained in a parabolic region in the complex plane independent of h > O. Consequently, each 0 E (0, rr/2) admits constants C > 0 and M >_ 1 independent of h, with Ah - C of type (0, M ) in Xt,. Proof:
Split A ( ,
) as
A(....,,) = A~ v) + (D,,. v).
(1.27)
where
A~ v) = i,j= s ~s a~j(~)~~ ~~
dz +
fo a ~uv
dS
1.2. Ritz-Galerkin Method and
2
Ou
j=l Here, .4 o is a strongly coercive symmetric form on V x V, and B is regarded as a bounded operator B 9V ~ X. Let A ~ be the self-adjoint operator in Xh associated with .4 o JVj,xVj~The inequality 1 <- IIm z[
I](zlh - A~
(1.28)
is a consequence of a property of self-adjoint operators, where z r C \ R and Ih denotes the identity operator on Xh. This implies
IlAOh(ZI h -- AO)-X)(~hllx ~ I[~)(~hllx-[-[Z[. [l(ZIh- A~ <
1 + II
z[
~)~hllx
[Xh x_< ~ l l X h l l x
for Xh E Xh. Heinz' inequality now gives that
l[(A~ 1/2 (ZZh -- Aoh)--lllxh,Xh ~ v/ilz[I/2 -
lira zl
On the other hand, the relation Ah = A~ + Bh holds with Bh = PhBIv h so that Bh : Vh ---* Xh is uniformly bounded:
IIBhllv~,x~ <_ c~, where 6'2 > 0 is a constant. We have
iix
o
for Xh r Xh. This implies
II(AO)- 1/21IVh,Xh <
C3 = 6 -1/2
similarly to (1.19). Given Xh E Xh, we have I[(zlh-
AOh)-lj~h)~h[[v ~ C2 II(A~) 1/2 (ZZh- Ao) -1 ]Bh)(~h[[x -<
v~C2C3 Izl 1/~ Jim zl
IIx~llv
M Izl 1/2 =
ii m z]
IIx~llv.
1. Elliptic Boundary Value Problems and FEM
10 We also have
(ZIh
--
A~) -1 (ZIh
-
Ah)
=
I h
(ZIh
--
--
Aoh) -1 Bh.
(1.29)
Therefore, if z E C satisfies
Mlzl ~/2 < - 1
(1.30)
IIm z I - 2' then z E p(Ah), and furthermore
follows for Xh E Vh. This shows the first assertion. Identity (1.29) implies (ZIh -- A,~ -I = (ZIh --
Ah) -I- (zIh - AOh) -I Bh (zZh - Ah) -I
Given Xh E Xh, we have
II(zZh-A )-'x,,llx _< (Ixhl ~ + C~
II(z:,,-
Ah) -1 Xh IV)" lira z1-1
_< (llxhllx + 2c~ II(zz,, -
A~ Xhllv)- Im z1-1
<
[IxhlIx
1+
]-i-m z I
IIm z1-1
Therefore, inequality (1.30) implies 3
II
<
2
and the second assertion follows.
1.3
Finite E l e m e n t M e t h o d ( F E M )
We proceed to the numeric.al method for boundary valuta probh.'m (1.8) with (1.9) or (1.10). We take X = L2(~) and V = H~I(~'2) or V = H~(f~), where f~ C R 2 denotes the convex polygon in consideration, and suppose that the associated bilinear form ,4 is strongly coercive. Finite element method is a kind of Ritz-Galerkin approximation applied to variational problem (1.22), where f E X. In tile simplest case, f~ is divided into small closed triangles
11
1.3. Finite Element Method
(simplexes) with the size p a r a m e t e r h > 0. The following condition is always supposed: If T1 and T2 E Zh satisfy f = T1 N T2 -r fi, then f is a face or a vertex of both T1 and T2. The set of such triangles is denoted by zh so that we have h = m a x { a ( T ) l T r zh}, where or(T) denotes the diameter of T. Each T E rh is called an element. Radius of tile inscribed circle of T is denoted by p(T). Let Vh be the set of continuous functions linear on each T E zh. It holds t h a t
Vh C V
dim (Vh) < +oc.
and
Given a continuous function v on f~, we have vertices of T E 7h are equal to those of v. It by 7rhv. T h e following theorem is due to J.H. {~-h}h>0 is said to be regular if there exists a
a unique is called Bramble constant
(1.31)
function in Vh whose values on the the interpolant of v and is denoted and S.R. Hilbert, where the family r, > 0 such t h a t
~(T) > . ~(T) for any T C fh and h > O. Theorem
1.2. If {Th} is regular, the inequality IIv -
~hVlIL~(T) + h
IV(v -
~hV) llZ~
(1.32)
holds for v E V N H2(f~) and T C Th. In particular, we have I1~ - ~VlIL~<~) + h IIV (v -- ~ ) 1
L~(~) < C h ~ II~IIH"<~)
(1.33)
with a constant C > 0 independent of h. Proof: Inequality (1.32)implies (1.33) as ll~ - ~ 1 1
~
h~
~
L2
TGTh
<
c~h4~llvll
~
E IIv -- '~hV TGTh
I21(T>
=c~h~
TCTh
To prove (1.32), introduce the reference simplex B={(x~,x~)l
z, > 0 ,
x~_>0, x , + x ~ < l } .
Given v C H2(B), let 7rv C 7)1 be the interpolant, so t h a t 7)1 denotes the set of linear functions and 7rv = v at each vertex of B. Any w E L2(B) is associated with the linear functional F 9H 2 ( B ) --~ R defined through
F(~) = (~ - ~v, ~ ) ,
1. Elliptic Boundary Value Problems and FEM
12 where ( , ) hence
denotes the L 2 inner product on B. The inclusion H2(B) C C(-B) holds and
II~-vll~,o~(~) __4c II~,llm(B)
(1.34)
follows. This gives
IF(v)l ~ I wll~(~)Ilvllm(~). On the other h a n d FIp 1 = 0 so that F is regarded as a bounded linear operator on (2 ~-- H 2 ( S ) / ~ ) 1 . W e have
wll~(~)IIv
F(v)l ~ C (
(1.35)
Q
1
with II~IIQ -- inf I v + qllm(B) l q ~ Px ~. We claim that there exists an absolute constant 7 > 0 satisfying ,#
II~IIQ _ -~ Ivl~,,
(1.36)
for any v e Q. Here, recall that Ivl2,B = ~-~1~1=2IID~'olI,,~(~) we take p C 7)1 satisfying (v + p , 1) --
9 In fact,, given
v e H2(B),
(0 (v+p),l)= ( ~y (v + p) , l) -- 0.
Poinca%-Wirtinger's inequality holds on B so that (v, 1) = 0 gives
Ilvll~(~) ~
"[1
IVI1,B
with an absolute constant 71 > 0. (Actually, given v C H i ( B ) , extend it on Z} = {(x, y) [ Ix + y[ _~ 1} by the even reflection. Then, we see that 7~-1 is taken as the second eigenvalue of - A with the Neumann b o u n d a r y condition on Z}.) This implies also Iv + P[ l,B by ( ~
+ p), 1) :
( ~ ( v + p), 1) : 0
<
71'1, + pl
2,B
with a constant "/1 > 0. Therefore,
IIv + pll L~(B) --< "T1 Iv "Jr-PI1,B <-- ")'1")/1 ['v +
PI2,B -
follows. Inequalities (1.35) and (1.36) are summarized as
IF(v)l ~ cG IIwlIL~(B)I'vl~,~ or,
I~ - ~-vllL~(~) __ cG I~1~,~.
"T~'T'~ I'uI~,B
13
1.3. Finite Element Method Take w E H i ( B ) and repeat the above argument for F ( ~ ) = (~ -
~ , ~)~,~ - X ~ (z)~( ~ - ~ ) , Z)~
9
I~1=1 Because Try E 791, inequality (1.34) implies
We obtain similarly
vl~,~
I V - 71"731,B --~ C
and hence
Iv - ~11/-/,(.~)
_< c I~1~..
(1.37)
follows for any v E H 2 (B). We proceed to the family {%} of regular triangulations of a polygon ~ C R 2 and take T C % arbitrarily. There exists an affine mapping 4) with (F(T) = B and its Jacobian J satisfying 5 < J / ( h 2) < ($-1 where ~ > 0 is a constant independent of h. This follows from the regularity of {%}. Let v E H2(T) be given, and set ~ = v o ~-1 Writing ({, r / ) = ~(x, y), w e have
~V
09~?
09~
< Ch 2 (
+ 0~0~
02V
02V
+o--~y + ~
on B. It follows t h a t IIv - ~hvll 2L2(T)
--
< -
L2(B)
C2J Ivl~,B < - C2J" C2h4J-1 Ivl 22,r
and the first inequality of (1.32) has been proven. Regularity of {rh} gives also that
on T, where w = tb o ~. Hence we have I v - ~h~121.r < C I ~ - ~ 1 ~ similarly. The proof is complete.
< C I~I~.B <
C h~ Ivl~.z []
W h e n f~ has a curved boundary, some modifications of 7rhu are necessary near there. If V = H~(f~) and f~ is convex, we may take a suitable polygon f~h C 12 approximating it and just put 0 on f~ \ f~h. The m e t h o d of curved element is valid for the general case, and assures the existence of Vh and ~rh subject to (1.31) and (1.32) although details are not described here.
14
1. Elliptic Boundary Value Problems and F E M
From the general theory developed before, stability (1.25) and estimate (1.26) follow for the approximate solution Uh = RhU and the error eh = Uh -- U, respectively. We have
I I R ~ - ~11~(~)~
Chll~llH~(~)
(u 9 H2(FZ)f"l V)
(1.38)
and lim R h u - U Iv = O hJ.O
(u 9 V) ,
(1.39)
(v
(1.40)
because H2(f~) C'l V is dense in V. Similarly we have lim IlPhV -h~O
v Ix = 0
9 X)
.
The elliptic estimate IA-l f IH2(~) <- C Ilfl L2(a)
(f e X)
(1.41)
follows form the convexity of f~. This implies the following error estimate:
I~,~ - ~1 ~=(~) ___ c h ~ II~IIH~(~)
(~ E H2(~)71 V)
(1.42)
In fact, putting ek = R k U - u, we have
I1~11~=(~)
-
A ( ~ , A *-leh) = A(R,~u - u, A *-~ eh)
=
A ( R h u,
--
u,A*-'
e_.h
--
~rhA*-1 c h)
___ c IIRh~ - ~llv" ](~h - 1)A *-~11~ <- C h,~ II~ll.~(n)IIA*-' ~hllH~(~) --< Ch~llu I/~(~) II~hll~(a)" This means (1.42). This kind of calculations is called the duality argument. Similarly we have
IlRhu-
UlIL~(~) --< ChllU[lHl(gt)
(U 9 V ) .
(1.43)
Relations (1.24) and (1.42) imply
IIA; 'Ph - A - 1
<- ch?,
(1.44)
L~(~). Note th~U inequality (1.44) is weaker than (1.42) in the case of V = H~(~), where D(A) involves the boundary condition. (See (1.15).) Henceforth, the family of triangulations {rh}h>0 is always supposed to be regular. where II " I ~ , x de~ote~ the o p e r a t o r ~orm in X -
1.4
Inverse Assumption
We say that the family {rh}h>o of triangulations satisfies the inverse assumption if there exists a constant ~'1 > 0 such that p(T) > .~h
1.4. Inverse Assumption
15
for any T E "rh and h > 0. This is equivalent to assuming
Ilxhllc~(z) < c h-2/" Ilxh IL,(T)
(1.45)
for T C ~-h and Xh E Vh with a constant C > 0 independent of h > 0, where p E [1, eel. It also gives
I VxhllL~(r) --< Ch -~ IlxhllL~(r)
(1.46)
and hence
follows. In this section, we show several inequalities derived form this assumption. Theorem
1.3. There exists/3 > 0 such that
(1.47)
IIA~llx.,x~ < ~h -2 if
{Th} satisfies Proof:
the inverse assumption.
In the case that .4( , ) is strongly coercive and symmetric on V x V, we have
Cllxhllv similarly to (1.19). Given Uh, Vh E Vh, we get from (1.46) that
I(Ahun, vh)l
--
<
k~,~ u h , ~ h
vnj
c II~hllv-IIv,~llv
< c h -~
Ilu,~llx IIv,~llx,
and hence (1.47) follows. In the general case, we split . 4 ( , ) as in (1.27). Namely,
A(~, ~)
A~
:
~) + (B~, ~),
where .40( , ) is a strongly coercive and symmetric form on V x V and B is regarded as a bounded operator B " V ~ X. It follows that An = A~ + Bn, where A ~ is associated with .4~215 from
and Bh = PhB[vh. Inequality (1.47) now follows
and
IBh~hllx
~ C
II~hllv ~
C h -1
for Xh C Vh. Recall that lrh denotes the interpolation operator.
II~llx
1. Elliptic Boundary Value Problems and F E M
16
T h e o r e m 1.4. If f~ is a convex polygon, we have (1.48) with a constant C > 0 independent of h. Proof." Because f~ is convex, we have A -1 (L2(~)) C H2(f~) C C(~) and AhTrhA - l " L2(Vt) ~ Vh is well-defined. Let Ph" L2(at) ~ Xh be L 2 projection and Rh "HI(Vt) ~ Xh Ritz operator. We have (1.24) and hence AhrchA -t = A,~ (Trs - Rh) A -1 -b Ph = Aht~h (Trh -- I ) A - ' + Ph
IIP~IIL~(~),L~(~)<--l
follows. Here, In use of (1.46), we have
[(Ahxh, r
is obvious.
= I(VXh, V~,,)[ <_ C h -1 IlVx,~ll~(~)II~,,,ll~(~
for Xh, ~2h E Xh. This means
for Xh C Xh. We obtain
IIA~R~ (~
-
I)A-X,o
Ic2(n)
-< Ch-1 IIvR
< ch-' IIv
- i)A-Xvlig=(al
_< c I[A-Xv[[,_,~(~) _< o Ilvll,_,~(~) by (1.25), (1.33), and (1.41). Therefore, (1.48) follows and the proof is complete.
[]
Recall that Xh denotes the space Vh equipped with the topology induced by X = L2(fZ). T h e o r e m 1.5. The L 2 orthogonal projection Ph " X ---, Xh satisfies the estimate
P,~llv, v ~ c if the inverse assumption holds for {Wh}. Proof:
Since
IlPhllx,x < 1, we have only
to
show that
for v E V. To this end, let /).h" V ---, Vh be the Ritz operator associated with
A(,~, v) = ./~ (vu. w + uv) d:~.
(1.49)
17
1.4. h~verse Assumption
Since Phlvh = Ih, we have for Xh = R,hV E Vh that
IIV(Ph-/)vllg~(a)
=
I I V ( P h - I ) ( v - Xh)IIL~(~)
<
I I V P h ( v - Xh)llL2(~) +
IIV(v- Xh)llL~(a)
_< oh-' IIP.(.v- X,,)IIL=(~) + Ivl .,(~) + )(~hllHl(a) Ch-ll v - x,,ll~=(a) + Ilvl/_.,(,-,) + IIx,, 1.1(.).
<-
Here, inequalities (1.25) and (1.43)imply Xh -- VIIL2(a ) ~ Ct~ ]Iv HI(a),
and
I Xhl Hi(n) <-- V ]Hi(U)
respectively. This gives (1.49).
[]
Similarly to (1.39), we obtain lim IIP, v h;0
vllg - 0
(v ~ v)
by (1.49). The following lemma of J. Descloux has its own interest. L e m m a 1.6. If {Th}h>o satisfies the inverse assumption, there exists a constant C > 0 provided with the following property: Let To E rh and f~o C f~ satisfy To N gto
=
0.
(1.50)
Then it holds that
IIPhVllL~(~o)~
exp
( - c dist (~o, To)h -I) II~ll.(~)
(1.51)
for any v E Le(f~) with supp v C To. Proof: Letting Ro -- To and R1 the union of triangles in Th \ {To} adjacent to To, we define Rk inductively as the union of triangles in Th adjacent to Lk = Ue
dist (Ro, Rk) < C ~ ( k - 1)h
(k = 1 , 2 , . . . )
(1.52)
with a constant C1 > 0. Given ~o c ~ with (1.50), we take k* satisfying rio C Dk., where Dk = Ue>kRe for k = 0, 1, 2 , . . . It holds that dist (To, Dk.) < dist (To, Rk.+l) < Ck*h. If v E L2(f~) and )(.h E X h satisfy supp v C To and supp Xh C Dk U Rk
= Dk-1
for
it follows that
(P~, x.) = (~, xt~) = o.
k >_ 1,
(1.53)
1. Elliptic Boundary Value Problems and F E M
18 We can take Vh E Xh satisfying
Phv vh =
0
on Dk on t2 \ Dk-1,
by setting Vh = Phv at one or two vertex (vertices) and Vh = 0 at the other vertices (vertex) on each T E ~-h contained in Rk, appropriately. In this case, with a constant ~/> 0 independent of h it follows t h a t
IlvhIIL~(T) < ~ IIPhvlIL~(T) for any T E Wh. We obtain
0 - (Phv, vh) -- IIPhvll ~L2(Dk) + s
(Pj~v)vj, dz
k and hence
ak - l i P , v i i ~L2(Dk) <- IIPhvll~=(.~)IIv~ll~<.~) _< ~ IIPhV IL2(Rk) = "7 (ak-1 -- ak) , for k = 1 , 2 , . . .
Writing ")/
~e
-2C2
1+~ with C2 > 0, we can deduce that
ak. <- e-2C2k'ao -< e-2c2~" IIPhv IlL2(a)2 _< e-2C2k" IIv IL2(n)2 . We have (1.51) as
iiPh~ll ~L~(~o) < < _
~ < ~-~c~. IlVlIL~(~) IIPhVlIL~(D~.)-exp (-2C2C~Xh,-'dist (To , ~o))" [['vll2L2(f2)
by (1.53).
[]
L e m m a 1.6 implies the L q boundedncss of the L ~ orthogonal projection Ph : X ---* Xh as follows. Theorem
1.7. If {Th} satisfies the inverse assumption, the inequality
IIP,,ll~(~),~(~) < c
(1.54)
LI(~'~),LOO(~'~)= <'~, PhV>LI(~-,~),LC~(~,~)
(~.55)
holds for q E [1, oo]. Proof."
We have
19
1.5. L ~176 Estimate
for u C L2(f~) and v C L~(ft). If (1.54) is shown for q = oo, then equality (1.55) implies (1.54) for q = 1. Therefore, (1.54) follows for 1 <_ q _< oc by Riesz-Thorin's interpolation theorem. In this way, the proposition is reduced to the case q = oc. Suppose that the m a x i m u m of Phv is attained on To, where To E Th. Given T E ~-h, we set VT =
v 0
on T, otherwise.
Then, v is decomposed as v = }--~-Te,hVr, SO t h a t
Ilrhvllg~(~)- IlPhvllg~cZo) < ~ IIP',VTIIL~CTo). TE ~-h
By virtue of Descloux's lemma, we have
IIPhvT Ic~(ro)
IIP, vTI c~(ro)
<
Ch-1
<
C h -1 exp
( - C dist
< C'exp (-C"dist
(To, T) h -~) IIVTIIL~(T)
(To, T)h-')}IVTI}L~(r~).
Recall Rk (k = 0, 1, 2 , . - . ) in the proof of L e m m a 1.6 with inequality (1.52). Because {Th} is regular and satisfies the inverse assumption, the number of triangles of Th contained in Rk is dominated by C k 2. We have
TCRk
~ C ( ~-']~2e-Ck)k -~
"V"L~176
and the proof is complete.
1.5
L ~~ E s t i m a t e
We have shown t h a t Ritz operator Rh " V --+ Vh associated with the finite element m e t h o d described in w satisfies
IIR~vl Hi(f-I)~ CIIVlIHI(a)
(v ~ V)
and
for j = 0, 1 and k = 1, 2. It should be noted that the L 2 stability
IIR,,~II~(~) < CIl~ll~(~)
(~ ~
v)
1. Elliptic Boundary Value Problems and F E M
20 cannot be expected.
In fact, if this is true, we have 1
by (1.24), where S denotes the closure of an operator S. In the adjoint form, this implies
I[AA;1p~[Ex,x <_ C, which is impossible because of Vh 92 D(A). J.A. Nitsche and R. Scott obtained the inequality
(v ~ v )
IRhvl Loo(n)<-- C log h[ Ilvl LO~<~>
(1.56)
for the case that f~ is convex, V = H~(f~),
and
A(u, v) = f V u . Vv dx, lit
under the inverse assumption. There, Nitsche adopts weighted norms, while Scott makes use of the discrete Green's function for the proof. Those methods will be described in the next section. Since Rhlvn = Ih, (1.56) implies
_< c (Itogh + 1)
u
-
x,~ll~(~)
for Xh C Vh. Tile inequality
I1~,,~- vllg~(~) --< Ch ~ I~llw~,oo(a) is shown similarly to (1.24) for v E V N W2,~176
so that we have
IlRhu - ullL~(n) _< Ch21log hlll,,,llw~.o~.
(1.57)
In this way, boundedness of Ritz operator implies the error estimate of the approximate solution. Furthermore, the optimality of (1.57) is known; we have a domain f~ C R 2, a triangulation {Th}, a constant c > 0, and u E Cg(~) such that
ashl0. Closely related topics are discrete maximum principle and L p error estimate. The former is originated by P.G. Ciarlet. A square matrix A = (aij)l_
aii > 0
(1 <_ i < n),
a~j <_0
and ~L~aij_>0 j=l
(l_
(i ~r j ),
1.5. L ~~ Estimate
21
In the case of the finite element discretization for (1.8) with (1.9) and s D) = - A , this condition holds if any T E rh is an acute or right triangle. Then the discrete maximum principle
f
>
0
~
RhU > 0
(1.58)
follows for f = Au. This property keeps to hold for the inhomogeneous boundary value problem. a. H. Schatz showed a general form when s D) = - A and {Th} satisfies the inverse assumption. On the other hand, P.G. Ciarlet and P.-A. Raviart showed that this type of maximum principle implies the estimate 2
IIR~ ~(~) ~ c~ ~
IIf~l ~(~)
(1.59)
k=0
for p > 2, where u = A - i f with
f= fo+
~aA OXk"
k=l
The optimal Lp error estimate follows from W I'p stability of Ritz operator by R. Rannacher and R. Scott"
IIm~llw~,~(~)_ c II~llw,,~(~)
(1.60)
Here, p E [2, oo], f~ is a convex polygon, and {Th} satisfies the inverse assumption. Those facts will be described in the next section. Now by Sobolev's imbedding theorem, inequality (1.60) implies
IIRhulIL~(a) --< c~ II~llw,,~(~)
(1.61)
for p > 2. This give~ O(h) conv~rg~n~ of I1~ - ~11~(~) by
Furthermore, inequality (1.61) gives (1.59) if t2 is convex, because then the W ~,p regularity of N.G. Meyers holds for I P - 21 << 1:
A-1
f0 +
<__Cp~
~xk k=l
Wl,p(f~)
IIAIILp(n)-
k=0
We can summarize as follows. Let f~ be a polygon, s D) = - A , and {~-h} a family of regular triangulations. Denote by Rh the Ritz operator associated with (1.8) and (1.9). Then, the discrete maximum principle (1.58) holds if any T C Th is an acute or right triangle. Its generalization is valid whenever {%} satisfies the inverse assumption. If f~ is convex furthermore, inequality (1.60) follows for p e [2, oc]. Either (1.58) or (1.60) implies (1.61), and the latter gives the O(h) error estimate in L ~176 norm.
1. Elliptic Boundary Value Problems and FEM
22
W h e n lower ~ r m s are added to L;(x, D), the discrete maximum principle is more difficult to verify. For instance, if s D) = - A + c(x) with c(x) > 0, any angle of triangles in wh is requested to be _< z r / 2 - e with some e > 0 for (1.58) to hold. However, inequality (1.61) is stable under such a perturbation. The following theorem is concerned with problem (1.8) with (1.9). There, neither the convexity of f~, nor the inverse assumption on {Th} is necessary. T h e o r e m 1.8. If any T E Th is an acute or right triangle, inequality (1.61) holds for p > 2, whenever aij = a(x)(~ij and strongly coerciveness (1.21) holds for s D) in (1.8) with (1.9).
Proof." Splitting A ( , ) as in (1.27), we have
~fa
0u0~
j=l
Let Wh be the set of continuous functions on f~ linear on each T E Th. It follows that v~ = w h n v . Given Xh C Wh, define JhXh E Wh by
JhXhlp = max { Xh[p, 0} for P E :Z-h, where Zh denotes the set of vertices of the elements belonging to Th. We show the following. Lemma
1.9. Under the assumption of the theorem on {Th}, it holds that A ~ (~h
JhXh, Jh~h) >_ 0
-
for any Xh E Wh. Proof: We have only to show V ( X h - JhXh)" VJhxh > 0
(1.62)
on each T C Th. Note that V (Xh -- Jhxh) and V Jhi~h are constant vectors on T. We shall show
0
Oz--~
(~,~ -
0
J,~x,,) . ~ d h x , ~ >_ 0
for i = 1, 2 by taking an appropriate rotation (1.62) is invariant under such a transformation If Xh _> 0 or Xh _< 0 on T, inequMity (1.62) vertices P1, /"2 and Xh < 0 at the other vertex may take so that P1 and P.) are on the xl axis Then, for i = 1 we have 0 --(xh-
Oxi
(1.63)
of the variable x. The left hand side of and then the proof will be done. is obvious. Suppose that Xh _> 0 at two P:~. Under the above transformation we and that P:t in the region where x9 < 0.
,]hXh) = 0
23
1.5. L c~ E s t i m a t e
and (1.63) follows. For i = 2, we take the vector el = PaP1 and e2 = P--~. JhXh >_ 0 on T and Jhxh = 0 at Pa so t h a t
We have
ej . V ghXh >_ 0
for j = 1, 2. Now the a s s u m p t i o n for T guarantees for e = ( 0 , - 1 ) t h a t 2
e --- E a j e j j=l
with some aj > 0 (j = 1, 2). Therefore, e - V J h X h > 0 or equivalently, 8 ~dhXh c9x2
< 0.
We have Xh -- JhXh <_ 0 on T and Xh -- JhXh = 0 at P1 and/~ 8 8x2
Similarly we get
(xh- &xh) <_ o.
This implies (1.63) for i = 2. For the reverse case t h a t Xh <__ 0 at two vertices and Xh > 0 at the other vertex, we have only to replace the roles of Xh and Xh -- JhXh each other. Inequality (1.63) follows and the proof is complete. [] We continue the proof. Given u E V and k >_ 0, we set Xh = R h u -
k E Wh
and
r / = JhXh C Vh.
Note t h a t r/Ion = 0. Uniform ellipticity (1.7) implies ~1 IIV~IIL~(~) < A ~
~) =
- A ~ (R,,~ - ~, ~) + A ~ (Rh~, V),
where
.4 0 ( / h ~ -
~, ~) = .4 0 ( R h u - k - ~, ~) = A ~ (Xh - &Xh, &Xh) >_ 0
by L e m m a 1.9. We have
.40 (Rh,,, ~) =
.4(Rh,,,
~) -
(SRhu,
~) =
A(u,
~) -
(eRhu,
~)
so that
~1 IIv~ll[~(~) _< a(~, ~ ) -
(sR~,
~)
Set w = supp 7. T h e n the first term of the r i g h t - h a n d side is e s t i m a t e d as
(1.64)
1. Elliptic Boundary Value Problems and F E M
24
In use of Poincar6's inequality on f~, we obtain
A(u, ~) <__ C IIV~IIL~<~)(llVUllL~<~)+ I~IIL~<~))
' ' ( IIWlIL~(~)+ II~llL~(a)) _ CIV~IIL~(~)I~1~-~ 1
1
(1.65) for p > 2, where w I denotes the volume of w. We claim the following. Lemma
1.10.
We have
for any v E V. Proof: Sobolev's and H61der's inequalities imply
for 2 < q' < oo. If we take q ' = p and r = (1 - 1/p)(1/2 - l / p ) -1 > 1, we have 1 -
-
rq
=
1
1
2
p
and
q r ' = 2,
where 1/r + 1 / r ' = 1 and 1/q + 1 / q ' = 1. Then,
_< cI..,I ~ IIBvll~,~(~) V~IIL~(~) - Cl~,l~-~ IIBvlI~(~)IV~IIL.~(~I. 1
1
In use of bj, c C L ~ ( ~ ) , we get
_ c ~l&-~ IIVvl L~(~)IIV~l~(~) by Poincar~'s inequality. Let us continue the proof. In fact, inequality (1.25) reads;
IlVR,,ullL~(~) _
=
/~h u
in L e m m a 1.10, we obtain I
l
1
1
_ c I~1~-~ IlVullL~(~)IIW/llL~(~) _ c 1~1 89IlWllL,(~)IIV~llL~(~>.
1.5. L ~ Estimate
25
We have
or
IlV~ll~(~) _< c I~1 89 IlWll~.(~) by (1.64)and (1.65). Now, H61der's inequality gives
IIV~IIL~(~) --< I~1 ~- 89IIV~IIL~(~) for 1 _< q < 2. On the other hand, Sobolev's inequality is applicable for r/ because r/loa = 0. That is,
II~ItL,* (~) --< C IIV~IIL~(~) for
1/q* = 1 / q - 1/2 with a constant C = C(q) > 0. Combining those inequalities with
I1~11~_< I~11-~" I1~11..(~), we conclude that
I1~11.(~) --< C I~1= IIW I.(~)
(1.67)
with ~ = 1 + ( 1 / 2 - 1/p) > 1. Here, we make use of the following lemma. Lemma
1.11.
The inequality
ITI I1~11~(~) ~
3
II~ll.(y)
(x.68)
holds for each T C Th. Proof: Note t h a t ~ >_ 0. There is a vertex P0 E T attaining the m a x i m u m of ~ on T. Let us identify T with the triangle :# in R 3 put on x3 = 0. Then tTlt IITIIILoO(T)indicates the three dimensional volume of the cylinder
On the other hand, because ~7 > 0 on T, the value
11~711L~(T)dominates the volume of
the three dimensional simplex composed of T and P0. The latter is 1/3 of the three dimensional volume of S. This shows (1.68) and the proof is complete. [] For t _> 0, we set
p(t) = Isupp Jh ( R h u - t)l. From the definition of Jh follows ~ ~ p ( t ) dt
~ TCTh
ITI IIr/llLoo(T) 9
1. Elliptic Boundaly Value Problems and FEM
26
Therefore, by (1.67) and Lemma 1.11 we get
Z
~ p(t) dt <
Z
I~,(~)
3 I1~11.(~) = 31~
TCTh
_<
3c ~1 ~ IWl Lp<~) =
3c
IlWll~,<~)p(k)~
for any k >__0. The non-negative and non-increasing function
H(k) =
p(t) dt
satisfies
H(k) _< ~ (-H'(~:)y
(k > O)
with ~ = 3C IlVUllLp(a I. This implies that
2/-1
-
2/-
1
nl/-~"
The left-hand side is non-negative and consequently, H(k) = 0 if k > 2/ -7-1
1
1
H~-~(O)~c~.
Noting
H(O)-
p(t) dt < ~ ( - H ' ( 0 ) ) ~' < ~p(0) ~' < n ~0 ~176
I~1 "~ ,
we obtain H(k) = p(k) = 0 if k _> k*, where k*-
2/ 2/-1
if~l~-I ,~.
In other words, Rhu(x) _< k * = constant x
IIV'~ll~(~)
(~ e ~).
Similarly we have
and the proof is complete. Inequality (1.59) with p > 2 is t)roven similarly, and then the following theorem is a consequence of tile duality argument. It is regarded as a discrete version of the L 1 estimate of Brezis and Strauss.
1.6. L p Estimate
27
T h e o r e m 1.12. Under the assumptions of the previous theorem, each q E [1, 2) admits a constant C > 0 independent of h > 0 satisfying
(1.69) for vh E Vh. Proof." Take gi E C~~
(i = 1, 2) and set
F(~) =
g'-5-~,
- ~ i=1
dx.
We define Uh E Xh by A(Uh, Xh) = F(Xh) for Xh E Xh. For this problem we have 2
I~IIL~(~) ~ C~-~ Ig~ IL~'(~/ i=1
with q' > 2 as is described above. Vh E Vh, and hence
dx i--1
It holds that (AhVh, Uh) = A(Vh, Uh) = F(Vh) for
IF(vh)l ~ IILhVhIILI(~)IluhlIL~(~)
0Xi ] 2
c ~ IIg~ll~,(~)IIAhvhlIL~(~) i=1
follows. We get
with 1/q + 1/q' = 1, and the proof is complete. Note that inequality (1.69) is a consequence of (1.59), and the assumptions made in Theorem 1.12 are not essential. For instance, it holds if/2 = - A and {rh} satisfies the inverse assumption.
1.6
Lp Estimate
In this section we show (1.60), supposing that /2 = - A , gt C R 2 is a convex polygon, and {Th} satisfies the inverse assumption. We apply the methods of the discrete Green's function and weighted norms. Because
117rhv - v[ w~,~(a) < Ch l v w~,,(a)
(1.7o)
I. Elliptic Boundaly VMue Problems and F E M
28
holds for v e WI'p(f~)N W2(f2), inequality (1.60) gives that
I l R h u - Ullwl,,tn)
=
[](Rh - I ) ( u -
<
( c + 1)llu - ~hullw,,~(.) < C h Ilullw~,~(.>
7rhu)llwl,p(a)
for ~ c W ~ " ( a ) n W~"(O). Here, we take the duality argument. Given p E [2, oc), let p' E (1,2] be the dual exponent satisfying 1/p + 1 / p t = 1. Because g2 is convex, the operator -A'
Wo~'p' (ft) f3 W 2,p' (f2) ~ L p' (a)
is an isomorphism. Any u e Wo'P(t2) N W2,P(t2) admits v e Wd'P'(t2) f3 W 2'p' (f2) satisfying - - m ' v -- [ R h U -
"lt[p-2 ( R h ' U -
lt)
in f2
and
It follows that
=
( v ( R , ~ , - ~), v (~ - ~h~)) p--1
<- c h~ II~,llw~,." This gives an error estimate of the optimal order,
IIR,,~-',,ILL.(.) _< G h ~ II'.,llw~,.(~>
(1.71)
for p e [2, oc). Given Xo E ft and ~c >> 1, we introduce the weight function cr = axo,h by ~(x) : (I 9 - xol ~ + ~:~h ~) '/~ A quick observation guarantees [D'cr~ I < C,,,cr "-1;~1
(1.72)
for Ifl[ = 1 , 2 , . . . and 7 E R with Cz,~ > 0 independent of x0 and h. Also, the inverse assumption implies Z 0 ~ T ~ Tit
===~
(7 ~,~ ]1
011
T
(1.73)
and also max T~rh
maxT o" } < C minT cr --
(1.74)
1.6. L p Estimate
29
uniformly in x0. Then, inequality (1.32) gives (1.75) TETh
for/3 e R 2 and v e H~(f~), satisfying
f a-2-~dx
vlT e
~1~+
-< A~ (
/0
He(T) for any T e Th. Finally, N2h2)-1-?/2 dy
@2 + t~2he)-l-~/2r dr ~ h -~
(1.76)
forT> 0 ashl0. Here, we provide an a priori estimate for -Av=f+V.b
with
inf~
u = 0
on cOf~.
(1.77)
If f = 0, we have
Ilvll~,(~) ~ c~ Ilbll~(~)
(1.78)
for p > 2. To see this, we take the dual exponent p' = p~ (p - 1) e (1, 2] of p e [2, ~ ) . Because ~t is convex, v C / 2 ( f i ) implies
W -- (--A) -1 (172P-2V) e W o 'p' (~~)f] W 2'p' (~-~) and
Ilwllw~.p,(,-,)<_ c I1~1p-1 .(~). In use of Sobolev's inequality we have
I1~11~,,(~) -
(w, w)
-
(b,
Vw)
< Ilbll ~ .llWll,_,~ (~) - G IlbllL~(~) 9II~ll~,~,/~) p-1 G IlbllL~(~) 9I1~11~,(~) 9 -
Lp+
(a)
This means (1.78). The following inequalities are also made use of. L e m m a 1.13. Let ~ C R 2 be a convex polygon and v e H~(~) solve (1.77) for f E H~(~) and b C Hl(~) 2. Then, taking O < 7 << l, we have
jaa2+3lD2vl 2 d x <
Cf~2+~lV.bl 2 d x + C 3 h - 2 ~ a 2 + ~ l b [
2 dx
(1.79)
if f = O, and ~ a-2-~ [D2v]2 dx < C~h-2 f a2-~ lV f, 9 dx i f b = O.
(1.80)
1. Elliptic Boundary Value Problems and FEM
30
Proof: To prove (1.79) for f = 0, we note _<
d~ + c~ s ( ~ IV,,I ~ + ~ - ~ d ) &
-< I1" (0"1+22v)
+
+
Generally, the identity p lVgl ~ d ~ - G
V g . V (pg) d , -
(Ap)
(1.81)
dx
holds for g E H~(D) and p e C2(~). We have o/n o-~ IW, I~ d.
= s v,,. v (~,~) d. + E1 s (AG'Y, v2 dm =
<
(v. b)~
Cs
d. + ~
2
(zX~') ~ d~
dz+C, fna'-2v2 dz
by Schwarz' inequality and (1.72). Therefore, (1.82) follows. Here, the second term of the right-hand side is estimated from above by H61der's inequality and (1.76). We get 2--[
o-~-2.,, ' d .
_<
9
o- - ~ - '
-v(2-'-t) 2--gTT-
Ilvll"L 2_.k2 < c.,.h "Y (Ft)
d:.:
-
I1',, I:"L
~
In use of (1.78) we have
c~ Ilbll L ~
+__f_a 2
< c~ <
(~)
~=+~lbt~ dx
C.~h-~'---~ '+~"
(2+'y) 2
~-~
o.2+..r Ibl 2 4.:
Therefore,
j~ c,'r-2'r d'r < C.rh-2 s o2+'r bl 2 dz
.__2__
(a)
1.6. L p Estimate
31
holds and hence inequality (1.79) follows. We proceed to the case b = 0. We note that -
A
9
Wo ' <'- (~2) R W2' ~---~(~2)
L1---~(~)
~
is an isomorphism if 0 < 7 << 1. Because f E Hl(f~), we can apply Sobolev's inequality aS
<
C[Ifll ~
< CllVfl[
2
a-2~-~ dx
<_ C
a 2-~ IVfl 2 dx
~2-~[V/i 2 dx
< C~h-~ Then, inequality (1.80) follows as
-< c~h-2 fa ~2-~ tvfl 2 dz. The proof is complete.
[]
We are ready to give the following. T h e o r e m 1.14. Inequality (1.60) holds if s = - A , {Th} satisfies the inverse assumption.
f~ C R 2 is a convex polygon, and
Proof: For the moment we write a = O/Oxi, where i = 1, 2. Let Fh C f~ be the union of all faces of the triangles in Th, and Xo E f~ \ Fh. There exists T C ~-h with x0 E T ~ Then we take ~xo C C ~ ( T ) satisfying 5xo _> 0,
fT~O dx = 1,
[Ok~xol<_Ckh-2-k
(1.83)
for k = 0, 1, 2 , . . - , where Ck is a constant independent of z0 and h. We have
O)(,h(Xo) = (OXh,~xo)
()(.h E Vh)
(1.84)
because OXh is constant on T. On the other hand there is gxo E V satisfying
(Vg~o, Vv)
=
@~o, 0v)
(v E V)
(1.85)
by Riesz' representation theorem. It is nothing but the regularization of the 0 derivative of the Green's function rfor - A .
32
1. E l l i p t i c B o u n d a r y Value P r o b l e m s a n d F E M
By (1.84) and (1.85) we get 0 ( R ~ ) (~o)
=
(o ( R ~ ) , ~ o ) = ( v R ~ , Vg~o)
=
( w . v R , gxo) = ( w . Vg~o) + ( w . v (R~ - I)g~o)
The first term of the right-hand side of the above equality is estimated by (1.83)"
<_ IIOulIL,(T) " c A -~ TI ~ = Ch-~ IIO~ IL,(T) " The second term is estimated in use of crxo = Crxo,h. We have
I(W. v (n. - I)~xo)l
~-o~-~ I W ' d~
__
~--2 9
~-o~-~ d~
(f~ a~o2+-YIV (Rh -- I ) gxo 12d x )1/2 <
C~h-
~,(p-2) (~ Mh
~ o ~ ~ IV~l" dx
)l/p
where Mh -
sup xoEfl\Fh _
Crzo
--
12
We obtain
la (Rh~)(xo)l _< Ch-~ Ila~ IL,(~) + c~ h-'(~:~---~'Mh.
,,~-o~-~ IV,t ~ d,
for Xo E T E rh. This inequality implies
and hence TETh
follows. Inequality (1.70) has been reduced to Mh < C~ h # 2 . We have the following lemma, where a stands for axo.
(1.86)
33
1.6. L;p Estimate L e m m a 1.15. If ~ ) ) 1, it holds that
L ~+~ IV (R,~- z)gl ~ d~ <_C~h~L ~+~ ID~gI~ d~
(1.87)
for any g E H2(~) n Hl(f~). Here, Lemma 1.13 is applicable by (1.85). In use of (1.83) and (1.73), we have
dz + C~h -2
_
~2+v~2 dx
< C,yh -6 f a 2+'Y dx ,~ C~h ~-2. JT This implies (1.86) by Lemma 1.15. Now we show Lemma 1.15 and complete the proof of
(1.70).
Proof of Lemma 1.15: Letting tb = a 2+~ (Rh - I) g, we have by (1.81) that L V (Rh -- I) g- V (~h - I)'~ dx = - L V (Rh -- I) g. V r dx
L
= -
1L
~ + ~ Iv (R~ - I)gl ~ d~ + ~
(Zxo ~+~) [(Rh - I) 9] ~ d~.
In use of (1.72) we get L a2+~ IV (Rh - I) g 2 dx
<_ - L v ( R h - I ) g . V ( r r h - I ) t b d x + C ~ ~ a ~ [ ( R h - Z ) g ]
12
-< -2
o~+~ Iv (R. - I) gl ~ d~ + ]
2 dx
~ - ~ - ~ IV (~h - I)r ~ d~
+C~ f ~ [(~, - Z) g]~ d~. Ja
This implies
L o-~+'Iv (R~ - i) 91~ d~ - L ~-~-~ Iv ( ~ - ,)~1~ d~+ C~L ~ [(R~- ~)g]~ dx. Here, the first term of the right-hand side is estimated by (1.75). Because
D2Xh = 0
on
T E ~-h
(1.88)
34
1. Elliptic Bounda W Value Problems and FEM
for Xh C Vh, we have
TCTh
=
o~h~-~ TCrh
s
~
al
&,
where the right-hand side is equal to
In use of a u _> ~2h9, we see that this term is dominated from above by
Taking ~ >> 1, we arrive at
a u+" V ( R h - I ) g [ 2
dx<_C,h 2
dr2+'[D2g
+C,
cr'](Rh--I)g
dx
(1.89)
by (1.88). We show that the second term of the right-hand side of (1.89) is absorbed into the left-hand side by the duality argument. We take __ (__/~)-1 [0.3, (/~h -- I) g] and apply the latter part of Lemma 1.13. This gives
f ~-~-, ID~l~ d. X
_< c., (,~h)-" s ~
IV (o-, (~,,- z)~)1" dx
_< c., (,~h) -~/~ (o-~-, IV (~,,- I)gl ~ + o--, [(R,,- I).~]2) d,. Here, (1.75), (1.88), and (1.90)imply
/
=
<_
m
- L v ( ~ - i ) v . v ( ~ - i) 0 d,
(~
,~+" IV (R,,. - I)g[2
dx
)1/2(~'2
o- 2 - ' ~
IV (~,, - z),,1" d,
_< c.,h=.fo,~-~-, ID~vl ~ d. _< c-,,~-= f (o-~+" IV (R,,- I):jl ~ + ,-,-~[(/~,,- I)~]~) d.. an
)1/2
(1.90)
35
1.7. Asymptotic Expansion Letting n >> 1, we have
Inserting this to (1.89) and again letting ~c >> 1, we have (1.87) by (1.89). The proof is complete. []
1.7
Asymptotic Expansion
Both (1.57) and (1.71) are optimal error estimates. This suggests a delicate asymptotic behavior of uh(x) as h I 0. However, Q. Lin and others observed that some kind of uniform structures of {%} make it simpler. In this section, we describe the simplest case. In the following theorem, ~ C IR2 is a convex polygon. The family {7h} of its triangulations is supposed to be uniform in the following sense: Any T 9 % and T' 9 rh are congruent, and any 7'1 9 rh~ and T2 9 rh2 are similar. T h e o r e m 1.16. Let u = u(x) 9 C4(~) solve (1.1) with (1.2), and {Uh} be its piecewise linear finite element approzimations. Then, with some e 9 C1(f2) it holds that u,~(x) : ~,~u(x) + h~e(x) + O (h ~ Ilog h ~)
(1.91)
as h ~ O, uniformly on ~. Inequality (1.91) implies an a posteriori error estimate 1 (uh/2 -- II,h) + 0
II,hI2 -- 7gh?..t -- - - ~
(h 4 Ilog h I2)
uniformly on f2. Also Richardson's extrapolation guarantee's the super-convergence I1 (4uh/2 5
--
uh)
I -- 7FhU LOO(a)
0
(h 4 Ilog hi2).
The following lemma is elementary. L e m m a 1.17. There is C > 0 independent of h > 0 such that for any f satisfying flx=O,h = O, it holds that
/'
f ( x ) dx =
1-2
f " ( x ) dx + h4
/o'
with some b 9 C[O, h[ satisfying[b[ n~(o,h) <-- C. Proof: Letting g = - f " , we have
f(x) =
/0"
/~(~, y)g(y)dy
b(x)f(4)(x) dx,
9 C4[0, h]
(1.92)
1. Elliptic Boundary Value Problems and FEM
36 for
.) (0 < y < x) .(hh__y)
y(h-
K(..y) =
(z < y < h).
h Then, it holds that
1/ot~~ ( h -
~0 h f(~) dx = -2
(1.93)
~)g(x) &
Writing G ( s ) = g(sh), w e seek B E C2[0, 1] satisfying -B"(s)
= H(s) =
1
1
12
2
s(1 - s)
(0<s
and
B G ' - B'G s=o,1 - -
(1.94)
O.
Here,
fo H(s)ds = 0 and (1.94) follows from B(0) c o s a - B'(0)sin a = B(1)cos/3 + B'(1) sin/3 = 0 with some a,/3 C [0, 2re). Therefore, there is a solution B(s) with the estimate IIBlloo _< C. Then, b(x)= B ( x / h ) satisfies h2
1
-h4b"(x) = 1---2- -~x(h- x)
(0 < .r. < h)
with it
[ b o ' - b'~]o = o
In use of (1.93) we have
f(x) dx+T
f"(x) dx
= =
x(h-x)h.~
= h4 The proof is complete.
h 2 g(x) dx
~ " ( ~ ) v ( ~ ) d~ = h ~
t,(~),4'(~) d .
Jo"
b(x)f(4)(x) dz. []
1.7. Asymptotic Expansion Proof of Theorem 1.16:
37 First, we introduce the discrete Green's function g)o c Vh by
x~(*0) = (Vg~o, Vxh) for
Xh E Vh and
Xo E Q. Because
(~hPutting
Vh = g,ho E Vh, we
Uh = Rhu,
.,,~)(*0)
= =
we get
(v ( ~ - ~.~), vg~0) (v ( ~ - ~.~), Vg'x~o).
have f
TErh
TErh
T
where D2vh = 0 on T E rh is m a d e use of. There exists a triangle K C R 2 such t h a t any face of any T E % is parallel to some faces of K. Let t~i (i = 1, 2, 3) be the three different unit tangential vectors of OK with t~l, ~2, and t~a form a triangle in counter-clockwise. For T E %, ti = ti(T) denotes the unit tangential vector on 0 T parallel to g~, with tl, t2, and ta are in counter-clockwise. We have either t~i = ti or gi = - t i , exclusively for i = 1, 2, 3. Therefore, the value p ( T ) = +1 is defined by # ( T ) = t~i.ti for i = 1, 2, 3. T h e face o f T parallel to ti is denoted by L~ = L~(T), and n~ = n~(T) stands for the outer unit normal vector of T on L~. If h and i = 1, 2, 3 are given, the length of Li(T) is independent of T E % and is denoted by hi. Finally we write ai = O/cggi and Di = O/Oti. Under those notations, it holds t h a t 3
I(vh) = E i:1
E
Ii(T)
TETh
with
I~(T) = fL i(T)
Here,
OVh/ani(T)
( u - 7chu) Oni(T) Ovh dS.
is a constant. Applying L e m m a 1.17, we have
h~ ~ D~u OVh I~(T) = --i-2 ,(w) an~(T) dS + h4ji(T) with
Ji(T) = fL~(T) bir(s)Dnu~,O'v~,on~kl)dS, where
b~(s)
is a continuous function satisfying
IIb~Ll~ -< c
1. Elliptic Boundary Value Problems and FEM
38 Let
L{(T) = L{(T') for some T, T'
E Th
with T -r T'. Generally,
Ovh
8vh
On~(T) :/O,~(T') there, but D,vh is continuous and For each T E Th, it holds t h a t
b~(s) : biT,(S).
Ttl = -hlh2 --~a
(t 1" t2t 1 - - t2)
with a = IT], so t h a t we get
II
) = h 4j 1 T( ) _
tt.t2
D~u'OlvhdS-ffL,(T) D21u'O2vhdS ) 9 l(Z)
An elementary calculation gives 2a
nit3 = hlh3'
n2t3 =
2a h2h3'
and
rt3t3 = O.
We have
D~u . CVh dS + ~
h, 1
CD'~u. 02v,, dx
2
and the identity
h3h2D
D~u . Dl vh dS
h~ ~ 2(f) D~u. D2vh dS + #(T)i-4
h4h'2h348a2/ r D:~D~u . D2vh dx
holds true. Because {Th} is uniform, wc have
hi = Cih
(i = 1,2, 3)
and
a =
aoh2
with some constants Ci > 0 (i = 1, 2, 3) and a0 > O. It holds t h a t Te',-h
~'(r) : -h ~c~C2c3[ 48a~
=
h?
c~c2q, 48a~2
.,,,
D:,D~u. D2v,, dx + C?h'4",'~eT,,']' (T)
D2D3D~u.vhdx+C~h. 4 Z J ' ( T ) . 7"E Th
The terms I2(T) and I3(T) are treated similarly. Letting
Di=Di,
if
i=i'
(mod3),
1.7. Asymptotic Expansion
39
we get 3
I(vh) = h2 ~ D4u "Vh dx + h4 E c4ji' i=1
where D 4 __
3
1
48ao2 E
C4Ci+lCi+2Di+lDi+2D~
i=1
and TCrh
dS.
Oni(T)
i(T)
Therefore, it holds that
(~
- ~)
(~o)
:
h~ s D~
9g~o d~
+h c4Z 3
i=1
T6Th
b~T(S)D4u 9 Og)~ dS.
The first term of the right-hand side of (1.95) is equal to
- A e = D4u
in f~
and
e= 0
(1.95)
O~(T)
i(T)
h2eh with eh = Rhe
for
on 0~.
Here, we have
Ileh ~IIL~(~) --< C h= (llog hi + -
1)IID4UIIL~(a)
by (1.57). To handle with the second term, we suppose xo ~ Fh and recall the function (~o used in the proof of Theorem 1.14. Then, the regularized Green's function {TxoC H2(~2)N H~(f~) is defined by
We have g~ho =
Rh[Txoby
(1.84). In use of ,
E
TCTh
i(T)
OOxo b~r(s)D~uoni(T) dS
=0,
we obtain
E fL rCTh
i(T)
b~(s)D4u 9 Og~~ dS[ Oni(T)
!
TE'rh
Oni(T)
i(T)
<_ c ]IDP~IIL~(~)Z fo Iv (9~o- 0~o)l dS. TETn
T
1. Elliptic Boundary Value Problems and FEM
40 A form of the trace theorem on
B = {(x~,~)
I ~ >_ 0,
~ _> 0,
x~ + ~ < 1}
is indicated as
II/IILI(OB < C (tl/IILI(B)+ t]~7/IILI(B)) In use of the affine mapping ~ : T ---+ B of w
(/E WI'I) .
this gives
Ilfll~(r/- C (h -~ II/IIL*(T/ + I v Z ~l(r/)
(f e W~'~(T))
uniformly in h. We obtain
[Ji I <_C ]lD4UllL~(n)(h -1 ]IV (9ho- 0zo)IIL,<~) + I D29zO[lL,
The estimate
h -1 ]IV (gh0 -- ~{qxo)IILI(~) +
Ilog0xolIc,(a) _< c ([loghl 2 + 1)
(1.96)
holds uniformly in xo. Proof." Write 6 = 6xo, 9 = 9xo, and 9 h = gho for simplicity. We shall make use of the estimates obtained in the previous section with 7 -+ 0. First, note that inequality (1.82) is applicable with C7 = O(1). Letting
~(~) = (Iz-
~ol ~ + ~ h ~) 1/~
we have s
]D2~I 2 dz_<
Cs163
(1.97)
Inequality (1.87) is also applicable with C~ = O(1). We have 4~2+'YIV(gh-0)l
2
dz
(1.98)
Both estimates (1.9r) and (1.98) are uniform in z0. The first term of the right-hand side of (1.97) is estimated as follows, where z0 E T E %:
To handle with the second term, we take the Green's function gx(y) for - A with the Dirichlet boundary condition. We have
0 < 9(z) = [ ag~dy <_C(logcr(z)l + 1). Ja
1.7. Asymptotic Expansion
41
Therefore, we have the following, where ~ C R 2 and R > 0 denote a ball containing f~ and its diameter, respectively:
~ ~-b
~ d~ ~ c s ~ - ~ (flog ~ + ~) d~ _< c
~ - ~ (IXog~ + 1) d~ _< C
< ( l l o g ~ + 1) d_~.
f
P u t t i n g r ~ = s, we have
ov-2~ 2 dx
<
_
C
f0
s (3,-1 flog 812 4- 1) d8
1
7-~_<~.
C
Thus we obtain
2
h-2 11o_1+~7(gh_ ~) 11~2(Ft)+ I10-1+~D2"qlIL2(~) < C
(h1~ +)
or,
h-1 iiv (gh_ ~)IILI(~) + IID2~IILI(~) I10.-1-~ [IL2(ft){h-1 110_1+~7(gh ~])llL2(gt) -t-I101+-~D2011L2(FI)} __.~ C l[O'--2--~/llLl(a) 9 ~ -t- h "]' Here, we take 3` = [log h1-1. It holds t h a t
l
3,3
= I~oghi ~
and
h ~ = e-1
T h e n inequality (1.76) is refined as follows:
C ~0R (r2 --I- t~h2)-1-~ rdr
f(d0
~t~)-1R
:
c(~h)-~
-
C3` -xh - ~ - C l l o g h .
We have shown (1.96) and the proof is complete.
(~ + ~)-1
_~
~d~
[]
1. Elliptic Boundary Value Problems and FEM
42 Commentary
to Chapter
1
1.1. Standard monographs on the theory of real analysis and functional analysis are Hille and Phillips [179], Agmon [2], Yosida [410], Kato [205], Needs [286], Lions and Magenes [247], Mizohata [267], Adams [1], Brezis [63], Tanabe [379], and so forth. The items referred to in this chapter are well described there, such as Sobolev spaces, Poincard's inequality (1.4), Riesz's representation theorem in Hilbert spaces, interpolation inequality (1.13) of the trace operator, Lax-Milgram's theorem, m-sectorial operator, the triple of Hilbert spaces V C X C V' (sometimes called the Gel'fand triple), holomorphic semigroup, type (0, M) operator, and the closed range theorem. Concerning the fundamental properties of the numerical range of operators, see Kato [203] besides the monograph Elliptic regularity (or elliptic estimate) for the case that f~ is bounded and 0f~ is smooth, is described in Agmon [2], Lions and Magenes [247], Mizohata [267], Brezis [63], Gilbarg and Trudinger [158], and Tanabe [379]. For the domain with corners, see Ne~as [286], [287], Grisvard [164], and Dauge [107]. Role of the convexity of the domain for H 2 regularity of the solution may be taken from the following example. T h a t is, u = r ~ s i n m 0 = Re (z~ + zz2) ~ is harmonic in the sector 0 < 0 < c~ and satisfies the null Dirichlet condition on the boundary if mc~ = 7r. Here, the convexity of this sector is indicated as c~ <_ 7r, or equivalently m _> 1. This condition means u ~ E L 2 near the origin. Similarly, we have W2'P(f~) regularity when f~ is a convex polygon and 0 < p - 2 << 1. More detailed profiles of solutions on conical domains were presented in Kondrach'ev [226]. The inhomogeneous boundary value problem such as (1.8) with u =
'u0
on 0f~
(1.99)
for given u0 E Hl(f~) is formulated as to find u C Hl(f~) satisfying ' u - u0 E V = H01(f~) and .A(u, v) = (f, v) for any v C H0~(f~). All those problems are generalized to the case that s D) has complex-valued coefficients. If we replace the base space V by a convex closed subset of a Hilbert space, and take the variational problem of minimizing a proper, coercive, and lower semicontinuous functional on V, then the variational inequality arises as the Euler equation. A typical example is g={vesl(fl) l rio ~_>0} and 1
/
,~
In this case, we get the unilateral boundary value problem -A'u
+ u = f
in fl
with u_>0,
O'lt ~_>0,
071, u.~-0
onO~.
Variational inequalities can thus formulate free boundary problems. For more details, see Kinderlehrer and Stampacchia [224], Baiocchi and Capelo [35], and Friedman [130].
Commentary to Chapter 1
43
1.2. Inhomogeneous boundary value problem (1.8) with (1.99) is discretized as finding satisfying U h -- UhO E V and .A(Uh, 'Vh) --- (f, Vh) for any Vh C Vh, where Vh C V, ~rh C ~2", and Uh0 9 ~'rh are approximations of V -- H01(gt), 19 = HI(~), and u0 9 V, respectively. The proof of Theorem 1.1 is based on the idea of Agmon [2]. Estimate (1.28) holds for any self-adjoint operator, and the proof is described in standard monographs of functional analysis. Fractional powers A 0 of a non-negative self-adjoint operator A in a Hilbert space is defined through its spectral decomposition. They are extended to m-sectorial operators. Kato [202], [200] showed an inequality of Heinz' type for them. See also Tanabe [378] for this topic. The question D ( A 1/2) = V is referred to as Kato's square root problem. If A is selfadjoint, it holds obviously. T. Kato showed D ( A ~) = D(A~o) for /~ 9 [0, 1/2), where A0 denotes the symmetric part of A described in w J.L. Lions [246] showed that D ( A 1/2) = V actually holds if A is associated with the second order elliptic operator and Hl(f~) C V c Hl(f~). For the general case, however, there are counter examples. See McIntosh [263] for later developments on this problem. Uh E ~rh
1.3. Inequality (1.32) on the interpolation operator 7oh is due to Bramble and Hilbert [45], [46]. For Poincar6-Wirtinger's inequality, see the references to w on real analysis or functional analysis. The interpolation operator 7rh associated with piecewise linear trial functions has the property II~hV --
~llw~,~(a> --< C~ h~-k Ilvl w~,p<~)
for v 9 W I ' p ( a ) A W2'P(gl) and k = 0, 1 if gt C R n is a polygonal domain, p > n/2 (i.e., W2'P(a) C C(~)), and {7h} is regular. Curved elements were introduced by Ciarlet and Raviart [87], ZlSmal [418], [419]. Many works were devoted the theory of approximation such as Yamamoto and Tokuda [409], Ciarlet and Raviart [86], Ciarlet and Wagschal [89], and so forth. For details and later developments, see Ciarlet [83], [84] and the references therein. Some of the earlier studies on the finite element method were Nitsche [296], Bramble and Schatz [55], and Bramble and Nitsche [47]. See also the survey articles Babu~ka and aziz [19] and Bramble [44]. The duality argument is sometimes called Nitsche's trick. Finite element approximation by piecewise linear trial functions for second order elliptic boundary value problems is valid even for three dimensional problems because of the inclusion H2(gt) C C(~). All results presented in this chapter keep to hold in that case, unless otherwise stated. From the proof of Theorem 1.2, we see that the family of interpolation operators {Trh} has the property limh,0 llTChV-- vll V = 0 for v 9 wl,p(f~)ffl v with p > n. Concerning the method employing polynomials of higher degrees as trial functions, we refer to Ciarlet [83], [84] and Brenner and Scott [60]. The method of a posteriori error estimate was proposed by Babu~ka and Rheinboldt [21]. Note that all error estimates presented in this section are a priori. Recently, the method of p- or hp- finite elements by I. Babu~ka is attracting interest, because the development of the hardware makes it clearer that triangulating f~ is not trivial. There, the triangulation is basically fixed and the degrees of trial functions are
1. Elliptic Boundary Value Problems and FEM
44
made higher. See Schwab [346] for details. For the finite element method applied to other problems including variational inequalities, see Galligani and Magenes [156], Glowinski [161], and so forth. Approximation of non-parametric minimal surface was studied by Johnson and Thom6.e [194]. For the parametric case, Tsuchiya [389], [390], [391] showed the convergence of the discrete minimal surfaces constructed by the finite element method. 1.4. Descloux's iemma was shown in [111]. Some generalizations of Propositions 1.5 and 1.7 were given by Crouzeix and Thom~e [102]. Related works are also referred to there. For Riesz-Thorin's interpolation theorem, see Adams [1], Folland [127], and so forth. Other inequalities driven from the inverse assumption are given in Ciarlet [83]. 1.5. The L ~ stability was proven by Nitsche [297] and Scott [349]. R. Scott showed it for - A + 1 under the Neumann boundary condition. Related works were done by Natterer [285], Nitsche [298], [300], Schatz and Wahlbin [345], Rannacher [321] and Finzi Vita [125]. See also the monographs Ciarlet [83], [84], and Brenner and Scott [60]. A related topic is the local estimate of the finite element solution. Several papers were devoted to it: Nitsche and Schatz [302], Bramble and ThomSe [59], Bramble, Nitsche and Schatz [48], Nitsche [299], Bramble and Schatz [56], Schatz and Wahlbin [341], [342], [343], [344], and Thom4e [382]. We also have the survey article Wahlbin [401]. Discrete Green's function was introduced by Frehse and Rannacher [128]. We have also the following result on the optimality of (1.57). Namely, if f~ C R 2 is a disc: f2 = {x I lxl
t2
The general maximum principle of Schatz [339] replaces the right-hand side with the term multiplied by a constant C = Ca k 1. Let Zh and Bh be the sets of vertices subject to {Th} located on gt and 0t2, respectively, and let wa is the piecewise linear function on t2 for a E :Z-hU Bh defined by
w(b) --
1 (b = a) 0 (b e 1-h U B,, \ {a}).
In Ciarlet and Raviart [88], it was shown that inequality (1.100) follows if the matrix
Commentary to Chapter 1
45
A = (A(Wa, Wb))a,bEZhUl3h is of non-negative type similarly, which means that .A(wa, wa) > 0
(a E Zh U 13h) ,
A(Wa, Wb) <_ 0
(a E Zh, b E 13h) ,
and
E r bE~hU~h
Wb) ~ 0 (a E Zh).
The notion of the m a x i m u m principle was brought in the theory of matrix in this way; the strong m a x i m u m principle for the matrix A is expressed as
Au=v
with
v>0
~
u>0 m
and max uj < c max uj
jENo
(1.101)
jEN+
with c = 1, where u = (uj), v = (vj), No = { i l v i = 0}, and N+ = { i l v i > 0}. Matrices satisfying such a condition were studied by Stoyan [358]. If c = 1 is not prescribed, inequality (1.101) is referred to as the general m a x i m u m principle. It is actually equivalent to the general m a x i m u m principle of Schatz if the matrix A is associated with the finite element approximation concerning the inhomogeneous b o u n d a r y condition. The W I'p regularity for the elliptic boundary value problem was shown by Meyers [264]. The W I'p stability of Ritz operator was obtained by Rannacher and Scott [323]. For Sobolev's imbedding theorem (or Sobolev's inequality), see the standard monographs introduced in the references to w Theorem 1.8 was shown by Matsuzawa and Suzuki [262], based on the idea of Suzuki and Fujita [370]. There, the m e t h o d of H a r t m a n and Stampacchia [166] was applied. The result holds also for the N e u m a n n boundary value problem. In the three dimensional case, estimate (1.61) holds with p > 3 if {Th} satisfies the following: any face Zl of any
simplex T E 7h is an acute or right triangle and the line containing the vertex Po (~ T1 of T and perpendicular to T1 passes through T1. As is described at the end of this section, in both cases of two and three space dimensions, inequality (1.59) also holds true with p > n under the same assumptions for {Th}, and therefore, inequality (1.69) follows for q E [1, n / ( n - 1)). There, the convexity of fi is not necessary. Those facts were shown in Matsuzawa [261]. The continuous version of T h e o r e m 1.12 was obtained by Brezis and Strauss [65]. We have
Ilvllwl,~(~/~ Cq IIA~IIL,<~)
(1.~o2)
for q E [1, n / ( n - 1)) if f~ C R n is a Lipschitz domain. Actually, it is the dual form of an estimate by Stampacchia [356], where p > n: n
A -1
/0+
__ j;;~ L~(fl)
p--0
46
1. Elliptic Boundaly Value Problems and FEM
1.6. Inequality (1.70) follows similarly to Theorem 1.2. The proof is described in standard monographs such as Brenner and Scott [60] and so forth. Even if Rh is replaced by rrh, inequality (1.71) keeps to hold. Furthermore, we see that the order O(h 2) is best possible to control Ilrrhu--Ullg~(n ). Regarding such a situation, we say that (1.71) is
optimal. 1.7'. The asymptotic expansion of the approximate solution was noticed by a series of papers by Q. Lin and his co-workers: Lin and Lu [241], Lin and Tao [242], Lin and Wang [243], Lin and Zhu [245]. See also the survey article Lin and Xie [244]. Relaxations of the uniformity of {rh} were also considered there. Note that if oqf~ is smooth, the elliptic regularity guarantees u E C4'~ from f E C2'~ However, because f~ has corners, it is desirable to reduce the assumption on the smoothness of the solution in Theorem 1.16. For this point, see Blum and Rannacher [40]. For the finite difference operator, there is a close relation between the asymptotic expansion of the approximated solution and the general discrete maximum principle. For this topic, see Wasaw [403] and BShmer [42]. The idea for the proof of Lemma 1.18 is due to Blum, Lin, and Rannacher [39]. Frehse and Rannacher [128] showed (1.96) for the case that 0f~ is smooth with the right-hand side refined as O (llog hi). Asymptotic expansion for the parabolic problem was studied by Helfrich [172].
Chapter 2 Semigroup Theory and FEM
In the following three chapters, we study the discretization of time dependent equations, and the present chapter is devoted to the simplest case. Those problems are described in Thom~e [383], [384], but our analysis is based on the resolvent of operators. Similarly to the continuous case, theory of semigroups on Banach spaces establishes the wellposedness of the discretized problem. Error analysis is also made on the same framework, in use of the complex function theory for the study of approximate operators. It may be worth mentioning now that the semigroup is a family of exponential functions of an operator.
2.1
Evolutionary
Problems
In this section we describe how the abstract theory of semigroups is applied to the continuous case, initial b o u n d a r y value problems for parabolic and hyperbolic equations. Unless otherwise stated, we continue to suppose that f~ C R 2 is a convex polygon. Take the typical initial b o u n d a r y value problem, the heat equation ut - Au -- 0
in gt • (0, cx~)
(2.1)
in t2
(2.2)
with the initial condition ult= o -- Uo(X)
and the boundary condition u = 0
on Or2 x (0, cx:~).
(2.3)
This problem is uniquely solvable if the initial function u0 is not so singular. Even if u0 is not continuous, for instance, if only u0 E L2(gt), the solution u(t, z) is quite smooth for t > 0. This fact is sometimes referred to as the smoothing property, and is reflected by the semigroup of operators which we now start to describe. As is mentioned in the previous chapter, the differentiation s = - A with boundary condition (2.3) is realized as a self-adjoint operator A in X = L2(f~) satisfying
D(A) = H~(a)n H~(a). 47
2. Semigroup Theoly and F E M
48
Then, system (2.1)-(2.3) is formulated as an abstract Cauchy problem for the unknown function u = u(t)" [0, oc)-+ X. Namely, du
dt
+ Au = 0
(t > O)
(2.4)
with
~(o) = ~o.
(2.5)
If we introduce {e -tA }t_>0, the semigroup (of operators) generated by - A , then the solution of (2.4) and (2.5) is simply given by
u(t) = ~-'Auo,
(2.6)
as the general theory of E. Hille and K. Yosida guarantees. However, since the present A admits of a complete orthonormal system of eigenfunctions, we can define e -tA through the eigenfunction expansion in the following way. Let {~}~--0 be the complete orthonormal system of eigenfunctions of A associated with tile eigenvalues {A~}~ n=O, 9 A~,~ =AnOn
( ~ , ~ m ) = S n m = ~1
t0
(n=m)
(~ # ~).
Then any v E X is expanded as oo
(2.7) n=O
for ~n = (v, ~n). The element e-tAr E X may be defined by (3O
e-tAr = E
~ne-t~"~n
(2.8)
(t )_ 0).
n=O
Then e -tA is regarded as a bounded operator oil X. In fact, from (2.8) we have oo
oo
II~-~vll~(~) = ~ 1 9 , ~ - ~ 1 ~ = ~ - ~ n=-0 OO
9hi ~
n----0
E 19,i n--0
In particular, e -tA is a contraction:
t2.9/ We have e-tAcpn = e-ta"y)~ by (2.8). This implies e - t A . e - s A y -- e - ( t + s ) A y
2.1. Evolutionary Problems
49
for v = ~ n and hence for any v E X. On the other hand, e-tA[t=O =--- I (the identity operator) is obvious. Thus the family of operators {U(t)}t>_o has the semigroup property, where U(t) = e-tA:
u ( t ) . u(~)
=
u ( t + ~)
U(O)
=
I.
(t, ~ >_ o)
In view of (2.9), we can say that {e-tA}t>_o is a contraction semigroup. strong continuity of t ~-+ e-tAr, lim lie-tAr -- filL2 = 0 tl0 (n)
(V E X )
Moreover, the
'
is proven. Thus the family {e -tn}t>_o forms a strongly continuous semigroup, referred to as the (Co) semigroup. The condition for v E X of (2.7) to belong to D(A) is oo
19~,, ~ < +o~, n=0
which is equivalent to the convergence of oo
fi/nAn~n
in X.
n=0
In use of the elementary inequality
0 < /~e-tx = t-l(t.~e -t~) <_ t -1
(t,~ > 0),
we have
LAe tAv 2
L2(a)
II -
E/~nAne-tAn~n
-
n=O
12 L2 (Ft)
0(3
-- Z
oo
I~l ~ I~o ~ - ~ I~ <- ~ - ~ Z
n=O
'~l ~ = ~-~ ' ~ll ~-(~,
n=O
This means e-tAv E D(A) and
IIAe-tA Ix,x <- t-1
(t > 0).
Inequality (2.10) describes a certain smoothness in X of e-tAr. Moreover,
d
_tAr
__
d oo n=O oo
:
-- E/~n~ne-tAn~n n=O
n=O =" - - A E - t A v
(2.1o)
2. Semigroup Theory and FEM
50
follows for t > 0. This means (2.4) for u(t) = e-tAr. We have seen t h a t (2.6) holds true, i.e., e-tAu0 solves the abstract Cauchy problem (2.4) and (2.5) for any u0 E X . From (2.10) follows also that
IlA~-tAllx, ~
IIAe-~A . Ae-5 A
-
II
<_ Ae-5 A
X,X
X,X
<_ 4t -2.
T h u s e -tA carries any v C X into D(A2). Similarly, for any n E N = {1, 2, 3 , . . - } and t > 0, the o p e r a t o r A~e -tA is b o u n d e d and is subject to the estimate
11An~-~A Ix,x <- M n t - n
(t > O)
with a constant Mn > 0. Consequently, for any v E X the function
t E (0,-bOO)
H
e-tA'v E X
is differentiable any times and d"
dt----~e-tAv = (--A)ne-tAv holds. We hereby notice the smoothing property of {e -tA}t>o. T h e above argument applies to the case where b o u n d a r y condition (2.3) is replaced by N e u m a n n condition 0u
On
on cgft.
= 0
T h e general case is treated as follows. Let A" realization of
s
D(A) C X = L2(~) ~ X be the L 2
= s162 D)
o :
- E
i,j=l
)a__
~
-~z~a{j(z Oxj + E
o bJ(Z)-~x j + c(x)
j=l
with either (2.3) or
Ou One
+ au = 0
described in w It is associated with the bilinear form A ( , V = HI(f~) or V = H~(f~). We have
D(A) = ( u C H2(f~)
o,,
+a'u=0
(2.11)
) of (1.11) on V x V, where
onOf~
}
if V = Hl(f~), and
D(A) = H 2 ( f ~ ) 2 / H I ( F I ) - {u r H2(f~) [ u = 0
on Of~}
2.1. Evolutionary Problems
51
if V = H~(t2). Consider the parabolic equation
ut + s
D)u = 0
in f~ x (0, oo)
(2.12)
with initial condition (2.2) and boundary condition (2.3) or (2.11). Taking v(t) = e-tan(t) instead of u(t), we can replace A by A + AI for A E R. This allows us to assume the strong coerciveness of the bilinear form A( , ) with a > 0. After the complex extension, we have
a~ A(~, v) > a II~ll~ --
(~
V
V)
(2.13)
(~, v ~ v)
(2.14)
E
besides
I,,A('Lt,,'u)[ ~ C1 I~llv IIv tv
with a constant C1 > 0. The operator A defined in this way is m-sectorial in X. It is of type (0, M) for some 0 E (0, rr/2). Based on inequality (1.16), we can define the bounded operator e -ta by the integration (Dunford integral) for t > 0:
Here, ~ = x/-L--1 and F is the boundary of E0~ = {z 0 < [ a r g
zI <
01}
for 01 E (0, rr/2), coming in from + o c e ~~ passing through the origin, and going out to + o c e -*~ It is a kind of Cauchy's integral representation, and some deformations of F in C \ E0 are allowed without affecting integration (2.15). We can show that {e-tA}t>o forms a semigroup. Actually, strong coerciveness (2.13) implies the contraction property:
_< 1 Its smoothing property is described as follows. First, for any *to E X = L2(t2), u(t) = e-tAu0 E C([0, ec), X) solves (2.4) and (2.5). Furthermore,
IIA - lIx,x holds for c~ >_ 0 with Ms > 0.
(t > O)
(2.16)
Regarding them, {e -tA}t>O is called a holornorphic
semigroup. Here, if c~ r N, A s denotes the fractional powers of A, a densely defined closed operator on X. It is given by A s = (A-C~)-1 with the negative powers A -~ defined through
A-~ -- 27r~1/ z_~(zi _ A)_ldz,
(2.17)
2. Semigroup Theory and F E M
52
where the b r a n c h z -~ is taken so as to be real w h e n z > 0. In particular,
IIA-~IIx,x < c follows for c~ >_ 0. If 0 < c~ < 1 and v E D(A), it also holds t h a t
AO~v
__ sin rrc~/0 ~ 7f
po,- 1 (#I + A ) - 1 Av dlt #~-1 [1 - # ( # I + A) -1] v d#.
sin 7rc~
(2.~s)
71
We have
Aa e -tA =
1 f r z~e-tZ(zI - A ) - l d z
2rrz and (2.16) is a consequence of (1.16)"
IIAO~-'~II~, x
_<
1
. 2
j0 ~176 r~
2rr
1 sin(01 -- 0o)
91
_tr cos 01
r
dr
A~-le-XC~176 dA = M~t -~
t-~ 7r sin (01 - 0o) We also have
]A~ [e-tA - e - s A ] ][x,x ~ C ~ ( t - s)s -~-1
(o < s < t)
(2.19)
for c~ > - 1 . In fact,
IIA~ [~-'~ - ~-~] I1~,~
___
~
.., 1 fr'~e-SZ
[e-(t-s)z
- 1] ( z I -
A) -1 dz X,X
_<
c
<~_~co~0, ( t - ~)~& = c,,(t
~)~-~-~
r
From (2.16) follows t h a t
]lAe-'Allx,x < C t - ' < Us -1
and
I Ae -tA X,X <--
C8 -1
C o m b i n i n g those inequalities with (2.19) with c~ = 1, we have
[[A [~-,A _ ~-~] Jlx,x ~ C(t- ~)~-~-' for/3 (K [0, 1]. In t e r m s o f {g-tA } t > 0 ' t h e solution u - - u ( t )
du d--[ + Au = f(t)
(0 < s _< t)
(2.20)
of tile i n h o m o g e n e o u s equation
(0 < t <_ T)
(2.21)
2.2. Semi-discretization
53
with (2.5) is formally given by
u(t) = e-tAuo +
e-(t-s)Af (s)ds,
(2.22)
where f : [0, T] ~ X is a continuous function. This is justified, for instance, if f = f(t) is H61der continuous and u0 E X. Actually, then u = u(t) in (2.22) belongs to U E C ([0, T ] , X ) N C 1 ((0, Z ] , X ) and solves (2.21) in X. It is referred to as the strong
solution. The hyperbolic equation is also dealt with the semigroup theory. It is associated with a semigroup in X = L2(f~), strongly continuous but not holomorphic. This will be described when we discuss its approximations.
2.2
Semi-discretization
In this section we consider the finite element approximation applied to the initial boundary value problem for (2.12). Assumptions made in the previous section concerning f~ and s D) are kept. The boundary condition may be either (2.11) or (2.3), but to fix the idea we describe the latter case. The initial value u0 = Uo(X) is taken from L2(f~). Thus, in terms of the m-sectorial operator A introduced in w the problem to be approximated is written as abstract Cauchy problem (2.4) with (2.5) in X = L2(f~), which is denoted by (IVP) later. Since the solution u = u(t) is given by u(t) = e-tAuo, we may say that our aim is to study the approximation of the semigroup {e -ta }t_>0. In this section we discuss the semidiscrete finite element approximation. There, the space variable x E gt is discretized by the finite element method, while the time variable t E (0, T) is kept undiscretized. Triangulating f~, we introduce the finite dimensional space Xh as in w It is composed of elementwise linear functions vanishing on the boundary, for we are considering the case of the Dirichlet boundary condition. The family of triangulations is denoted by {7-h}h>0 , h being the largest diameter of triangles in ~-h. Always it is assumed to be regular, and the inverse assumption is introduced when it is necessary. The semidiscrete approximate solution uh is the function uh : [0, ec) ~ Xh determined by the following conditions, where Xh C Xh: d
d~ (~(t), x~) + .4 (~(t), x~)
=
(uh(O),x~,) =
0
(t > 0)
(u0, xh).
(2.23) (2.24)
In use of the operator Ah : Xh -~ Xh defined by Ah(Vh, Xh) = (A~vh, Xh)
(Xh e V,,),
relation (2.23) is written as
dub + A,~u, = 0 dt
(t > 0).
(2.25)
2. Semigroup Theory and F E M
54
In terms of the L 2 orthogonal projection Ph " X --~ Xh, on the other hand, condition (2.24) is equivalent to
~ ( 0 ) = P~0.
(2.26)
Initial value problem (2.25) and (2.26) for Uh is denoted by ( I V P ) h . uh = uh(t) of ( I V P ) h is given by
The solution
~,~(t) = ~-'~,, P,,u0.
(2.27)
For practical computation, one takes the standard basis { ~ h l a E 1-h } of Xh defined by qD2(b) = ~1
(b=a)
[o
(v e Z,, \ {at),
where 1-h denotes the set of vertices subject to ~-h locating in ft. Then, setting Uh(t) ~- E ~h(t)~a, a h aEZh
we determines ~j~(t) through the ordinary differential equation d~
-.
Mh?-/+ K,,~ = 0.
-'*
(2.2S)
a
Here, ~ = (~h I a E Z~,) is an N vector-valued unknown function with N = dim Xh. Mh and /s are N x N matrices defined as follows: Mh = (Mab(h))a,bEZh
with
Mab(h) = (qo~, q~2),
I(h = (I(ab(h))a,bEZ~
with
Kab(h) = ,,4 (qpha,~ ) .
However, representation (2.27) makes it easy to pick up some operator theoretical features of Ah on Xh uniformly in h. We have proven Theorem 1.1 in this direction. Recall that there, crucial facts follow from the consideration on the numerical range. In particular, the operator Ah in Xh is of type (0, M) with 0 E (0, 7r/2) and M _~ 1 indeI)endent of h. The operator e -tAb is well-defined on Xh. Moreover, we have e_tA,, =
1_~27rZ JfF e - t Z ( Z i h -
Ah)-ldz
(2.29)
with the path F in (2.15) and the identity oI)crator Ih on Xh. Consequently, e-tAh Ph = ~ 1
/r e -t~ (ZIh
- Ah)-i Ph dz
follows. A corollary of (2.29) is the estimate
IIAT,~-'~"ll~h,~,, _< n ~ t -~ with Ms independent of h and t > 0, where c~ > 0. The main result of this section is the following theorem. 12,cflccting the smoothing property of parabolic equations, the semidiscrete approximation admits the error estimate of the optimal order for t > 0, even if the initial function u0 is not smooth.
2.2. Semi-discretization
55
T h e o r e m 2.1. Given uo E X = L~(f~), let u and Uh be the solutions (IVP)h, respectively. Then we have
of ( I V P ) and
Ilu(t) - uh(t)llL~(a) + h Ilu(t) - uh(t)iIHl(a) _< Ch2t-llluolls~(a)
(2.30)
f o r t >O. The proof will be done in use of (2.15) and (2.29). We have
eh(t)
=-- u(t) -- Uh(t) ----
27rl f r e-tz [(zI -- A) -1 - (zIh -- Ah) -1 Phi Uo dz,
and therefore, inequality (2.30) is reduced to an estimate on the resolvents. Here we need more detailed studies on the numerical range u(A) of the bilinear form A ( , ). Recall that 0 C (0, re/2) is defined by cos0 = 5/C1 for the positive constants 5 and C1 in (2.13)and (2.14). For 01 E (0, rr/2), the sector is defined by
P~0~ = {~ I 0 _< larg zl _< 01}. L e m m a 2.2. There exists 50 > 0 such that for any v E V and z E C \ Eol , the inequality
Iz Iv ~ + Ilvl ~ _< ~0 Iz vii 2x -
x(v,
v) I
(2.31)
holds. Proof." Take 09 C (0, 0 1 ) and set cos02 = 52/C1. It holds that 0 < 52 < 5. Letting % = 5 - 5 2 > 0, we have ReA(v,v)
- % I1~11~ > > --
cos 0 2 '~ I~11~ ~os o~ 1,4(v,,,)l V
:
C 9 I"
I1,,11~
for v C V. Given ~ c u(A), we have v E V \ {0} satisfying
Call C = ((v). Inequality (2.52) reads;
for # ( v ) = Ilvll~//II~ll~. Therefore we have ~(v)E ~, where
Take z C C \ Eol. Then, an elementary calculation gives dist (z, 7-g) >
Izl sin (01 - -
02) -Jr-~0#(V) sin 02.
(2.32)
56
2. Semigroup Theory and F E M
Consequently we have
Iz Ilvll
- A(v,
v) l :
iz -
(v)l
___ Ilvll~.. di~t (z, ~)
( sin,ol o , osinO
1
2
2
-> ~o (Izl Ilvllx +lvllv)
with 6o = {rain ( s i n ( 0 1 - 0~),%sinO~)}-i > o. This means ( 2 . 3 1 ) a n d the proof is complete. [] An immediate consequence is the following. L e m m a 2.3. Given z E C \ Eol and uo E L2(~), we have
Izl Ilwll~(~) + Izl '/~ I1~1 .,(~) < ~0 Iluol ~(~)
(2.33)
and
IIw H~<~)~ C II~ollL~<~>, where w = ( z I -
(2.34)
A) -1Uo.
Proof: In use of (2.31) we have
Izl IIWllL~<~) ~ ~ + Ilwl.,<~)
< _ 60 I((zI - A ) w , w ) l = ~o I(~o, w)l < ~o luoI L=(~)'lIwllL=<~).
(2,35)
From the first term of the left-hand side of (2.35), it follows that
I:l ~ll~(~) ___~o Iluo I~(~). Applying this to the right-hand side and picking up the second term of the leR-hand side of (2.35), we obtain (2.33). Inequality (2.34) follows because A is of type (0, M)" [[w[ .2(a) _< C Awl L2(~/- c I((A - ~I) + ~I)(zI - A)-luol[L~(~)
_< Cll~o I~(~) + I~ II(zI
- A)-luollL~(a)
The proof is complete.
Cll~oll~(~) []
We are ready to give the following. Proof of Theorem 2.1" Taking z E F, we put w : (zI-
<
A) -1 u0
and
Wh = (ZIh -- Ah) -1 PhUo.
2.2. Semi-discretization
57
We have z ( ~ , ~) - A ( ~ , r
= (~o, ~)
for V)E V and z(~,
x~) - A ( ~ , , , x~) = (~o, x,,)
for Xh r Vh. Because Vh C V, we get
~(~ - ~ , ~ ) + A ( ~ - ~ , x,,) = o
(x~ 9 v,,).
(2.36)
Write rh = Wh -- W and apply (2.31) and (2.36). We have for any Xh E Vh t h a t
II*ll
I
I
=
5o I ~ ( ~ , ~
- ~) - A(~, ~
=
5o ~(~,,, ~ , , - ~ ) - A(~,,, x , , -
- ~)1 ~)1
<_ ~o ~11/~1 ~ I~(~). lzl ~/~ I ~ - ~11~(~) + ~oC~ I1~11~(~). I1~ - ~11~r 9 P u t t i n g Xh = Rhw C Vh, we have
Izl I1~11~r <_ Ch
+ I1~11~Hi(a)
(Izl 1/2 Ir,,llL~(~)"Izl ~/2 IlwllHl(a) + IluhllHl(a)"
IlwllH~(~))
by (1.43) and (1.38). In use of the inequalities in L e m m a 2.3, we get
Izlll, lL ,o, §
H'(~) <-- C
( Izl
IlrhllL~(~) + IlrnllH,(~)
) hlluollL~(~),
or
Izl
~
< Ch ~ I1~oll~
Thus we have proven
I1~11,,(~) < Chll~ollL~(~)
and
I1~11~(~) < Ch Iz1-1/2 I1~o11~(~)
uniformly in z E C \ Eol. In the operator theoretic way, they are expressed as
II(zI-
A) -~ - (zZh - A,,) -~ P,,llx,v <- c h
(2.37)
and
I I ( z I - A) - 1 respectively.
(ZIh -- Ah) -~ Phllx, X < Ch [z1-1/2 ,
(2.38)
2. Semigroup Theory and F E M
58 Inequality (2.37) gives Ileh(t)l H,(a)
<
I II
27r
--
1
-
92
2n
/o
- A) -1 - (Zlh -- Ah) -1 Phi
~ - , . c o s o , . Ch
uol[.x(~) dzl
,,,,"uoll,_~(~)d,-
Cht-~luolls~(a).
=
Thus w e have estimated the second term of the left-hand side of (2.30). To handle with the first term, we make use of the duality argument. In fact, we have Ii~hll~(n) =
sup
I(r,~,g)l
Given g E L2(f~) \ {0} and z E C \ Eo,, let
v = (zI-
A*)-lg
and
Vh = (ZIh -- AI*~)-1 S ~ g
and apply inequalities (2.37) and (2.38) and their analogies with A and Ah replaced by A* and A~, respectively. We obtain I(rh, g)l
=
Iz(n~, ~) - .A(n~, ~)l
=
Iz(r,~, v -
<
Izl Ilrhll,.,~(r~)I1~ - ~hll,_,:~(~) + el I~,,
vh)-
.A(,'h, ,_,- v,~)l
<
Ch zl '/~ Iluoll,_,:~(r~/ 9h Iz -'/~ IIg I,,~r
=
C h~
I1"~o11~(~/ gll,_,~Ir~/"
.~(~)
IIv - ~hlt_,,,(a)
Ch I1"~o11,,~(~/" h IIg ,_.,~(~)
This implies
[l(zI,~- A h ) - ~ P h - ( Z I - A ) - l l l x , x < Ch 2
(2.39)
for z C C \ E0, and (2.30) follows as
<
--
27c
e-tZ] ]l [(zI -- A) - 1 - ( z I h - Ah) -1Ph] Uo L2(f~)Idzl
< ch?t -1 Iluo I,r.,~(r~). The proof is complete.
2.3
Fractional
Powers
This section is devoted to a few remarks on tile fractional powers of Ah. We continue to suppose that the bilinear from A( , ) is strongly coercive. The at)proximate operator Ah
2.3. Fractional Powers
59
in Xh is of type (0, M) with 0 E (0, rr/2) and M >_ 1 independent of h. Recall that the negative powers A -~ and A~ ~ for a > 0 are defined through Dunford integrals: A_ ~ =
1 fr z-a(zI27r~
A)-ldz,
Ah_~ = 2rcz 1 f r z-C~(ZIh -- A h ) - l d z . We show an error estimate due to T. Ushijima for those operators. The inequality
II(P,- 1)A-~IIx,x <_ C h 2~ holds for a = 0, 1 by the fundamental property of the orthogonal projection and the elliptic estimate. Hence it is also true for c~ E [0, 1] by Heinz-Kato's theorem. Therefore, the following theorem implies
I PhA -~ _ A,-;~Phll <_ C~h ~.~ for a c (0, 1). Theorem
2.4. For each a E (0, 1), the inequality
I A-~ - A;~162
(2.40)
<- c o b ~~
holds true. Proof." We have
II(zz-
A) -~ -
(zi~ - Ah)-IphlIx.x
_< II(zz- A)-lllx,x +
II(zI.
-
C
A~)-~P~II~,x <_ zl
for z C C \ E01 , which combined with (2.39) gives
ll(~zfor/3 E [0, 1]. Divide F into F1 =
A-~-A-s
A) - 1 -
=
1 ~r l+r~ 2m
=
I+II,
we have h- 2
dO
<
Ch2Z
l~t~-e
I~ < h -~} ~ d r~ = {~ ~ r l ~1 > h-~) 9 Putting
el
{z C
(ZIh - Ah)-lP~llx.x
z -~ [(zl-
A) - 1 - ( e l h -- A h ) - l p h ] dz
2. Semigroup Theory and F E M
60 with ~ > a, and
l[III x,x <- C
_2r -a-l+B" h2ndr = C~h-2(-~+n)h 2n = C~h 2~
with/3 < a. Hence (2.40) follows.
[]
We proceed to the elliptic-Sobolev inequality. The elliptic estimate and the interpolation theory guarantee
IIA-~III,
___c /llL~ia)
for a E [0, 1]. On the other hand, some forms of Sobolev's inequality are indicated as
II/llc~(a) <_ Cpllfllw,,p(n) for p > 2 and
Ilfllw,,.(n) _< C<,,pll/llH~o(n) for ~ > 1/2 ~nd 0 < p - 2 << 1. B ~ c ~ u ~
D(A <') = H~<~(a) n Hd(r~) for ~ ~ (~/2, 3/4), it
follows that
for e > 0. We shall show that an analogous inequality for Ah holds uniformly in h. In the following theorem, convexity of f~ is not necessary. T h e o r e m 2.5. Let {Wh} satisfy the inverse assumption and Then, e > 0 admits a constant C~ > 0 satisfying
II
I
I xh L~O(a) _< C< Ate< Xh L2(n )
for Xh c Vh. Proof."
We have (1.61) and hence
IIn,,~ll~.~(~) _< C.Ilull~,,.(<,) holds for p > 2. Recall (1.24):
RhA-1 = Ahlph. Then we get
IIAz'P,
_< G
IA-'SIIw,,.(~)
aij
=
a(x)(Sij in s
( 41)
2.4. Full-discretization
61
follows for )/h E Vh. Here, the right-hand side is dominated from above by
for some c~ E (1/2, 1), so that estimate (2.40) is reduced to
for/3 E [0, 1/2). Its adjoint form is expressed as
_<
<242)
We show (2.42) and complete the proof. First, take the case that the bilinear form . 4 ( , ) is symmetric: A* = A. As is shown in w then we have D(A ~/2) = V,
C-1 f Iv ~ I[A'/2IIIx <_c Ifllv
(f E v),
and i/2 uniformly in h. Therefore, inequality (2.42) for fl = 1/2 follows from Theorem 1.5 as
IIAI: hA
X
<- c
<- llA " lIv <_
Inequality (2.42) is obvious for fl = 0. Therefore, it holds for fl E [0, 1/2] by Heinz' theorem. As for the general case, we take ,4 o = (A + ,4*)/2. Then the m-sectorial operators A0 and Aho in X and Xh associated with A ~ on V x V and Vh x Vh, respectively, satisfy
ILA 0 A ILxx <- c for fl E [0, 1/2]. On the other hand, the inequalities
IiA~A-s
< C~
and
II oA llx,x < C~
(2.43)
are proven for fl E [0,1/2) by T. Kato. Here, C a > 0 is independent o f h > 0 and the proof is complete. []
2.4
Full-discretization
In this section we consider approximation methods for (IVP), where the space variable x E f~ is discretized just as in the preceding section, and the time variable t is discretized
2. Semigroup Theory and F E M
62
by a uniform mesh: t = nT with T > 0 a n d n = 0 , 1 , 2 , - - . For the sake of simplicity, we approximate d u / d t in (2.4) by the simplest backward or forward difference quotient. Thus the approximation which we are going to discuss may be called the difference finite element approximation.
Backward
Difference Approximation
First, we deal with the following approximation with backward difference for the time variable:
U~h(t + T) -- u~(t)
+ Ahu~h(t + 7) = O
(2.44)
with u~(0)
=
S;Uo,
(2.45)
where t = n r for n = 0, 1, 2 , . . . The approximate solution u~ is a function from the discrete time t = n r into Vh. It is expressed as
u~(t) : (I,, + ~-Ah)-nP,~,,~o
(2.46)
for t = n r , although for actual computation one discretizes tile time variable in (2.28). Approximate problem ( 2 . 4 4 ) w i t h (2.45)is denoted by (IVP),~,~. Our main concern is tile rate of convergence of ut~ to u. However, we mention the stability of ( I V P ) "r,h u first T h e o r e m 2.6. The fully discrete approximation ( I V P ) T~h u with backward difference in t is absolutely stable. In fact
~ II'~01~(~) Jl,,.,,.(t)llc~(~) r holds for t = nT. This theorem follows immediately because (Ih + TAb) -1 is a contraction on Xh, which is a consequence of
R, (Ah'~,,,,v,,.)= Re A(~,,, v,,.)>_ o. The rate of convergence reflects the smoothing property of parabolic equations. have tile following. T h e o r e m 2 7. Let eh.(t) = "u(t)_. T 9
f o r t = nT.
.
T,h
"
Then we have
We
2.4. Full-discretization
63
Proof." P u t C ( 1 ) ( t ) = u ( t ) -- U h ( t ) = e - t n u o
-- e - t A h P h U o
and ~(2)(t)
=
uh(t) - u~(t) - e-tAhphuo -- (Ij~ + TAh)-nPhUo
--
ts /: Ph u o
for t = nT. We have e~(t) - s(1)(t) + c(2)(t). T h e o r e m 2.1 says t h a t
II~(1)(t)ll.(~)
_< Ch2t-1 ~o11~(~)
for t > 0. In vi@w of IIP~llx~,~ ~ 1, w@ h~v@ only to ~how that
IIK I ~,x~ - I I
+
IXh,Xh~ CTt-1
TAh) -n
(2.47)
for t : nT > O. For this purpose, we write
~ d ((I~ + ~A~)-"~ -"(~-")"~) ~
Z =
n
=
n
sA~(Ih + s A h ) - n - l e - n ( ~ - s ) A h d s 7 ^A3/2
Z
~h
( I h ~- sAh) -(n+
1)
9A h / 2 e - n ( r - s ) A h d 8
and use an inequality, the discrete analogue of (2.16):
IIA~(I~ + ~Ah)-kllx~,~ <_ C~(k~) -~,
(2.48)
where k > c~ > 0 and s > 0. This can be proven by means of the Dunford integral in the following way. In fact, letting Fk(z) = (kz)~(1 + z) -k, we have
Fk(sAh)
= - -
(ksAh) ~ (Ih + sAh) -k 2ml f r Fk(sz) (zih
-
where F denotes the p a t h of integration mentioned in w
Ah) -1 dz, For z E F we have
[1 + sz[ ~_ 1 + spcos01 for p = ]z]. If the smallest positive integer greater t h a n c~ is denoted by k0, it holds t h a t
I~ + ~zl ~ >_ 1 + k(k - 1 ) - - - ( k - k0 + 1) (spcosOx)k o k0!
2. Semigroup Theory and F E M
64
by the binomial theorem. We can choose a constant % > 0 satisfying l1 + szl -k <_ (1 + 7o(ksp)k~ - '
(z C F ) .
Because the inequality
[l(z/~-
A~)-'llx.,x ~ ~
Clz1-1
holds on z E F, we have
llFk(sAh)ll ~
C
=
C
(ksp) ~
jfo ~176
(1 + 7o(ksp)k~ -1 dp p
7]~ (1 + %r/k~ - ' dr/ 71
Here k0 > a, and the right-hand side is finite. Inequality (2.48) follows. Now we can show (2.47) as
llK/:llx~,x~ ~
T
n
o•0 /o"
s . C((n + l)s)-3/2. I. C(n(T - s))-'/2ds
<_ c ~
~ . c(~)-~/~(~(~
_-
_C
=
CT___
- ~))-'/~d~
s_1/2( r _ s)_l/2d s
n
nT
=
Cwt -1
T h e proof is complete.
Forward Difference A p p r o x i m a t i o n We proceed to the forward difference approximation and define the approximate solution
U~h(t) E ~ by
.,,,,:(t + ~) - .,;,(:)
+ Ahu,~(t) = 0
(2.49)
with
~d (o) = Shoo, where t = n~- for n = 0, 1 , 2 , - . . . by ( I V p ) F h . We have
(2.5o)
Discrete initial value problem (2.49)-(2.50) is denoted
u~(t) = (Ih - wAh)nPhUo
for t = nT. As in the usual difference approximation, the stability consideration is more crucial. In this connection we claim tile following.
2.4. Full-discretization Theorem
65
2.8. Suppose the condition (2.51)
IIAhllx~,x~ ~ 2cosO for T, h > 0 small, where 0 denotes the acute angle defined in w COS 0
5 =
C1
Then ( I V P ) ~ h is stable. Precisely we have
II~T~(t)ll~(~) _< 211~olt~,.(~)
fort
:
(2.52)
nT.
Therefore, under the inverse assumption for {~-h}, scheme ( I V P ) ~ h is stable if ~-/h 2 is sufficiently small by Theorem 1.3.
Proof of Theorem 2.8: First, we study the numerical range of Sj~ = Ih -- ~-Ah. Take Xh r Vh satisfying [IXhllL2(a)= 1. Then, we have ( s t xh, xh) = 1 - r r
with
Here, I(S/~Xh, Xh)] <-- 1 because
I~r - I~ (AhXh, Xh)l _< T IIAhllx,~,x~ ~ 2cos0 and [arg0<)[ = [arg r
0.
This means . ( s ; ) c {r e C l Ir
1}.
According to a theorem by T. Kato, the numerical range of (S/~)n keeps to stay in the unit disk for n = 1, 2 , . . . . This implies
II(S~)nlIX~,Xh ~ 2 by a fundamental property. Because
< ( t ) = (s;)nP.'.o, we have (2.52). The proof is complete. As to the rate of convergence of (IVP)~h, we have the following.
66
2. S e m i g r o u p T h e o l y a n d F E M
Theorem
2.9. A slightly m o r e s t r i n g e n t t h a n (2.51), a s s u m e l i m s u p ~-II&llx.,x. T,h~O
~,, (svP)[~
< 2cos0
(2.53)
Th~n fo~ th~ ~ o ~ e~(t) = u ( t ) - u,](t) = e - t A u o -- (Ih - " r A h ) n P h u o
we have
le,:(t)llc~(n) _
C(h 2 +
~-)t-11 uollc~(a),
where t = nm with n = 1, 2 , . . . . Proof. We again split the error as ej~(t) = c(1)(t) + e(2)(t), where
~(1)(t)
=
e - t A u o _ e-tA~ PhUo
and c(2)(t) : e-tah PhUo - (Ih - "TAh)nphuO = K~Ph'UO. It suffices to show
(2.54)
IIKT, Ixh,x,, <- o~-t-' for
t
=
n'r.
We can derive the integral representation of t(#~ as K,[
=
-
fo""-~s ( ( I h
=
n
=
n ~o r ~_A3/2 h trI t , - s A h )
lo r s A ~ ( I h -
- sAh)ne -''('-~)m")
ds
sAh)n-le-n('-s)A"ds d8. n - l " ' ' h A1/2,~-n(r-s)Ah "~
In order to estimate the integral above, we make use of (2.16) and its analogue [IAT,(/h- sAh)"ffx,,,x,< _< C~(ns) -~ to be proven below, where (~ > 0 and 0 < s _< r . Then inequality (2.54) follows as
I I~
Xh,X h
/o"
'~
/7,
S" C .
((/1,-
=
cnl/2(?),
--
1)-3/2
<
C n -1 = C,-rt -I
1)8) - 3 / 2 . C(Tt(T - - s ) ) - l / 2 d s
/o r 8-1/2(7 - __ 8)-1/'2d8
(2.55)
2.4. Full-discretization
67
We give the p r o o f of (2.55). We choose ~c > 0 a n d p > 0 satisfying
IIA~llx~,x~ (1 + ~) -- ~ < 2cosOl. This is possible because of (2.53). T h e n we i n t r o d u c e a positively oriented c o n t o u r c o m p o s e d of the following two p o r t i o n s (as sets), where R = # / s : F (1)
{re•176 I O<_r<_Fg},
=
r (~) :
{n~'~
-o, <_ o <_ o,}.
We have
_
#
I Ah xh,x, - r ( 1 + ec) < R. Therefore, p u t t i n g Fn(A) = (nA)a(1 - A) '~, we can represent F~(SAh) by the D u n f o r d integral:
Fn(SAh) = ~ 1
~r Fn(SZ)(ZIh- A h ) - l d z = ~ 1 (1(1) + i(2) ).
Here, I (j) s t a n d s for the c o n t r i b u t i o n to the integral from F (j) for j = 1, 2. Let 7 E (0, 2cos01 - #). If z = re +~~ E F (1) we have
O<_sr<# a n d hence I1 -
szl 2 = 1 +
s2r 2 - 2st cos01 <_ 1 - ' y s r
follows. This implies
IIz(1) ll~,x~
< C =
C
foo /o
( n s r ) ~ ( 1 - "ysr) ndr ~_ C r {a-le-'r{d{--
C(a1).
fo
(ns~/)ae -~ns~dr T
(2.56)
If z = RP ~ E F (2), we have
=
111(1 _ z
< -
lllx x
iZ -1
1-Izl
-~llAhllX~,x~
Because R = p / s _> p/w = (1 + t~)II&llx.,x~, this implies
II(zI~-A~)-lllx~,~
l+rc < _ R~
__
1
n-IIAhllxh,x~
2. Semigroup Theory and FEM
68 On the other hand we have
ll - szl < I1 - # e ~ < l = (1 + # 2 _ 2 # c o s 0 , ) 1 / 2
= (~2 < 1.
Those inequalities imply IIz(=)ll~,,,x, ' < c
f_o, ( , ~ R ) o < ' (1 + ec) ~ d o = o ( n ~ ) o < ' < c 2 =) 0x
- -
/~/';
- -
(2.57) '
because n~($~ ---+ 0 as n --~ oc by 0 < 62 < 1. Inequalities (2.56) and (2.57)imply (2.55). The proof is complete.
2.5
Inhomogeneous Equation
We consider the semidiscrete approximation applied to the inhomogeneous equation
du dt
--+
Au = f (t)
(0 <_ t <_ T ) ,
with the initial condition
u(0) = ~o e L ~ ( ~ ) = As is mentioned at the end of w the exponent 0 E (0, 1)"
[f]e,L~(~)
=
x.
suppose that f 9[0, T] ---+ X is H61der continuous with
f ( t ) - - f(s)llL2(a)
sup t,sE[O,T],t:/:s
t - - 8l 0
Then, we have a unique solution u E CI+~
Similarly, the approximate solution uh"
dUh + Ahuh dt ~h(0)
T] ---+ X) expressed as
[0, T] ---+ Vh defined by
= Phf(t) =
< +oo.
(0 <_ t < T)
Ph,,o
(2.59)
is expressed as
"uh(t)
=
c-tAh'ltO -Jr-
(2.58)
e-(t-s)A~phf(S)ds.
To estimate Ilu(t)- Uh(t)llL2(fl), we introduce the error operator
Eh(t) = e -tA - e -tAm Ph
2.5. h~homogeneous Equation
69
acting on X. We have
IIEh(t)llx,x ~ 2 because e -tA and e -tAb are both contractions under our assumptions. Furthermore, the estimate
(t>0)
I]Eh(t) Ix,x <-- Ch2t-1 follows from T h e o r e m 2.1. We split the error eh(t) = u ( t ) - uh(t) as
eh(t) = Eh(t)Uo + O~Ot E h ( t - 8)f(8)d8 --
C (1) (t) -t- C (2) (t) -Jr"C (3) (t),
where c(1)(t)
=
Eh(t)uo,
e(2)(t) =
E h ( t - s ) ( f ( s ) - f(t))ds,
e(a)(t)
E h ( t - s)f(t)dt.
=
We have [IC(1)(t)l[L2(n) _< Ch211uOllL2(f~)t-1,
[]C(2)(t) I[L2(ft)
~
'ot C h 2 ( t - 8) -1 [[f(s) - f(t)llL2(a ) ds
<-- Ch2 [f]O,L2(n)
(t- s)-l(t-
s)~
<- Co h2 [f]O,L~(a) tO, and furthermore, c(a)(t)
=
= = Here, recall (1.44):
/ote-(t-s)A f (t)ds _ fot e-(t-*)A,, phf (t)ds [A-le-(t-s)Af(t)] 8:ts=o--[Ah le-(t-8)&P'~f(t)] s--O s=t [Ahle-tdhph- A-le -tg] f ( t ) - [AhIPh -- A -1] f(t).
2. Semigroup Theory and F E M
70 Then
we have only to estimate
Ahle_tAhp h _ A_le_tA =
1 fF z - le-tz 27r~
[(ZIh - Ah)- 1 Ph - ( z I -
A) -1] dz
(2.60)
in the operator norm. Here F = F (1) U F (2) is a positively oriented contour obtained by deforming the original one:
P (2)
{re-t-iO11 t -1 ~ r < -~-(x)}.
--
Let j = 1, 2 and I(J) stand for the contributions to the integral in (2.60) from F (j). We get
II/(')I x,x
< --
1
"2
/0 'r te_cos
Ch2. 1
1
d~s =
Ch 2
t
by (2.39). On the other hand, we have
IIz<2>l
x x
--< 27I"1. 2 ~t ~_1!c-trc~ :
Ch 2 foe J1
Ch2 dr
!e_pcosO ' d p = Ch 2. P
Summing up them, we get the following. T h e o r e m 2.10. As to the error eh(t) = u ( t ) - Uh(t) committed by the semidiscrete finite element approximation (2.58)(2.59) for inhomogeneous problem, we have
II~h(t)llL~(~)_
to
f o r t >O.
2.6
Higher Accuracy
As is described in the commentary for w1.3, one may use more sophisticated finite element spaces for Vh and try to obtain approximate solutions with higher accuracy. In order to have a look into this direction, we again consider (IVP):
du +Au=0 dt
(t>0)
with
u(0) = u 0
We study its semidiscrete approximation ( I V P ) h
dub d---t + AhUh = 0
(t > 0)
with
Uh(0) = Phu0,
2.6. Higher Accuracy
71
but assume t h a t the finite-dimensional subspace Vh has the following properties: There exists an integer s _> 2 such that if v C HS(ft) A V, then for some vh in Vh we have
IIv - vh Ig~(a) < ChS-ll vllg~-~(n)
(2.61)
IIv - v~IIH,(~) <_ ChS-1llvl HS(~),
(2.62)
and
where C is independent of v and h. If s > 2, we suppose that conditions (2.61) and (2.62) with s replaced by 2 hold true as well. Confirm that all assumptions on A and A( , ) remain unchanged. Then we have the following. Theorem
2.11. The inequality
Ilu(t) - uh(t)llL~(a) _< C
h
Ilu01L~(~)
holds for u0 E L2(gt), where t > O. Let
~(t)
= u(t) - ,,~(t) = ~-~A~0 - ~-~P~u0.
The first step for the proof is the following. L e m m a 2.12. Take z satisfying X = L2(f~). Then, the inequality
01
~
]arg z] < 7r, and put w = ( z I -
A ) - l f for f E
s-2
Izl ~/~ IlwllH~-~(a) < c y ~ Izl ~/~ [fl HS-2-k(a) k=O
(2.63)
k=O
holds. Proof: Introduce the operator B = A + e z01 02 c~y2
(2.64)
in L2(Q) for Q = f~ x R 1, which is verified to be uniformly elliptic. Letting r = V/]Z, we set v(x, y ) = ~(y)eZrYw(x) with ~ E C ~ ( R ) satisfying
r
1 (lyl-< 1/2) o (lyl->~).
The elliptic estimate guarantees __
s-
L2(Q)
9
2. Semigroup Theory and F E M
72 Here, we have
ii~ll 2Hs(Q) -lal+k_<s
fl/2 lal+k<s J
[2
I(~r)~e '~y ds. IID~wll~,~(~) 89
lal+k_<s
a=o
On the other hand, we have B y = ~(y)s
(2zr~'(y) + ("(y))w
f ( x ) + e '~
with s-2
II~(y)e'Wf(x)lI~-2(Q) <- c E ( 1
+ r) 2k Ilf 2
k=0
and s-2 __ C ~~(1 + r)2(k+l) I wll 2 Hs-2-k ~Hs-=(Q) <
1l(2~('(y) + r
(f~) 9
k=0
Because
I VIIL~tQ) <-- 2 Ilwl L2(n) --< 2]zwllL2tn) <_ C IlfllL2(n), if Iz] _> 1, then we have S--2
r=~ Ilwl 2H,_k(n) --< C E ( 1 + r) 2k k=O
fll~,-=-~(~)
k=O s-2
+ C E(1
+ r)2(k+l) ] w I H,,-2- k (a) + C ]l/ll 2L 2 ( U )
9
k=O
We get (2.63) for r = ]z] 1/2 sufficiently large and this proves the lemma because it is obvious for ]z] small. [] In the second step, we recall (2.31)"
iz iivll 2
2
I
2
Making use of (2.61), (2.62), and (2.63), we get the following.
I
(v c V).
73
2.6. Higher A c c u r a c y L e m m a 2.13. If f E Hs-2(Q), we have
8--2 IlrhllL~(a) ~ Ch ~ ~
Izl k/2 IlfllH~-~-~(~),
(2.65)
k=0
A) -1 - (zIh - Ah) -1 p h f .
where rn = ( z I -
Proof: We have
Izl I ~,~11~L~(~) + II~hllH~(~) ~ < ~ lz
5 I(Z~, ~ -- ~h) -- A ( ~ , ~ -- ~ ) l
< (Izl II~ll~(~)IIw - w,~ll~(~) + c [[rhllHl(a)Ilw- WhilH,(a)), where Wh C Vh satisfies (2.61) and (2.62) with v and Vh replaced by w and Wh, respectively. This gives that
_< c x / ~
hS-l~[ IlwllH.-l(~)
I1~11~(~) 9
+ c l l ~ l l . l ( a ) h ~-~ llwll/.(a)
ca 8-1 (V/~zI I[rh[[L2(f~) -~- [[fh[IHl(f~)) " ( ~ / ~ [[W[IHs-l(fl) -Jr-[IW[[Hs([-I)) . In use of L e m m a 2.12, we obtain
v'~ IlrhllL~(a) + Ilrhll/-/l(a)
8--2
<
ChS-1~ Izl k/2 IlfllH.-~(a) 9
(2.66)
k:0
Inequality (2.66) implies (2.65) by Nitsche's trick and (2.61) and (2.62). The proof goes in the same way as in w and is left to the reader. [] By means of (2.65) we can estimate the error eh(t) = u(t) - Uh(t) : ~ 1 /~ e_tZ [(Z_[- A) -1 -- ( Z I h - Ah) -1Ph] Uo dz.
We obtain s-2
tic(t)-
~(t)ll~(~) <__ChSt -~ ~-~t ~ - ~ I1~011.~(~) k=0
for u0 ~ H ~ - 2 ( a ) . To relax the assumption u0 E HS-2(~), we follow the idea of H.P. Helfrich. Proof of Theorem 2.11:
Introducing the error operator
Eh(t) = e -tA -- e -tAb Ph,
(2.67)
2. Semigroup Theory and FEM
74 we have
Eh(t) = -Eh(t/2)Eh(t/2) + Eh(t/2)e -~A + e-~AEh(t/2) by the semigroup property. First, applying (2.67), we get s-2
<_ Ch2(t/2) -1
~2,11 e-~A'uo ' LI.e(a)
(t/2)---vf=O
A standard elliptic estimate and the smoothing property of {e -tA }t>o now imply
s--2
<_ ChSt -1
L2(n)
-rA~e-~Auo II S t - -"~'(ll'' g=O
II'
+ e-~Auo
L2(n)
s-2
z ~-~-,( ~
IIL2(~))
)
g=0
s-2
<_ Ch-~t -~
[[UO[[L2(a ) -71-ChS
E t-~
[['ttO[[L2(a ) .
g=O
Thus
we
have
IIEh(t/2)e-~All
X,X
( O < t _< 1).
<_ ChSt-~
(2.68)
Next, taking the adjoint operator, we see that
holds with
e(t/2)" = e-~
- e-5
Ph.
Here, replacing A by A*, we have
similarly to (2.68).
Finally, the results in w
are summarized as IlE,,(t)llx,x
This gives that
IIE~(t)llx,~ _< c
(0
X,X
rain (1,
<
Cmin
h,2 ) I' V "
~)IIEh(t/2)llx,x+rain (1, (--~t) ~)]
(2.69)
(2.70)
2.6. Higher Accuracy
75
for 0 < t < 1. Inequalities (2.69) and (2.70) imply
I [ E h ( t ) l l x , x < - C m i n { 1, (--~~t) s} and establish the theorem. As an example of the fully discrete approximation with higher accuracy, we consider the Crank-Nicolson scheme which defines the approximate solution u~(t) E Vh by
U~h(t + T) -- U~h(t)
+-~1AhU~h(t-+ T) + -~1 Ahu,:(t)
= 0
with u~(0) = Phuo. It is represented as
U~h(t)= ( I h - - ~3-Ah) ~ (Ih + 1
-~TA h) PhUO -~
for t = nr. A condition for the stability is limsup 7-IIA,
Ix~,x~ < + ~ .
(2.71)
r, hiO
This is met if a kind of the inverse assumption assures for 7-/h 2 to stay bounded. Now, assume that Vh is subject to (2.61) and (2.62) with s = 4. Then we have the following. T h e o r e m 2.14. Under the assumptions stated above, the error Ch = u(t) - U~h(t) corn-
mitted by the fully discrete Crank-Nicolson approximation is established as
II~h(t) I1~(~) _<
C h4 + 72
t~
where t = nT- > O. Similarly to Theorem 2.9, the proof is reduced to estimation of the corresponding Dunford integrals. This part is left to the reader, and we only show the following. L e m m a 2.15. Under the assumption (2.71), we have
h
Ih + q Ah
<_ C(~q) -~/~
(2.72)
Xh,Xh
and I[Ah c~/2~-~(~'-q)AhIIXh,Xh <_ (n(T _ q))-~/2 for n E N and O < q <_ T.
(2.73)
76
2. Semigroup T h e o w and F E M Proof:
Let r IIAhlly~,x~ ~ Mo < + ~ .
= 1-
1-r 1+r
I1+r
( _< exp
ICI ~ Mo, we have
Given r = r e~~ with
-1+
41r (11+r + 1 1 - r 2cos01
)
2MocosOa + M 3 r
9
(2.74)
We introduce the positively oriented closed curve F q/2 for 0 < q _< r enclosing the spectrum of Ah. Namely, F q/2 = F~/2 U F~/2 as a set, where _
q
qMos p
=
[p <_01
.
We have
(T~qAh)5/2(Ih--~Ah) I~ + ~A,~ n
l(fF
=
2m
=
Io+I1.
~/~ +
fF)~/~
(--:l 11+~~ n ( z I h -
(nqz)5/2
A~) - l d z
M~king the change of variable by putting r = (q/2)z, we have 1 (2nr I~ = 2~--~m ~
1 -r 1+(
2r q
Ah
- de q
and hence
II1o x~,x~
< c -
j;
12nr
-~'r162
~
< C r
/0
r a / 2 e - ~ d r < +oo
.
follows from (2.74). Similarly, we have Ix =
(2ni)5/2
1- r
]
iIh - At~
1+r
- dr q
with 2 ~Ih -- Ah
<-- 114o
Xh,Xh
and 1-C 1+(
<exp(-2M~176 ( 1 + Mo) 2
for r C F~. We obtain (2.72) as
,,ii,[Xh,Xh< 201.25/2SUp{p5/2exp ( . -
The proof of (2.73) is similar.
~>o
. . . )} 2pcos01 (1 + Mo) ~
q 2 Mo
.
q
[]
2.7. L ~ Estimate 2.7
77
L ~176E s t i m a t e
We consider heat equation (2.1) with boundary condition (2.2) and initial condition (2.3), where f~ C R 2 denotes a convex polygon: ut-
in f~ • (0, oc),
Au = 0
on ~,
~1~=0 = ~ 0 ( ~ )
u = 0
on cOf~ x (0, oc).
Taking X = L2(a), V = H~(a), and
A(u, v) = ~ V u . V v dz, a self-adjoint operator A in X is associated with ,4( , ) on V x V. Above problem is reduced to Cauchy problem (2.4) with (2.5) in X, denoted by (IVP):
du d--[ + Au = 0
(t > 0)
with
u(0) = u0
Let Vh C V be the finite-dimensional space introduced in ~1.3 and let Xh be the space Vh with the inner product induced from X. We suppose the inverse assumption for the family {rh} of triangulations of 12. The semidiscrete approximation of ( I V P ) is given by
duh + Ahuh = 0 dt
(t > 0)
with
uh(0) = Phuo,
where Ah denotes the self-adjoint operator in Xh associated with AIvh• and Ph : X Xh the orthogonal projection. Let {e -tA }t>0 and {e -tAb }t_>0 be the semigroups generated by A and Ah in X and Xh, respectively. Then,
Eh(t) = e -tA - e-tahph denotes the error operator. We show t h a t L ~162 stability of the approximate solution holds in the following sense. Theorem
2.16.
We have
for Xh E Vh and t > O. For the proof, we introduce the discrete delta function (5 = d~o,h E Vh and the discrete fundamental solution F = r(t) = r~o,~(t) ~ vh:
(~o,~, x~) = r~o,h(t)
=
x~(~o) (x~ e vh) e-~A~o,h.
Given Xh E Vh and x0 C f~, we have (r~o,~(t), x~) = (e-~a~5~o,~, xh) = (hxo,h, e-~A~Xh) = (e-~A~Xh) (ZO)
2. Semigroup Theory and FEM
78 so that
[le--tAhxh LOO(n)<_sup IIr~o,~(t)ll~, IIx~l~(~) xoEgt
follows. Inequality (2.75) is reduced to
~up
t_>O, xoE~
IIr~o,,,(t)ll~, _< C I ] o g h l .
Take the weight h m c t i o n introduced in w
with t~ = 1:
: ~o,h(~) = (1~- ~ol ~ § h ~) 1/2 Similarly to (1.76) we have
I1~-111*L2(a) =
L cr-2dx < C ~ooC rerdr + h2
-
-
c
logh,
and hence
IIr=o,h(t) L, ~
I1~-'11.(~)
9I~=o,hr=o,h(t)llL~(~)
<_ Clloghll/21~o,hrxo,h(t)llc~(n) follows. We have only to show that
sup
IIC~o,hr~o,h(t)llc~(n) <_ Clloghl 1/2
(2.76)
xoEFt, t > 0
We have assumed the inverse assumption and hence T h e o r e m 1.7 is applicable. particular, ]IPhIIL~(~),L~(~) <_C follows. Furthermore, we have the following. Lemma
2.17.
The estimate
holds for Xh E Vh. Proof:
Let ~ = 7Ch(crexh) and T C rh. We have D~Xh = 0 on T if [c~] = 2 and hence
+ h I1~ (~ ~,~
~)ll.(.
~
ch~
_<
c h ~ (llx,,I ~ ( ~ ) +
Ic~1_<2
II D~
II(,Vxhll~(~/)
follows. This implies
II ~
+ h IIv (~
- ~) I ~(~) -< c~ (~,~11~(~) + I1~~11.(~))
9
In
2.7. L ~ Estimate
79
On the other hand, we have
Ilv (~- P~(~))I1.(~) < ch-~ I1~- P . ( ~ . ) I1~(~) Estimate (2.77) is a consequence of those two inequalities. The weight function a = Crxo,h compensates the singularity of the discrete delta function (~ = (~xo,h. We have the following. Lemma
2.18.
The inequality
(2.78)
II~o,h~o,hllL~(~) ~ c holds true. Proof: Fix Xo E ft and put a = Crxo,h and 5 = 5xo,h. Let
for j >_ 1 and
~0={~e~l
I~-~01
Because or(x) <_ Ch2J for x E t2j, we have
j>_0 To estimate
II~llL~(a~, we take
7) E C~~
If x0 E T 9 Th, the inequality
l( (~, ~)l = l( (~, rhqo)l = I(rh99)(xo)l <- Ch-1
IIPh~IIL2(T)
follows from the inverse assumption. Here, L e m m a 1.6 is applicable and the right-hand side is estimated from above by C h -1 exp ( - C 2 j)
wll~(a,)-
Therefore,
11511L~(a,)--
exp ( - C 2 j)
holds and (2.78) is proven. The following lemma may be regarded as a discrete Sobolev's inequality with penalty term: Lemma
2.19.
We have
Ix:hi ~
(2.79)
2. Semigroup Theory and F E M
80
Proof." Let Xh E Vh. A form of Sobolev's inequality is indicated as
IlxhlIL~(~) < CP 1/2 IlVxhlIL~(~),
(2.80)
where C > 0 is independent ofp > 2. On the other hand, inequality (1.45) is a consequence of the inverse assumption:
[Ix,, [L~(~) < Ch-2/P [IX,,IIL~(n) P u t t i n g p = ]log hi, we obtain (2.79).
[]
We are ready to give the following.
Proof of Theorem 2.16: We show (2.76). First, we have 1d
2
2
= (OtF, o2F) + (VF, V(cr2F)) - 2 (VP crVVcr)
Because
(o~r, >) + (vr, v~) = 0
(~ e v,,),
we have ld
2
2
where
= ( o , r . . ~ r - ~).
n = (vr, v(~r
- ~)),
and I I I = - 2 (VP, a F V a ) .
Let ~b = Ph(a2F) E Vh. Then, I = 0 follows first. By means of tile inverse assumption and L e m m a 2.17, we next get
_< ch-' Ilrllg~(~)h (llrllg~(~) + II~rllg~(~))
Finally,
by I[Vcr[]L~(n) < C. Summing up those inequalities, we obtain d -
-
2
2
2
F[ t,'(a)
81
2. Z L ~ E s t i m a t e
This implies liter(t)
2 IIL~(~)
+
/0 II~Vr(~)ll ~
/0
2
2
and inequality (2.76) is reduced to
~ I r(~) IIL:(~) ds < c laog h
Jj
(2.81)
by Lemma 2.18. Because F satisfies A;IOtF + F = 0
(t > 0),
we have 1 d (AhlF, F ) + IIVll ~
2 dt
L2(a) = O.
Regarding r(O) = 6xo,h, we get 1 (A;1F(t),F(t)) + f0 t
IIr(,)ll~(~)ds
= ~1 (Ahl~zo,h)(X0).
Let
Gxo,~ -- A ~ 16~~,h. Because Ah 1 is positive definite, inequality (2.81) is reduced to Gxo,h(xO) <_ CI log hl.
(2.82)
Here, we note that G~o,h E Vh is nothing but the discrete Green's function: (VG~o,h, VXh) = ( V A - ; ~ o , h , VXh) = (~o,h, Xh) = Xh(X0) In particular, G~o,h(xo ) =
IIVGxo,hll~(~). From bemma
Gxo,h(xO) ~ C[ log h[ 1/2
(Xh e V~).
2.19 follows that
IVG~o,~ll~(~) - c (t log hi. Gxo,h(X)) 1/2
This implies (2.82). The proof is complete.
[]
Some error estimates follow from the stability Theorem 2.16 and Nitsche-Scott's inequality (1.57):
For instance we have the following.
82
2. Semigroup Z h e o l y and F E M
Theorem
2.20.
The inequality
Ilu(t) - uh(t)llc~(n)
_
Ch21log hi 2
9
{ sup Ilu(t)llw~,~(a) + (0
/o'Ilu,(s)l w~,~(a)ds}
(2.83)
holds. Proof:
Recall the error operator Eh(t) = e -tA -- e -tAb Ph.
In use of (1.24), we have O Or {e-(t --r)Ah PhEh(r) }
=
e -('-r)m'' [mhPh -- Phm] e -rd
=
e-(t-r)m"AhPh [I -- Rh] e -rm,
if 0 < r < t, and hence
PhEh(t) =
1'
e-(t-r)A"AhPh [I -- Rh] e-rAdr
follows. In particular we have
Eh(t)
= --
[ I - P,,] Eh(t) + PhEh(t) ( I - e-'A"Ph) [ R , , - I] e -tA
+
/o'
e-(t-")nhAhPh[Rh
-- I][e -rA -- e -tA] dr.
Integrating by parts, we obtain
-
/o
e-(t-")AhAhPh [ R , , -
I1 [e -'A - e -'A] dr r=0 +
j~Ot e-(t
Ph [Rh -- I] Ae-rAdr.
We have 3
Eh(t) = Z
~.=I
E#(t)
with
E~(t) = --c-'A"Ph[,e,,- ZJ, and t
E,{(t) =
fO
e -(t-~)A'' ~ [ R h - I]Ae-~Adr.
2.8. Hyperbolic Equation
83
From Nitche-Scott's inequality (1.57) follows that
On the other hand, Theorem 2.16 implies
IE~(t)~o I~(~)
<
ch~l lo~ hi ~ II'~0(t)l ..~,~(~),
IIE~(t)uollc~o(a)
< Ch21 loghl 2
Ilut(r)llw2,~(a ) dr.
Inequality (2.83) is proven.
2.8
Hyperbolic
Equation
This section is devoted to the hyperbolic equation
utt + s
D)u = 0
in f~ x [0, T]
with the initial conditions
~1,=0 = ~ (x)
and
utlt=o = uo(x)
in f~
and the boundary condition either u=0
o n c g f ~ x [ 0 , T]
or
cgu c9~
+ au = 0
on cgf~ x [0, T].
It is reduced to the evolution equation of second order in X = L2(f~),
+ Au = 0
(0 <_ t <_ T)
(2.84)
with
where/L = du/dt and ~ = d2u/dt 2. The finite element approximation is introduced in a natural manner. We shall develop an error analysis on the semidiscrete approximation, assuming that A is self-adjoint. Take the triple of Hilbert spaces V C X C V ~ given in w The bilinear form A = A( , ) on V x V is supposed to satisfy A* = A,
IA(u, v)l <_ C lullv Ilvl v,
and
A(~, ~) > ~ II~II~
2. Semigroup Theory and F E M
84
where C > 0 and 5 > 0 are constants. Then the associated operator A in X with .4( , ) on V x V becomes self-adjoint. Given Ul E V and uo c X, we consider evolution equation (2.84) with (2.85). It is transformed into a system of equations of first order. Let X =
X
and
(o i)
.4 =
A
0
"
Then, equation (2.84)with (2.85)is reduced to
dU d----[+ .4U = 0
(2.86)
(0 <_ t <_ T)
in X with
u(o) =Uo,
(2.87)
where U: As is described in w
(u)/L
and
U0:
( u' )u0 "
the relation D (A :/2) = V and the equivalence
I vllv ~ IlA1/~vll,,
(v ~ x)
(2.88)
follow from ,4* = .4. Therefore, norm in A' can be provided with
[[Ul[x=lluo[x+ IlA1/2Ul[[x
for U =
(ui).u0
This makes for .4 to be skew-adjoint, and consequently, to generate a (Co) group
{T(t)}te R in X by Hille-Yosida's theorem. In fact, z.4 in X is associated with the symmetric bilinear form B = B( , ) on l; x V, where V
and
~(u, w) =~ {-A(,,,,o, w,) + A(,**,,wo)} for V,o
ail(|
Therefore, if
Uo c D(A) -
D(A) ) V '
2.8. Hyperbolic Equation
85
the function U(t) = T(t)Uo is C (2.86) with (2.87). Furthermore,
1
in X on [0, T], takes the value in D(A), and satisfies
(0 < t <_ T)
IT(t) Ix,x < 1
follows. Let {Vh [ h > 0} be a family of finitedimensional subspaces of V introduced in w As before, Xh denotes the space Vh equipped with the topology induced from X, Ph " X --+ Xh the orthogonM projection, and Rh " V --~ Vh the Ritz projection. Let Ah be the m-sectoriM operator in Xh associated with A VhxVh" Then the semidiscrete approximation of (2.84) with (2.85) is given as
~,~ + AhUh = 0
(0 <_ t <_ T)
(2.89)
in Xh with Uh(O)
/Lh(O)= PhUo.
and
= ./~hUl
(2.90)
W e shall derive certain estimates on the error eh(t) = u(t)--Uh(t), noting the equivalence 1/2
I Xh v "' Ah Xhlix
(X,~ r Vh)
uniformly in h. Recall that
(v e D(Ak/2)) holds for k = 1,2 and j = 0, 1.
Helfrich's Method
In the same way as in (2.84) and (2.85), equation (2.89) with (2.90) is reduced to
dUb + AhUh = 0 dt
(0 < t < T)
(2.91)
in A'h with
Uh(O) = Uoh,
(2.92)
where
Xh= and
U~=
X~
(uh) i~h
0 Ah
-I) 0
'
Ah =
'
Gob -- ( RhUl Phuo ) "
2. Semigroup Theory and FEM
86
The operator - A h also generates a (Co) group {Th(t)}te ~ in A'h, and the solution Uh = Uh(t) of (2.91) with (2.92)is given by
u,,(t) = Th(t)Uoh. The inequality
ITh(t) I~.,~. _< 1
(2.93)
holds. Let
,t 01t -- ,iA uolix + ILA,+ IlIx for ~ = 0, 1/2, 1, where
=
Then
we
have
II .11o = II .11~
CXe--
Uo
D (A e)
.
and
IIT(t)Uolle <~ C IIUo
~.
We can trace the computations in the proof of Theorem 2.20. In fact,
0) 0
Ph
"X--,&
is the orthogonal projection, and oh(t) = T(t)
- Th(t)72h
denotes the error operator. First, we have
I [ Z - 7~,,]ch(t)Uol o -< -
inf
xEA),
c h ( t ) U o -- Xhl o
inf IIT(t)Uo-X~ o
xE A'h
<_ Oh. (liA'/~.a(t)}l~ + IIA'..(t)llx) <_ C/, IIUo .~. Next, the identity 8
a, ~ [ r h ( t -
~)?,,~,,(~)] = T,,(~- ,~)[A,,7.~ - P,,A] T(, ~)
2.8. Hyperbolic Equation
87
implies
72hCh(t) = f0 t Th(t- r)[72hA- Ah72h] T(r)dr
r=t = [Th(t- r)7)h [A -1- A;l~)h] .AT(r)]r= 0
- JootTh(t- r)72h [,14-1- Ahl~)h] A2T(r)dr
3 t~=l where
~h(t) = 72h [A - 1 - Ah~7)h] AT(t), C2h(t)
:
--Th(t)Y)h
[A-1- JJ~;l~)h]A,
Th(t- r)7-)h [A-1 - Ahl~)h] A2T(r)dr.
e~(t) = Here, we have A-1 - A-172h
( 0 a-1 )
=
I
=
(
0
-
Q0 --A;1)Q /~h I
0
0
0
I--ORh -A-I+Ahlph)O
with [[Ahlph-
A-~{Ix,v < Ch
II1- Rhllv,x ~
and
Ch.
Therefore,
IIA-~- Ahlpnllx_+x ~
Ch
follows. We get
1t4(t)Uollo
~
II~(t)Uollo
~
(t)Uollo
_<
Ch ch
11.4~ollo - Ch IIGII1 IIAUollo -
ChllUolll
ItA UOllo = cth llgoll,,
and hence the following theorem holds. Theorem
2.21.
The estimate
lu(t) holds true.
- uh(t)llv + I ~(t) -
G(t)llx _< Ch (tlllUol I1 + IllUo11 89
88
2. Semigroup T h e o l y and F E M
Proceeding to the L 2 error analysis, we set X-1 = The (Co) groups satisfying
V*
{T(t)},~ and IIT(t)llx_,,w_,
and
X-lh =
Vs
"
{Th(t)}te ~ extend to those on X_I and X-lh, respectively,
< 1
and
Ilth(t)l x_,.,w_,. -< 1.
(2.94)
(The proof is left to the reader.) Now, under the inverse assumption, (2.88) implies the equivalence
In fact, we have
IIA-'vllv
Ilvllv, ~
and
II~,,llvx ~ IIA;'~,,I[~,,
by A* = A, where v E V and X:h E Vh. On the other hand,
tlx,,llv, ~ ll)o, llvx
for
X:h E Vh
follows from Theorem 1.5:
IIP,,llv, v <_ c In fact, we have
II>c.llvz
=
supp{l(Xh, v)l I v c vh, Ivllv _< 1}
<_ tl)o, llv, = sup{l(Xh,v)llv ~ v, I1~11,. <_ ~} - ~up {1(~, P~)I I ~ e v, I1~11,. _< 1} ___ Ilxh Ivx sup{llP,,vll,, I~ e v, Ivllv < ~} <
I~,, Iv,: IIPhll,.,v.
Furthermore,
holds, and hence
Phllv,,v, <- c is satisfied. Finally, from Theorem 2.4 we have for v E X and Xh E Vh that
Ix,,- ~SI~, <_ c
IA-'/"(x,,
-
v)lf~
(2.95)
2.8. Hyperbolic Equation Taking Xh
89
= al/2
"'h Rh A-112v, we obtain (2.96)
inf [[Xh -- VilE, < C h l vllx .
Xh eVh
In use of Phlvh = Ih, we have from inequalities (2.95) and (2.96) that
II(Ph - I)vllv, < c h Ilvllx. Above estimates imply the following. First,
o)
73~ = ( P;~ 0
Ph
satisfies (2.97)
[l~oallx_l,X_l <- C. Next, from M-1 -- Ah17)0h =
I-
0
Ph
A-s 1Ph - A -1 ) 0
it follows that
(2.98)
]A -1 - Ah17) ohl[x_l,X_ 1 < - Ch because of
II[A;1ph- A-~] ~ollx : II
co~(t) = T~(t)~oh - T ( t )
in X_I, we repeat the computation given above for Eh(t) in Xo. In use of (2.94), (2.97) and (2.98), we get
IIco,,(t)go Ix_l <- C h
(tlgoll~/~ + IIgollo)
9
However, the identity
~.(t) = ~0~(t) + T.(t)(~0. holds with Ph - P0h = ( / ~ h k
- Ph
0
- p.)
o) 0
In use of
II(nh-
Ph)vll~ -IIP,~(n~ - z),,ll~ _< II(n~ - Z)vllx _< C h Ilvllv,
we get
IlPh - Poh[l~_l-~xo -< Ch. By means of (2.93), we have thus proven the following.
90
2. Semigroup T h e o w and F E M 2.22. The L 2 estimate
Theorem
Ilu(t) -
uh(t)llx + IIA -1/2 (it(t) - ,~(t))ll x <__c h (t I u01 1/2 -lt" IIV0ll0)
holds true.
Energy Method Theorems 2.21 and 2.22 can be prove differently. In fact, weak forms for (2.84) with (2.85) and (2.89) with (2.90) are given as
(~, ~) + .A(u, ~) = o
(~ e v)
with ,,,(0) = ,,,,
,a(0) = ,,o
and
(a~. xh) + A(~,. xh) = o
(x,, ~ v,.)
with
~h(o) = ~h...
~h(o) = Phuo.
respectively. They imply the energy equalities
IIa(t)ll,~ + [ A'/~u(t)ll~x =
,,o11~ + IlA'/~u, =~
and
x,, +
~,~ ,~h(t)
Xh
= IlPhuoll xh = +
Ah'/= Rhul
Xh
Then the energy estimate of the error is derived in the following way. Let
.fi2~=
II,~h
-
P,,,;, 12x + A ( u h - Rhu, u h - Rhu).
Then, fh, = fh(t) satisfies fh(O) -- 0 and
- {(~, .a,,.- & a ) +
A(.., 'ah-
~,,.a)}
=
A (~h, [ P , ~ - n,,.].a) - A (.,,., [ p , ~ - n,,.].a)
=
A (~,, - ~,,,,,, p~[~ - R,,],,~)
_ f,, liP,,[/- nh]all~.
(2.99)
91
2.8. Hyperbolic Equation The inverse assumption guarantees (1.46):
(x~ c vh).
Ilxhllg _< oh -~ II>~llx
W e can estimate the right-hand side of (2.99) from above by
C h - l A I[S- s{~]~ x. We have L <_ Ch -~ I [ / - Rh]u Ix ~ Ch IIA~ Ix ~ Ch IIg011~,
and hence
fh(t) <_ Cht Uo
1
follows. This implies
u ( t ) - uh(t) v+ I/L(t)- <~(t)ilx
_< f,<(t)+ [R~- s]~(t)liv _< f h ( t ) + Oh (llAu(t)llx +
+
II/~(t)llv)
<_ Ch ( U0tll + IIU0111/2), or the energy estimate given in Theorem 2.21. The L 2 estimate is shown in the following way. Set
~. ( u . - Phi) X + I 1 ~ - R,<~II=X" We have g~(0) = 0 and
gh[Th = -=
( a h 1/2 (~th- Ph%t),A-s
Phil) nu (~th- RhU, ith- RhU)
{ (Uh, Ah-l[ith -- Phit]) -}- A (Uh, A;l[ith -- Rh~'t]) } -- {(/~, A-~l[ith -- Phit]) + A (U, A-~l[ith - Rh/L]) } A (uh, A-~[P~ Rh]/t) (u, A;~[Ph R,,]/t) -
-
,,4
-
= A ( ~ - R~, A-i~[P~- Rh]~) = ( ~ - t{~, P~[s- R~],a)
(2.100) Therefore,
{]h ~ Ca ]l'{t IV ~ Ch Uol 1/2, and hence
g,,(t) < Cth IIUoll~/~, follow.
2. Semigroup Theoly and F E M
92 Here we note
(2.101) with I111 -
R~]~
x -
c h Ill, IlK <_ c h IIUol 0 .
Furthermore, the equality
A-1/2[it h - it] = [Ahl/2P h - A -1/2] ( i t - ith)+ Ah 112
Phit)
(2.102)
holds with
II[A~I~
~-~1<~ ~/llx <_ _
c~l,u-~l,x c h (ll'al x + II~hl x) - Ch (llgollo + go,,I o)
<_ Ch llgol o. The L 2 estimate follows in this way:
Ilu(t)- ~,h(t)llx + IIA-~/~ (~(t)- ~h(t))ll x _< Ch (t goll~/2 + gollo) 9 This method gives the following. T h e o r e m 2.23. The estimate
l u(t)
-
uh(t)Ix + IA-~/2 ( ~ ( t )
-- ~ h ( t ) ) l l x
_< Ch ~ (t Ilgol 1 + Ilgo ll/2)
(2.103)
holds true. Proof." In fact, (2.100)implies
.0h _< [z - Rdallx _< c h 2 IIA'allx _< Ch? I Uoil, and hence I[I - ~,,.]*~llx
<- Ch 2 Au Ix <- Ch21Uol ~/~
holds in (2.101). On the other hand Theorem 2.21 gives that
1t[~; ~'~', ~*'~] (*~ *~',~ll~< ~,, Jl~- ~,,Jl~ < ~,~? (,tJ~olt, + jj~o,l,,~) in (2.102). Those inequalities imply (2.103).
C o m m e n t a r y to Chapter 2
93
C o m m e n t a r y to C h a p t e r 2 2.1. Abstract Cauchy problems on Banach spaces such as (2.4) with (2.5) are called evolution equations. Their wellposedness induces dynamical systems there, represented as semigroups. The generation theory of (Co) and holomorphic semigroups are described in Hille and Phillips [179], Yosida [410], and Kato [205]. The fractional powers of operators are defined through the associated negative powers expressed by the Dunford integral. For those topics, including the proof of inequality (2.18), see also Tanabe [378]. Integrability of inhomogeneous equation (2.22) is also discussed there. 2.2. Mathematical study on the finite element method applied to parabolic problems was initiated by Blair [38], Thomfie [381] and Zlgmal [420], [421]. Application of the semigroup theory to this problem was proposed by Fujita [139]. The proof of Theorem 2.1, based on the Dunford integral, was given by Fujita and Mizutani [144]. A different approach was taken by Helfrich [170], [171] for self-adjoint A, introducing the error operator Eh(t) = e - t A - e -tahPt~. He then applied the semigroup properties and a kind of duality argument. This method, sometimes called the real method, will be described in w167 and especially, in the next chapter in details. Related works were done by Bramble, Schatz, Thomde, and Wahlbin [57] and Baker, Bramble, and Thomde [25] for semidiscrete and full-discrete approximations, respectively. 2.3. Theorem 2.4 is due to Ushijima [397]. The interpolation theory is described in Folland [127] for instance. For concrete characterizations of the domains of fractional powers of elliptic operators, see Fujiwara [150], [151], Lions and Magenes [247], Henry [173] and the references therein. Discrete elliptic-Sobolev inequality is due to Suzuki and Fujita [370]. Inequality (2.43) was proven by Kato [199], [202]. 2.4. Relation to the contractility of the resolvent and the coerciveness of the bilinear form is described in Tanabe [378]. We followed the arguments of Fujita and Mizutani [144] for the proof of Theorems 2.7 and 2.8. Concerning the numerical range of iterative operators utilized in the proof, see Kato [203]. See also Kato [205] for other properties of the numerical range. 2.5.
Theorem 2.10 was shown by Fujita and Mizutani [144].
2.6. The proof of Theorem 2.11 is due to Fujita and Mizutani [145]. Related works were done by Douglas and Dupont [114], Dupont [117], and Wheeler [406]. The operator B defined by (2.64) was used in Lions and Magenes [247]. Various works have been done to analyze approximations with higher accuracy. In this connection see Raviart and Thomas [326] and its bibliography. 2.7. L ~ error analysis for parabolic equations was done by Nitsche [301], Nitsche and Wheeler [303], Schatz, Thom6e and Wahlbin [340], and Thom6e and Wahlbin [386]. We have followed the argument of Thom6e [383], [384] for the proof of Theorem 2.16. Lemma 2.17 is related to Nitsche and Schatz [302] on super-convergence. A sharp form of inequality (2.80) was given by Cazenave [71].
94
2. Semigroup T h e o l y and F E M
We can derive some error estimates for non-smooth initial data u0, taking account of the smoothing properties of {e -tA }t>0 and {e -tAb }t>0" Use of the discrete elliptic Sobolev inequality (2.41) is one way, but more fine estimates were shown in Thomde [383], [384] based on the energy method. This method will also be described in the following chapter. _
2.8. The error analysis developed in this section is nothing but the one summarized as the real method. As is described above, the duality argument involving evolution operators is due to H.P. Helfrich. As to the discretization for the time variable t of the semidiscrete approximation (2.89), there is a scheme called Newmark's/3. It is given as D,eu,~(t) + Ahu,~(t) + / 3 T 2 A h D ~ ( t ) = 0 for t = n r with T
Uh(0 ) : RhU~
and
,
T
u h ( - 1 ) = R h ' u ~ - wPh'Uo,
where r > 0 is the size parameter,/3 k 0 is a fixed constant, and
D...,(t) = ~-~ {~(t + ~ ) - 2,~(t)+ ,,~(t- ~ ) } For this scheme, Fujii [134] proved tile following. T h e o r e m 2.24. Suppose the inverse assumption and the T / h < t~7 in case 0 <_/3 <_ 1/4, where t~ > 0 is an absolute constant. TheTt, the energy estimate ,
T
T
lit(t) - uh(t)Iv + I ~(t) -- D~u,~(t)llx <_ C (h + T)IIUollo holds for t = nT < T.
For the proof, see also Ushijima [394], [397], [398]. There, Theorems 2.21 and 2.22 were also shown by a systematic and sophisticated use of Helfrich's method. Related works were done by Fujii [133], Baker and Bramble [24], Gevici [157], and Larsson, Thom~e, and Wahlbin [231].
Chapter 3 Evolution Equations and FEM
In the case of temporally inhomogeneous parabolic equations, the complex m e t h o d does not work so well by itself for the analysis of the finite element approximation. Here, we adopt the m e t h o d of Helfrich and then that of energy, and extend the error estimates of the preceding chapter to this case.
3.1
Generation
Theories
The present chapter is devoted to the temporally inhomogeneous parabolic equation; t2 C R 2 denotes a polygon, and /2 = /2(x,t, D) a second order elliptic operator with time-dependent real smooth coefficients:
c(~, t, D) = - ~
~-~ a~j0 (~, t)
0
2
+ Z~=~ bj(x , t) ~+~(~,t). o
i,j=l
Uniform ellipticity 2
i,j=l
is assumed, 51 > 0 being a constant. We study the parabolic equation
ut +/2(x, t, D)u = 0
in t2 x (0, T)
(3.1)
with the initial condition
glt=o = uo(x)
in ft
(3.2)
and with the boundary condition either u = 0
on at2 x (0, T)
(3.3)
or
0u -t- a u = 0 cO~,c
on 0t2 x (0, T ) .
--
(3.4)
In (3.4), cr = a(x, t) is a smooth function on 0t2 x [0, T], and 0/0r, L denotes the differentiation along the outer co-normal vector ~L: 0
_ ~-'2
n i a i j ( x , t)
i,j=l
95
0
3. Evolution Equations and FEM
96
where n = (nl, n~) is the outer unit normal vector on 0f~. Assuming u0 E X = L2(~), we can reduce equation (3.1) with (3.2) and (3.3) (or (3.4)) to the evolution equation
du
dt
+ A(t)u = 0
(0 < t < T)
(3.5)
with u(0) = u0
(3.6)
in X = L2(gt). Taking V = H~(f~) or Hl(f~) according as the b o u n d a r y condition (3.3) or (3.4), we put
.At(u, v) = i,j:,
aij(x, t)-~xj -~xi dx + J:'
bj(x, t)--VOzy dx
+ ffl c(x, t)uv dx + fon a(x, t)uv dS for u, v E V. An m-sectorial operator A(t) in X can be defined through the relation
At(u, v) = (A(t)u, v) ,
(3.7)
where u E D (A(t)) C V and v E V. As in the temporally homogeneous case of w relation
D (A(t)) = H2(fl) N H~(fi)
the
(3.8)
follows for V = H~(f~) and
D (A(t)) = {v C H2(f~)
+av=O
on0f~
(3.9)
for V = Hl(f~), if 0fi is smooth for instance. Generation theories of a family of evolution operators {U(t,S)}o<s
du dt
- - + A(t)u : 0
(s < t < T)
(3.10)
with 'u(s) = 'Uo C X
(3.11)
3.1. Generation Theories
97
is given by u(t) = U(t, s)u0. Consequently, the semigroup properties
U(t,r)U(r, s) = U(t, s)
(0 < s < r <_ t <_ T)
and
(0 <_ t <_ T)
U(t,t) = I follow. Furthermore, we have
U(t, s) = - A ( t ) U ( t , s)
and
-~sU(t, s) = U(t, s)A(s)
(3.12)
in the strong sense in X for 0 _< s < t < T. It is worthwhile to give a short summary of those theories here. In fact, studying the finite element approximation of (3.5), we show some kind of stability of approximate solutions and smoothness of original ones. Both of them are established by re-examining the theories, and then error estimates will follow. Crucial assumptions are the following: (i) Each - A ( t ) generates a holomorphic semigroup with certain estimates uniform in t E [0, T]. (ii) A(t) is smooth in t E [0, T] in some sense. The family {U(t, S)}o<s
IA,(~, ~)1 s c~ II~llv II~llv
(u, v ~ V)
A~(~ , ~) > a l~ I~V - ~1~1 X=
(~ ~ v)
and -
-
,
(3 13) "
with constants C1 > 0, 6 > 0 and A E ][{ independent of t E [0, T]. We may suppose A = 0 in (3.13), taking v - e-~tu(t)instead of u - u ( t ) i n (3.5)"
x,(~, ~)>__ ~11~11~
(~ ~ v ) .
(3.14)
Then, A(t) becomes of type (0, M) with 0 E (0, 7r/2) uniformly in t e [0, T]; we have
C \ Eo C p(A(t)) for
ro = {~ c e l
o < arg I~l < o},
(3.15)
3. Evolution Equations and FEM
98 and M~ II(zI- m(t)l _ XlIx, X <_-~
(0 + e < _ [arg z I _< 7r)
(3.16)
for e > 0. Each -A(t) generates a holomorphic semigroup {e -~a(t) }s>0 uniformly bounded in t C [0, T]. We have
II~-~A(,)llx,x <_1 for 0 < s < + o c and 0 < t < T. Concrete expressions of the second condition, on the other hand, are slightly different according to the theories. In each of them, construction of the family gives some estimates on their members at the same time. For the moment, the operator norm on X = L2(12) is simply written as II" IIGeneration
T h e o r y of T a n a b e - S o b o l e v s k i i
Because of (3.8), D(A(t)) is independent of t in the case of V = H~(f~). Furthermore, the inequality I[A(t)A(s)
-~ -
111_
cIt-
sl ~
(t,~ E
[0, Z])
(3.17)
follows with c~ C (0, 1] from an integration by parts and the elliptic estimate. In such a situation, the family of evolution operators {U(t,s)}o<s
U(t,s) = e-(t-s)A(~) +
e-(t-r)A(~)R(r,s) dr,
(3.18)
where R = R(t, s) is the solution of the integral equation of Volterra type, I~,(t, S) --
I~ 1 (t, r ) R ( r ,
s)dr
= /~1 (t, S)
(3.19)
for /~,1 (t, S) -- --
(A(t)
- A ( s ) ) e -(t-s)A(s)
Those relations are obtained formally by substituting tile right-hand side of (3.18) into the first relation of (3.12). Eventually the unique solvability of (3.19) follows and {U(t,s)} defined by (3.18) becomes the desired family of evolution operators. Meanwhile the following estimates are proven, where 0 _< s < r < t _< T and 0 E (0, a):
(3.20) IIU(t, s)ll + [IA(t)u(t, s)A(s)-' I[ <- c,
[IA(t) [ u ( t ,
~) - u(,-,
~)] A(.~)-' II -< C o ( t
- ~)o(~ _ ~ ) - o
(3.21) (3.22)
3.1. Gel~el"ation Theories
99
G e n e r a t i o n T h e o r y of F u j i e - T a n a b e If V = Hl(f~), D(A(t)) varies as t changes, and the preceding theory does not work directly. However, the m-sectorial operator A(t) in X defined through (3.7) can be regarded as that in V', which is denoted by A(t). Its domain is independent of t: D(A(t)) = V, and furthermore, conditions (3.15) and (3.16) hold with A(t) and X replaced by A(t) and f( = V', respectively:
V',V' --
]Z q- 1
The coefficients aij, bj, c and cr are smooth so that the inequality IAt(u, v) - A~(u, v)l _< O It - sl ~
I1~11~I1~11~
(u, v E
V)
(3.24)
holds with c~ E (0, 1]. This implies (3.17) for A(t) in 2 . Therefore, from the preceding theory, {A(t)} generates a family of evolution operators in 2 denoted by {U(t, s)}0<s 1/2, it can be shown that
U(t,s)=O(t,s)x becomes a bounded operator in X and {U(t,s)} forms a family of evolution operators generated by - A ( t ) . This {U(t, s)} satisfies (3.20) and U(t, s)[ _< C in X.
G e n e r a t i o n T h e o r y of K a t o - S o b o l e v s k i i We can show that D (A(t) p) is invariant in t for p C [0, 1/2). symmetric part A ~ of At as in w
To see this, take the
Ato= 1 (.,4, + At).
At(~, ~) = A,(~, ~),
If Ao(t) denotes the self-adjoint operator associated with At~ it holds that
c - ' Ivllv _< IlAo(t)l/% Ix < C vllv.
(3.25)
As is described in the commentary to w the domain of A(t) p is equal to that of Ao(t) p if p C [0, 1/2). (The idea of proof will be obtained from the argument described below.) Here, the latter is independent of t if p = 1/2 as is indicated in (3.25), and p = 0 obviously:
D(Ao(t) '/2) = V,
D(Ao(t) ~ = X.
Then, Heinz' inequality implies the invariance of D (A(t) p) in t for p E [0, 1/2). We show that the inequality
IlA(t)PA(s)-p-/11
c t - sl
(t, s E [0, T])
(3.26)
3. Evolution Equations and F E M
100 follows from (3.24). In fact, given # > 0, we have (#I + A(t)) - 1 -
(pI + A ( s ) ) - ' = - (pI + A(t)) -1 Ao(t) 1/2. Bo(t, s ) . Ao(s) 1/2 (#I + A(s)) -1
with Bo(t, s) = Ao(t) -1/2 [A(t) - A(s)] Ao(s) -~/2
Inequality (3.24) reads; IA(t) - A(s)l v,v, <- C I t - sl ~ , or
liB(t, ~)11 ~ c It - ~1~ We make use of the following. L e m m a 3.1. If H is a non-negative self-adjoint operator on a Hilbert space X with norm
II. II, ~ e h~ve
~
~ #2~ IH~/2 (~I + H) -~ v[I 2 d # -
Cp IIH~
2
(3.27)
for p c [0, 1/2) and v E D(H~ Proof: Introducing tile spectral decomposition
ad E ( a) ,
H :
we have
#~o ]IH~/~ (#I + H)-'
= d#
-
~,=o @
a(~, + a ) - ~ d IIE(a)~ll ~
9
-
ad IIE(:~)vll =
=
ad IIE(:~)vll ~. a ~o-'
=
Co
#~o(# + a)-~da ~o(#
+ 1)-2d#
A2PdlE(A)vl 2-- CplIHPv 2.
This means (3.27).
[]
We are able to give the following. Proof of (3.26): We show
II[A(t) p - A(s) p] vii < c It - sl ~ IIAo(s)~
(3.28)
I01
3.1. Generation Theories for p E (0, 1/2) and v e D (Ao(s)a). Then, inequality (3.26) follows from (2.43). For this purpose, we may take v r D(A). Applying (2.18), we have ([A(t) ~ - A(s) ~ v, X) = sin 7rp
7r
/0
pP dp. (B(t,s)Ao(s) 1/2 (pI + A0(s)) -1 v, Ao(t) 1/2 (pI + Ao(t))-i X)
for X E X. In use of (3.25), inequality (3.28) follows as [([A(t)P - A(s)P] v, X)[
(/0 9
I[Ao(t)'/2 (#I + Ao(t)) -x Xll2x d•
It - s[ ~
= c~ IIAo(s)Ovllx. I l x l l x - I t - sl ~ . The proof is complete. Above properties of the fractional powers of generators, summarized as the invaria.nce of D (A(t) o) and inequality (3.26), guarantees the existence of the family of evolution operators in the following way if c~ + p > 1 is satisfied. Therefore, evolution operators exist if c~ > 1/2 again, in the parabolic initial b o u n d a r y value problem in consideration. For simplicity, we describe the case p = 1/m with m an integer. By taking an appropriate approximation of A(t), say, the modified Yosida approximation: Aa(t) = A(t)(1 + AA(t)o) -m with ,~ I 0, we can reduce the theory to the case where each A(t) is bounded in X. In fact, we can show that (3.26) implies
IlA~(OPA~(s) - p - Ill <_ C l t - sl ~
(t, s e [0, T])
with a constant C > 0 independent of )~ > 0. Existence of a family of evolution operators {Ua(t,s)}o<s
U(t, s) = s-lim Ua (t, s). M0
Then, {U(t,s)}o<s
llA(t)ZU(t,s)ll < C z ( t - s)-z
(O<_s
follows from ]lA~(t)Zu~(t, s)[I < C z ( t -
s) -z
(0 < s < t < T ) ,
(3.29)
where 0 _< p < a + p. We shall give an outline of its proof, dropping the suffix A for simplicity of writing.
3. Evolution Equations and FEM
102 Letting
D(t, s ) = A(t)PA(s) - p - I, we have
liD(t, s)
~
Clt
-
s ~
and m
A(t) - A(s) = Z
A(t)I-PPD(t' s)A(s)PP"
(3.30)
p=l
We get t
-
= ~=
e-(t-r)A(t)A(t)l-p"D(t,r)A(r)~~
dr.
(3.31)
We introduce the following notation: Given the operator-valued functions I(e = IQ(t, s) (g = 1, 2) on D = {(t, s ) [ 0 _< s _< t <_ r } , we define another I( = K1 * I(2 by (//(1 * 1(2)(t,s)=
K~(t,r)K2(r,s) dr.
For a > 0 and M > 0, we say t h a t K E Q(a, M) if the inequality
[ K( t,s)ll - M ( t -
s) a-1
holds. Then, if Kl C Q(ae, Me) for g = 1, 2 we have /s
* /(2 E Q ( a l +a2, B ( a l , a 2 ) M 1 M 2 ) ,
where B(a, b) denotes the beta function: B(a,b)
jfo 1(1
=
-
x )a- l x b-1 d ~ .
Let
W(t, s) = U(t, s) - e -(t-~)A(t)
and
Yq(t, s) = A(t)oow(t, s).
Equality (3.31) reads; m
(3.32) p=l
3.1. Generation Theories
103
where
Hq,p(t, s) = A(t)'-PP+qPe-(t-s)A(t)D(t,
s)
(3.33)
and m
p=l
for
Yp,_, (t, 8) = A(t)PPe-(t-s)A(t) To avoid a technical difficulty, we take (3.34) and transform (3.32) into a system of integral equations for
{Zql q = 1,.-. ,m}. That is, m
Z~ = ~
Hq,, , Z, + &,o
(3.35)
p=l
with m
(3.36) p=l
Consequently, Zq (q = 1 , . . . , m) is subject to the iteration scheme oo
Zq -- ~
(3.37)
Zq,i
i=0
with m
Zq,,+, = ~
H~,, 9&,,
(~ = o, 1 , . . . ) .
p=l
From definition (3.33) follows H~,, ~ Q (~ - qp + pp, M , )
with a constant M1 > 0, because (2.16) holds by (3.16)"
<
> 0),
(3.38)
3. Evolution Equations and F E M
104 where ~c >_ 0. F u r t h e r m o r e , we have
Zq,o E Q(1 + a - qp, Mo) for some M0 > 0. We can derive
Zq,i E Q (1 + (i + 1)a - qp, Mi) with A/Ii+l/~Ii = r n M o M i B (c~ + p - 1, (i + 1)c~) by an induction, and then Zq E Q ( 1 + a - q p ,
(3.39)
C)
follows from (3.37) with a constant C > 0. Inequality (3.39) now gives an estimate on Yq even in the case t h a t q is not an integer. Namely, (3.34) makes sense for any q >_ 0, and Zq,o is again given by (3.36). T h e n Zq is defined by (3.35) with p taking integers as before. Since Zq in (3.35) admits (3.39), inequality (3.34) gives an estimate of Yq for a - qp + p > 0, because Yq,O E Q(1 - qp, C) can be shown. Consequently, Yq E Q(1 - qp, C) follows. Writing qp = / 3 , we then obtain (3.29) from (3.38).
Generation Theory of Kato-Tanabe Let us define another bilinear form .At(, ) on V x V by
,=
j=l
s
/o ~
Then, we have
IA,(u,~)l _< C llullvllvllv, I.A,(.,,, v) - A,(.,,, ~)1 _< c It - ~1~ II~llv Ilvll~, for u , v E V with some c~ E (0, 1] and lira
sup
(At-
1
As)('u, v) - As(u, v) = 0 .
II~llv,llvllv
dA(t)-' dt
d A -
ds
(S)-1 ~
holds with K > 0 and a E (0, 1].
K
It-
s
(t, ~ E [0, T])
(3.40)
3.1. Generation Theories
105
2) The inequality
c9 ( z I - A(t)) -~
M~
(z e c \ r0+~)
---[;i
(3.41)
holds for c > 0 sufficiently small.
Proof of (3.41): Take z 9 C \ E0+E and v 9 X, and put w(t)=(zI-A(t))-lv. We show that it is continuously differentiable in X with the estimate
Ow x
-5i-
M~
(3.42)
-
satisfied. Recall
(8.48)
(z - A ~ ) ( ~ ( t ) , ~) = (~, ~) and (2.31)"
Jzl lJxJl~ + II~IIV~ - -<
~o I~ II~IIX~
- A ( ~ , ~)J ,
where X C V and z C C \ Eel. This implies also (2.33) as
I1~1 ~ < 6o Iv ~ /I~1
~nd
I1~11~ < ~o vlt~ /t~ ~/~
Differentiating formally in t, we get (Z -- , A t ) ( ~ b ( t ) , )(~) = A t ( w ( t ) , ) ~ )
(3.44)
from (3.43). Here, the right-hand side is a bounded linear functional of X C V. In use of (2.31), we can define ~b(t) E V through (3.44) conversely, by Lax-Milgram's theorem. Now we shall show lim w(t) - w(s) _ ~b(s)ll t--~s
t -- 8
: 0.
(3.45)
V
In fact, letting ~ ( t ) - ~(~) t--8
we have
(~ -
A~)(~(t), ~(t))
=
(Z -- r
-- W(8), x ( t ) ) -- (Z -- ,At) (~b(8), x ( t ) )
t-8
(z - A~) ( ~ ( t ) - ~(~), ~(t)) - As (~(~), ~(t)) t--8
+ (A~ - A~)(w(~), ~(t)) t--8
{1
((z - As) - (z - A~)) (~(~), ~(t)) - A~ (w(~), ~(t)) + (At - A s ) ( ~ ( ~ ) , ~(t))
}
3. Evolution Equations and FEM
106 Again by (2.31), we get 1
IIx(t)ll* V - -< ~0
(.4, - .4~) (~(~), z(t)) - A~ (~(~), ~(t)) + &c It- ~1. Iw(~) ,,. IIx(t) v,
or
IIx(t)llv < C I t - s
9~,(~)1 v +C
sup xEV
1
---zU t (.4, - .4~) (~(~), ~) - A~ (~(~), ~)
Ilxllv_
lim IIx(t)llv = 0 . t---* 8 We turn to the proof of (3.42). In fact, again by (2.31) we have
z IIw(t)ll x2 + I~(t) ~. _< ~o I(z - .A,.)(~(t), ~(t))l 5o 12., (w(t), w(t)) I _< C IIw(t) ,, IIw(t)ll,,. This implies
~i,(t) Iv < c IIw(t)Iv -
CIl~ll~ Izll/~
and
~/2 II~i~(t)lv/~ ......
IIw(t) Ix < Cllw(t)llv
Izl '/~
-
C lvllx
Izl
by (2.33). The proof is complete.
Proof of (3.~0): We take v C X, and set .f(t) = A ( t ) - l v E V. The existence of df(t)/dt follows similarly. Noting At (.f (t), X) = (v, X) , we define j~(t) by
where ~ E V. Then it follows that lim
t--~8
.f(t) - . f ( s ) - r t-s
=0. V
3.1. Generation Theories
107
Inequality (3.40) is a consequence of the stronger one:
Ilvll~,.
(3.46)
In fact we have
IIx ~ < A<(r - r x) = -Jlt(f(t), x ) - ( A t - A,)(j'(s), x ) + Jl,(f(s), x) = ( f % - Jlt)(f(t), x ) + J l ~ ( f ( s ) - f(t), x) - ( A t - As)(r <
x)
C Ilxllv (it - sl ~ IIf(t)llv + Ilf(s) - f(t)llv + It - sl IIr
or
iii(t)- ./(~)llv _< c (it- ~l ~ lif(,)il~ + ilf(~)- f(t)ll~ + it- ~l
Its( )ll ).
(3.47)
Remember that
IIf(t)llv <_ C IIv v, follows from
llf(t)ll~
_ .At(f(t), f(t)) -- (v, f(t)) < c
IlVllw, IIf(t)llw
.
Similarly,
~llJ(s)ll~ ~ A,(r
r
- -J~(f(s), r
< c }lf(s)llv IIf(s)llv
gives
Finally, we have
(5lif(s)- f(t)ll~
< .At(f(s)- f(t), f ( s ) - f(t)) = ( A t - As)(f(s), f ( s ) - f(t)) <
C It - ~1 IIf(~)ll. IIf(~) - f(t)ll.
and hence
] i f ( s ) - f(t)iiv <_ C I t - s I tlf(s)llv <_ C i t - s I livllv, follows. Plugging those inequalities into (3.47), we obtain (3.46). Subject to the above conditions on the differentiability of resolvents, the family of evolution operators {g(t,S)}o<s
u(t, ~) = ~-(~-~)~(~) + w ( t , ~)
3. Evolution Equations and FEM
108 with
w(t, ~) =
~-(*-~(*)R(~, ~) d~,
(8.48)
where R = R(t, s) is the solution of
R(t, ~) = R~(t, ~) +
R~(t, T)~(,-, ~) d~
(3.49)
for
0 + N0 R l ( t , s ) = -(~-t
)r
1 O~F e -(t- s)z -07(zI0 -- 27r,. A(t))-ldz,
(3.50)
F being positively oriented boundary of E0+~ for e > 0. Inequality (3.20) and IIg(t, s)ll < c are also derived from this scheme. More precisely, we have
IR(t, s)][ _< C
(0 _< s _< t _< T)
(3.51)
and
(0 _< s < ~ < t <_ T),
(3.52)
for 7 C (0, 1) and 5 C (0, a). They are consequences of those with R replaced by R1,
IIR~(t, s)ll <_ c
(3.5a)
and
II/~l(t, 8 ) - /l~l(r, 8)11 ~ C7,5((t- T)7(T- 8) -7 -[- ( t - r ) a ( r - s)~-a-1), respectively. This theory of generation is particularly remarkable as any assumptions on the domains of A(t) are not made. Consequently, it is no wonder that a little stronger assumption on the smoothness in t of A(t) is imposed.
3.2
A Priori
Estimates
In this section, we show a uniform estimate concerning the evolution operator associated with the discretized problem, which plays a key role in the error analysis. In the following lemma, V C H C V' denotes a triple of Hilbert spaces, A(t) an m-sectorial operator in X associated with the bilinear form At(, ) on V x V, smooth in t E [0, T] in the sense of the preceding section, and {U(t, S)}o<8
3.2. A Priori Estimates
109
L e m m a 3.2. Each/3 e (0, 1/2) admits the equality
d(t)U(t, s) - A(r)U(r, s) = A(t) [e -(t-~)A(t) - e -(~-s)A(t)] + A(t)ZZ~(t, r, s)
(3.54)
with Zz(t, r, s) subject to the estimate IIZ,(t, r, s)ll <
C,(t
-
r)(r
-
(3.55)
s) ~-',
where O < s < r < t < T. Proof: We shall take the reduction process first. In fact, from the construction we have A(t)U(t, s) - A(r)U(r, s) = A(t) [e - ( t - s ) A ( t ) --
e -(r-s)A(t)]
+A(t)" [A(t)l-'~ -(~-s)~(') - A(~)'-'~ -(~-s)~(~)] +A(t)" [ I - A(t)-ZA(r)'] A ( r ) l - ' e -(~-s)A(~) +A(t) a [ A ( t ) l - ' w ( t , s) - A ( r ) l - ' W ( r , s)] +A(t) ~ [ I - A(t)-~A(r) ~] A(r)'-~W(r, s), so that 4
/=1
holds with
Z~(t, r, s) = A(t)l-Ze -(r-s)A(t) - A(r)l-Ze -(r-s)A(~), Z~(t, r, s) =
[I - A(t)-ZA(r) ~] A(r)X-'e -(~-s)A(r),
z~(t, ~., s)
A(O'-~w(t,
:
Z~(t, r, s) :
s) - A(~.)~-~WO ., ~),
[I - A(t)-ZA(r) ~] A(r)'-~W(r, s).
The Dunford integral gives the expression
d(t)Ze_Tm(t ) _ A(s)e_~d(~ ) = 12m ofr z'-Ze-rz
[ ( z I - d ( t ) ) - ' - (zI - m(s))-'] dz,
while
]l(z~-
A ( t ) ) -1 - ( z I -
A(s))-'ll < c~ It- sI -
Izl
follows from (3.41). We have
IlA(t)~e -~A(t) - A(s)~-~(s)ll <_ c~ It - sl r - ~ < c
d--~-~lt-s l -
~ - ~ e -~"~~176 At
G I t - sl r ~-~
(3.56)
110
3. Evolution Equations and FEM
for/3 > -1. This implies IIz~(t, ~, s)ll < c~(t - ~)(~ - ~)~-~
Inequality
III-
A(t)-ZA(s);311 <- C ~ l t - sl
follows from (3.26). We have
~
C / 3 ( t - r ) ( l " - 8) ~-1
Inequality (3.51) gives IIA(t)l-ZW(t,s)lt
: <_ c~
A(t)l-ze-(t-r)A(t)R(r,s)dr ( t - ~)~-'d~ = C ~ ( t -
~)~
and hence llz~( t, ~, ~)11 - c , ( t - ~)(~ - ~),
follows. The proof of Lemma 3.2 has been reduced to
Ilz~(t, ~, s)l t = IlA(t)l-~w(t, s) - A ( r ) ' - ~ W ( r , s)l I <_ C,(t - r ) ( r - s ) ' - ' Equality (3.48) now gives 7
Z~(t,r,s) = A ( t ) l - Z w ( t , s ) - A ( r ) l - Z W ( r , s ) = ~
Z[3(t,r,s )
l=5
with Z~(t, r, s)
=
f
t
A(t)l-~e -(t-~)A(t) [/~(z, s) -/~(t, s)] dz,
z~(t, ~, ~) =
[ A ( t ) ' - ~ -('-z~('~ - A ( ~ ) l - ~ - ( ~ - ~ ( ~ ]
Z~(t, ~, s)
A(t)~-~-(~-z~(~)dz
=
+
9[P(~, .~) - ~(~, ~)1 d~,
. ~(t, s),
[ A ( t ) l - ~ - ( ~-z~(~ - A ( ~ ) I - ~ -(~-z~(~]
d z - ~ ( ~ , s).
111
3.2. A Priori Estimates In use of (3.52) we have
[IZ~(t,r,s)l[
<
IlA(t)'-Ze-(t-z)A(t)ll .
<_ c~.,
IIR(z,s)- R(t,s)ll dz
( t - ~)~-~(t- ~)~(~- ~)-~d~
+G,~
( t - z)~-~+~(z- s) -~+~-~ 9( z - s)~-'dz,
<- G .
( t - z)~-l+~(z- ~)-~+l-~dz. ( ~ - s) ~-~
=
~)(~- ~)~-1
G(t-
taking "7 e ( 1 - f l , 1) and 5 e (c~ - fl, c~). From (3.96) and (3.56)we have
_ G(t-
~)(~- z) ~-~.
Combining this with
}lA(t),-z~-I,-zl~r
A(~,)'-Z~-Ir-z/Ar II
-< G(~"- ~)~-', we get [IA(t)~-ze
-r
-
A(r)l-Ze-(r-z)m(r)[I < C ~ ( t - r)'~(r- z) - ~ + z - '
for r~ E [0, 1]. This implies
IIG( t, ~, ~)ll _<
I[A(t) ~-~-(~-~(~ - A(~)'-~-(~-z)~(~)l[- I1~(~, ~ ) - ~(z, ~)11 d~-
+c~,~
= G(t-
( t - ~)~(~ - ~)-~+~-1. (~ - ~)~(~- ~)~
,~)~(,~- s)~-',
3. Evolution Equations and FEM
112 by taking -7 e ( a - / 3 , 1) and (5 E ( a - / 3 , a) similarly. We have Z~(t, r, $)
=
[A(t)-fle-(t-z)A(t)]~:t r 9R(t, 8) -Jr-[A(t)-fle -(t-z)A(t) - A ( r ) - f l e - ( r - z ) A ( r ) ] : : :
9I~(t, 8)
11
/=8
with
Z~(t, r, s) =
[I - e -(t-~)m(t)] [A(t)-zR(t, s) - A(r)-ZR(r, s)],
Z~(t, r, s) = - e -(t-r)A(t) [A(r) -z - A(t) -z] R(r, s), zl~~ r, s) = - [A(t)-Ze -(t-~)A(t) - A(t)-f~e -(~-s)a(t)] R(r, s), z ~ l ( t , 1", S)
:
-- [A(t)-~e -(r-s)A(t) - A(r)-~e -(T-s)A(r)] R(r, s).
The inequality
IIA(t) - ~ - A(~)-~II <- G It- ~1 follows from (3.26) so that we have IIz~(t,
~, ~)11 <-
c~(t -
~)(~ - s)/2i'- 1 .
On the other hand the inequalities
Ilzy(t, r, ~)ll _< Cz(t- ,~)(,~- ~)~-' and
IIz~'(t, r, s)ll _< G ( t - r)(~- s)~-' follow from (3.96) and (3.56), respectively. In this way, Lemma 3.2 has been reduced to
[[A(t)-zR(t, s ) - A(r)-ZR(r, s)[ I _< Cf3(t- r ) ( r - s) z-I
(3.57)
Integral equation (3.49) reduces this inequality furthermore to IIA(t)-ZR,(t, s ) - A(r)-ZR,(,', s)l [ <_ C ~ ( t - r ) ( r - s) z-l,
(3.58)
R~ - Rl(t, s) being the right-trend side of (3.50). We show this fact first. In fact, in use of (3.49) we have m(t)-~R(t,
s) - d ( r ) - r R ( r ,
s)
= [ A ( t ) - O R l ( t , s ) - A(r)-aR~(r,s)] + +
/'
A(t)-;3R,(t,z)R(z,s)
dz
[ A ( t ) - O R l ( t , z ) - A(r)-aR~(r,z)] R(z,s) dz.
3.2. A Priori Estimates
113
The second term of the right-hand side is estimated by (3.51) and (3.53):
/*
liA(t)-Zll
9IIRl(t,
z) l-IIR(z,
~)11 dz <
c(t
-
~).
Similarly, the third term is estimated as
fs r IlA(t)-ZRl(t,z) - A(r)-~t~l (r, z) C
dz
9 IR(z,s)
( t - r ) a ( r - z)~-lds = C~3(t- r ) a ( r - s) l~.
Thus, (3.58) implies (3.57). We now turn to the proof of (3.58). Recall the symmetric part Ao(t) of A(t) associated with Ao. We have the following. L e m m a 3.3. The equality
[ 0_~( z l _ A(t))_ 1 - -~sO(zl - A(s))-l] = A(t)lo/2 ( z I - A0(t)) -1Bz(t, s)Ao(s) 1/2 ( z I - Ao(s)) -1 holds with
Bz(t, s)II ~
c I~ -
(3.59)
sl ~ ,
where z E EOl. Proof: Define A'(t)" V --, V ' b y J~t(u, v) = (A' (t)u, v) for u, v E V. It holds that
0 (zI - A(t)) -1 = - ( z I - A(t)) 0-7
-1 d'
(t) (zI - A(t)) - 1
We have
0 =
(zI(zI q-(zI
A(t)) -1
0
A(s))-I 1
(zI-
A(~)) -1A'(t)(zI -
+ (zI - A(t)) -1
A(t)) -1 [A(/~) - A(s)]
-
d(f)) -1 [m'(t)
-
(zI
[A(t) - A(s)]
9d ' ( s )
The equality
m'(s)]
-
(zI -
A(s)) -1
d(s)) -1
(zI -
A(s)) -1
(zI - A(s)) -1 d'(s)(zI
-
d(s)) -1
114
3. Evolution Equations and F E M
arises with the following terms" Bl(t, 8)
=
Ao(t) -1/2 ( z I - Ao(t)) ( z I - A(t)) -1Ao(t) 1/2 9A o ( t ) - l / 2 A ' ( t ) d o ( t ) -1/2. do(t) 1/2 ( z I - A ( t ) ) -~ do(t) 1/2 9Ao(t) -1/2 [A(t) - A(s)] Ao(t) -~/9
9Ao(8) 1/2 ( z I - A(8)) -1 ( z I - A o ( s ) ) d o ( s ) -~/2, B2(t, s)
=
mo(t) -1/2 ( z I - Ao(t)) ( z I - A ( t ) ) -1 do(t) ~/2 9Ao($) -1/2 [A'(t) - A'(s)] do(s) -'/2
A(8)) -1 ( z I -
9Ao(s) ~/2 ( z I Ba(t, s)
=
A o ( s ) ) A o ( s ) -1/2,
Ao(t) -1/2 ( z I - Ao(t)) ( z I - A(t)) -1Ao(t) 1/2 9Ao(t) -1/2 [A(t) - A(s)] Ao(s) -1/2. Ao(s) 1/2 ( z I - A(s)) -1Ao(s) 1/2 9d o ( s ) - ~ / e A ' ( s ) d o ( s ) - l / 2 ,
m o ( s ) ( z I - A ( s ) ) -~ ( z I - do(s))Ao(s) -Vg
Thus the lemma has been reduced to the following inequalities:
IlAo(t)-'/~ (zI
- Ao(t)) ( z I - A ( t ) ) -~ Ao(t)l/2ll <_ C,
IlAo(t)-l/2A'(t)Ao(t)-l/21l
<_ C,
< C It - ~1,
IlAo(t) -1/~ [n(e) - A(~)] no(~)-'/=ll
IIAo(t)-'/2 (zI - n ( t ) ) - ' (~I - Ao(e)) no(e)-'/=ll < C,
IlAo(t)
A'(~)] nol/=ll _< C l t - ~ l
~
They are direct consequences of (3.23), (3.25), and the following inequalities:
IA(t)l v,v, ~ C,
C -1 ~
C -1 ~ IIA0(t)Iv, v, < C,
I A'(t)llv, v, <_ C, A(t) - A(s)llv, v, < C I t - sl,
IIA'(t) - A'(.s)l v,v, < C I t - ,s ~. Details are left to tile reader. Now we show the following inequalities as consequences of Lemmas 3.3 and 3.1, where
9 ~ (0,1/2). A ( t ) -O~ A ( t ) -~ (9
(zI - A(t))
-1
0
- ~
(3.60)
<_ C o ,
( z I - A(s)) -1
A ( t ) o f f - - T A ( t ) - ' - A ( s ) o "ff-~A(s)- ~
< C It - s] ~ , < CI~ It -,sl "
(3.61) (3.62)
3.2. A Priori Estimates Proof of (3.60):
115
Inequality (3.26) (with a = 1) gives
l]A(t) ~ [A(t + e) -~ - A(t) -z]
tl -<
and hence (3.60) follows because
ate3A(t)_ ~ ---- 27rzl/r z-~-~c3 (zI -
A(t)) -1
dz
exists. P~oof of (J.61): If H is a positive definite self-adjoint operator in X, its spectral decomposition assures
IIH1/~ (zI - H)-~I _< c for z E E01. Inequality (3.61)follows from (3.59).
Proof of (3.62):
Noting
cOA(t)Z . A(t)_Z ' A(t)Z at~---A(t)-~ = cot we have
A(t)~ ff---iA(t)-~- A(s)~ ff---~A(s)-~ = - ata a(t)~. A(t)_ ~ + ff__~A(s)~.A(s)_ ~
[-~A(t)Z- osOA(s)Z] A(t)-z + CgA(s)3" A(s)-Z [IInequality (3.62) is reduced to
[ff---~A(t)~- ~sA(S)~J A(s)-~[[ < C~ t-s[ ~.
(3.63)
We have assumed the boundedness of A(t) so that (2.18) holds for any v E X. It follows
that
8 A(t)Zv =
sin 7r/3 ]i~176Zc3
7r
1
tt -~ (#I + A(t))- v dl~.
If we apply Lemma 3.3, we get
sin7cflTr/o~176
(B-u(t's)A~
(pI + Ao(s))-lv, Ao(t) 1/2 (pI + Ao(t))-lx).
3. Evolution Equations and FEM
116 Inequality (3.63) follows from
( OA(t)~-~
--~sOA(s)~l C
(fOOO#Z3 [Ao(8)1/2
( # I --t-
A0(8))-1 '/)112)1/2
IlAo(t) ~/~ ( . I + Ao(t)) -~ ~[l~ d.
:
It -
sl ~
c~ IIA0(~)~llx. I~llx" t - ~1~
and (3.25). We are able to complete the proof of Lemma 3.2.
Proof of (3.58): We have Oq e_(t_s)A(t) A(t)-ZR~ (t, s) = - A ( t ) - z - ~0qe -(t-~)A(t) - A(t)-Z-~s
=
O -~
_
(A(t)_ze_(t_s)a(t) ) + A ( t ) l - . e_(t_~)a( t)
0__ A + ot ( t ) - "
=
e
-(t-s)A(t)
1 Ot 0 f_ 2m r z-Ze1
27r~
(t-s)z
(zI - A(t)) -1 dz
Jfr Z 1-~e-(t-~)z (zI
--
A(t)) -1 dz + -~ 0 A(t)_ ~ 9e_(t_s)A(t)
1 f~ z-Ze -(t-~)zO (zI - A(t)) - l dz + o__ Ot A (t)-z . e-(t-~)A(t) 2m Ot Therefore, the equality =
16 A(t)-~Rl(t's) - A(r)-~Rl(r's) = E Ze(t'r's)
g=12
holds with
Z12(t, r, 8) =
1 ofr z-~e -(~-~)z [e -
zla(t, r, z) -
1 Jfr 27rz
~e-(~-~)z [0-o-i (z: - A(t)) -1 - ~0 (ZI-- A(t))-lJ d~,
Z14(t,r,z)
=
Z15(t, r, Z)
:
Z'6(t, r, s)
=
0-t0A(t)_ ~ " A(t);~ -~r0 A(r)_ ~ . A(r)f3] A(t)_~e_(t_s)A(t) ~O
A(r)_ ~ A(r);' [A(t)-Ze -(t-s)A(t)- A(t)-Zc-("-s)A(t)],
O A ( r ) - " . A(r) ~ [A(t)-/~e -(r-s)A(t) - A(r)-~ -(r-s)A(r)] Or
3.2. A Priori Estimates
117
Here, we have ~
,•'0
<
C
# - z e -(r-s)'c~176 9( t - r)# d# #
--
C/3(t-
1")(/--
<
C
p - Z e -(r-s)ac~176 . (t - r ) a d #
8)/5-1
and
IJZ'a(t,
r, s)
j~0~176 c~(t- ,-)~(r- s)~-'
:
Taking the adjoint operators in (3.62), we have
cot (t)-Z " A ( t ) z
A ( s ) - Z " A(s)Z
< Cz
It
- sl
This implies
]lz'4(t, ~, s)JI ~
A ( t ) - ~ . A ( t ) ~ - -~rA(r) -~ 9A ( r ) ~
IIA(t)-~ll. 9 II~-('-~)A(~)ll s) ~.
_< G(tSimilarly, we have
}I~ at
(t)-~ " A ( t ) z
-
and hence
9
[[A(t)-l~e-(t-s)A< t) _ A(t)-Ze-(r-s)Att)[[
C/~(t- r)(? ~ -- 8) ~-1 hdds by (2.19). Finally, inequality (3.56) is valid for/3 > - 1 and therefore
[IA(t)-Ze-(r-s)A(t) _ A(r)-~e-(~-sla(r)ll <
c~(t-
~)(,~-
~)~-'
follows. The proof is complete. Modifying the proof of Lemma 3.2 slightly, we have (3.54) with/3 = 0 and Zo(t, r, s) satisfying IIZ0(t, ~,
s)ll
<_
c ~ ( t - r)~(r - s) -~
for 0 < ~ < a. Combining this with (2.20), we get IIA(t)U(t, s) - A ( r ) U ( r , s)ll < G ( t
- r)~(r - s) - ~ - 1 + C~(t - r)~(r - s) -~
(3.64)
for 0 _< s < r < t _< T, where/3 E [0, 1] and ~c E [0, a). The proof is simpler and left to the reader.
3. Evolution Equations and F E M
118 3.3
Semi-discretization
As is described in w parabolic equation (3.1) with (3.3) (or (3.4)) and (3.2) is reduced to evolution equation (3.5) with (3.6). In the same way as in the preceding chapter, this equation is discretized with respect to the space variables x = (xl,x2). We triangulate f~ into small elements with the size parameter h > 0 and denote by Vh C V the space of piecewise linear trial functions. As before, Xh denotes Vh equipped with the L 2 topology. The operator norm II " Ilxh,x, is written as I1" II for simplicity. The m-sectorial operator in Xh associated with r h is denoted by Ah(t). Finally, Ph : X ~ Xh is the orthogonal projection. Then, the semidiscrete finite element approximation of (3.5) with (3.6) is given by duh --+ dt
Ah(t)Uh = 0
(0 < t < T)
(3.65)
with Uh(0) =
Phu0
in Xh. Because dim Xh < +oo and Ah(t)'s are smooth in t, they generate a family of evolution operators {Uh(t,S)}o<s
] Ah(t)Uh(t, S)ll + IlUh(t, S)Ah(S)ll <_ C ( t - s) -1 and
(3.66)
IIuh(t,~) l < c
hold uniformly in h for 0 < s < t < T. Here, the theories Fujie-Tanabe and KatoSobolevskii are applicable. The theory of Kato-Tanabe works also from the argument in w We turn to the error estimate for scheme (3.65). Employing the method of Helfrich, we show
Ileh(t)l x < Ch'2t-' '~011x
(0 < t < T)
for eh(t) = u ( t ) - uh(t), and extend the similar result of w introduced the error operator Eh = Eh(t, s) by
(3.67) For this purpose we
G ( t , ~) = u(t, s) - Uh(t, s)P,~. Obviously,
~h(t) = E,.(t. 0)..0 holds and inequality (3.67) is reduced to
IIG(t, s)ll < c h 2 ( t - s)-'
(o < s < t <_ T ) .
(3.68)
3.3. Semi-discretization Calculations of w equality
119 are valid even for temporally inhomogeneous generators. From the
O [Uh(t, r)PhEh(r, s)] = Uh(t r)[Ah(r)Ph - PhA(r)] U(r, s) Or follows
PhEh(t, s) =
~st Uh(t, r)[Ah(r)Ph
- PhA(r)] U(r, s) dr.
Introduce the Ritz operator Rh(t)" V ~ Vh through the relation (~r
x;, ~ v.).
(3.69)
The equality
Ah(t)Rh(t)v = PhA(t)v
(v E D (A(t)) C Vh)
holds similarly to (1.24). We have
Eh(t, s)
= =
[I - Ph] Eh(t, s) + PhEh(t, s) E~(t, ~) + EX(t, ~) + e~(t, ~)
with
E~(t, s)
=
E~(t, s)
=
E3(t, s)
=
/t
[I - Uh(t, S)Ph] [I -- Rh(t)] U(t, s), Uh(t, r)A~(r) [Rh(t) -- Rh(r)] U(t, s) dr, Uh(t, r)Ah(r)Ph [I - Rh(r)] [U(r, s) - U(t, s)] dr.
It sumces to show that
IIE~( t, ~)ll -< c h ~ ( E s t i m a t e of
t -
8)-1
( ( = 1,2,3).
E~(t, s)
We have shown inequalities (1.38) and (1.42) for v E V N H2(~):
II[Rh(t) - I] ~llv
<
II[Rh(t) - I] ~llx
<
Ch Ilvl ~.(~),
Ch~ll'~ll_,-,~(~).
Therefore, from the elliptic estimate follows that
[IE~(t,s)ll
~_ (I + l Uh(t,s)l I . IIPhll) . l[[I- Rh(t)]A(t)-lll . llA(t)U(t,s)l I ~_ C h 2 ( t - s) -1
(3.70) (3.71)
3. Evolution Equations and FEM
120
Estimate of E~(t, s) We shall show the inequality
(3.72)
ll[Rh(t)-/~h(S)]Vllx <--Ch 2 It- sl ~ IvllH~(a) for v E V N H2(f~). Then it follows that IIE~(t,s)ll
_
IIUh(t,r)Ah(r)ll
tl[Rh(t) - R~(~)] A(t)-' l[" IIA(t)U(t, s)ll d~ <
c
( t - s)-l+~h~d~. ( t -
= Ch2(t-
~)-~
Ch2(t- s) -1. To show (3.72), we recall the adjoint form .A~('u, v) = .At(v, u), and denote by/~h(t) the 8)-1+ c~ ~
Ritz operator associated with it:
.,4; (,Oh(t)~, ~:h) : A;(v, ~:h)
(,, ~ v, ~,, c v,,).
The inequalities
ll[/~h(t)--/]vlIv~ ChlvllH~(~)
and
I][~h(t) -/]vllx ~ Ch21v IH~(nl hold similarly to (3.70) and (3.71), respectively, where v C V N H2(~2). Setting z = [Rh(t)- Rh(s)] v E Vh, we have
llzll
= A, (z,
A, (z,
=
(At - As) ([1 - Rh(s)] v, [~h(t)A(t)*-lz)
=
( A t - Ms)([1-- Rh(S)] V, [/~h(t)- I] A(t)*-lz) + ( A ~ - A s ) ( [ I - R/~(s)] v, A(t)*-lz)
= ( A t - A s ) ( [ I - Rh(s)]v, [/~h(s)- I] A(t)*-lz) + As ( [ I - Rh(s)] v, [A(s) * - 1 - A(t) *-1] z)
:
(,At- ,As)([I- Rh(8)] "u, [il~h(8) - I]
A(t)'-lz)
+ .As ([I - Rh(s)] v, [I - Rh(s)] [A(8) *-1 - A(t) *-1] z)
_<
+ c I I [ / - Rh(s)] vllv I[[1 - ~h(S)] [A(s) * - ] - A(t) *-]] 6' It - sl ~ h 2 Ilvl g=(~) IA(t) *-'zl ,,=(.) + Ch 2 Ilvlln=(~) l] [A(S)*-I -- A(t)*-l] ~11.~(~)-
zll~
3.3. Semi-discretization
121
The elliptic estimate implies
IIA(t)*-x~llH~(~) ~ c Ilzll~ and
II [A(s) *-1 -
A(t) *-1] ZlIH2(a) -< C It - sl ~ Ilzllx.
Inequality (3.72) has been proven.
E s t i m a t e of E3(t, s)
First Case
If V = H~(f~), the duality argument of Helfrich is applicable and inequality (3.22) follows. For Uo E D ( A ( s ) ) =_ D we have
jfst IIUh(t, r)Ah(r)ll.
SO ollx
I11I -/~h(r)] A(t)-l[[
9IlA(t)[u(t, s) - U(r, s)] A(s)-lll 9IlA(s)uollx ds <_ Co
=
(t - s)-lhg(t - r)~
- s)-~
IIA(s)u01 x
Ch 2 IIA(s)uollx
with 0 E (0, c~). We can show
for g = 1, 2 similarly, from the second estimate of (3.21). (Details are left to the reader.) We have
IIEh(t, s)A(s)-~ll < Ch 2.
(3.73)
Now, the semigroup property of evolution operators implies
Uh(t, r)Uh(r, s) = Uh(t, s)
and
U(t, r)U(r, s) = U(t, s)
for 0 _< s _< r < t _< T. The identity (3.74)
Eh(t, s) = Uh(t, so)PhEh(So, s) + Eh(t, so)U(so, s) follows with so = (t + s)/2. The second term of the right-hand side of (3.74) is estimated as
IIE~(t,~o)U(so, s)ll
<_ IJE~(t, so)A(~o)-~ll'llA(so)U(so, <
C h 2 ( t - s) -1
s)ll
3. Evolution Equations and F E M
122 by (3.73). On the other hand, (3.69) gives [[Uh(t,S)PhEh(So, S)II =
[IU~(t, so)Ah(So)Rh(So)A(so)-XEh(So, s)ll
< [Iuh(t, so)Ah(So)Ph [Rh(S0) - I]A(so)-lEh(So, S)] + ]lUh(t, so)Ah(so)PhA(So)-IEh(So, s)ll C(t- s) -1 {] [/~h(SO)- I] A(8o)-1 I 9IIE,~(~o, s)ll +llA(so)-lEh(so, s)[I}. Because IIEh(So, s)ll ~ c follows from (3.66), inequality (3.68) is a consequence of
]A(t)-lEh(t,s)l
<_ Ch 2.
(3.75)
To prove this inequality, let
U(t, s) = U ( T -
s, T -
t)*
(Ih(t, s) = Uh(T - s, T - t)*.
and
Then,
{U(t,S)}o<s
{U~(t,S)}o<s
and
are nothing but the families of evolution operators generated by A(t) - A ( T t)* in X and Xh respectively. The relation
Ah(t) = A h ( T -
l~h(t, s) = l~l(t, s) - [Ih(t, S)Ph = E h ( T - s, T - t)* follows. Because
I1E,,/,sIAl,/ 11 is proven similarly to (3.73), we get (3.75) as
I[A(t)-lEh(t,s)[I E s t i m a t e of
Eta(t, s)
= ]lEh(t,s)*A(s)*-l][
<_ Ch 2.
Second Case
If V = H I ( ~ ) , D ( A ( t ) ) varies as t changes. We cannot expect the estimate
IIA(t)U( t, s)A(s) -1]1 <- C or IIA(t)[U(t, s) - U(r, s)]A(s) -1 II -< Co(t - r)~ In this case, we apply the method of telescoping as
5
(t- ~)E,~(t, ~) : ~ F~(t, s) g=l
- s) -~
t)* and
123
3.3. Semi-discretization
with the following terms:
fs t (r - s)Uh(t, r ) A h ( r ) . Ph [I - Rh(r)] [U(r, s) - U(t, s)] dr,
F~(t, s)
=
F~(t, s)
=
(t - r)Uh(t, r ) A h ( r ) " [Rh(s) -- R~(r)] [U(r, s) - U(t, s)] dr,
F~(t, s)
=
(t - r)[Uh(t, s ) A h ( s ) -- Uh(t, r)Ah(r)]
9Ph [I - Rh(s)] [U(r, s) - U(t, s)] dr,
f'
(r -- s)[U(r, s) - U(t, s)] dr,
F2(t, s)
=
--Uh(t, s ) A h ( S ) " Ph [I -- Rh(S)]
F2(t, s)
=
Uh(t, s ) A h ( s ) . Ph [I -- Rh(S)] (t -- s)
[U(r, s) - U(t, s)] dr.
We have to show that [IFeh(t, s)l I _< C h 2
(g = 1, 2 , . . . , 5).
(3.76)
Recall that Kato-Tanabe's generation theory is applicable to (3.5). Combined this with the elliptic estimate we can show the following for 0 _< s < r < t _< T, where 0 _
IlU(t, 8) - U(r, 8)IlL2(FI),H2(~2) ~_ C k { ( t - T)~(T -- 8) -~-1 -1- ( t - T)~(T -- 8)-1} , IlUh(t,r)Ah(r) -- U , ( t , s ) A ~ ( s ) l l
<_ Ck { ( t - r ) - ~ - Z ( r - s) ~ + ( t -
/s'
[U(t, s) - U(r, s)] dr
I[
L2(f~),H2(f~)
(3.77)
r ) - l ( r - 8)a},
(3.78) (3.79)
< C.
Those inequalities are proven in the following way. Proof of (3. 77):
Take u0 E L2(f~) and set u(-, t) = U(t, S)Uo. We have B(x,t,D)u(x,t)
=0
on 0f~ where 13 = O/Ovc + ~. This implies B ( x , t, D) (u(x, t) - u(x, r)) : [B(x, t, D) - B ( x , r, D)] u(x, r)
there. Similarly, we have
[A(t)U(t, s) - A(r)U(r,
s)] uo(x)
+ [s
r, D) - s
t,
D)] u(x,
r)
124
3. Evolution Equations and F E M
in f~. The elliptic estimate gives
II[U(t, ~) - u(~, ~)1 ~olIH~(~) -- I1~(', t) -- ~(., ~)11,~(~) <_ C I~(', t, D) (u(., t) - u(-, ~))11~(~) +C ILL(', t, D) (u(., t) - u(., ~))11,,,~(o~) <_ C I I [ A ( t ) U ( t , ~) - A(,~)U(r, ~)] ~o11~(~) +C I [L(', ~, D) - s t, D)] u(., ~)11~(~) +C lilt3(', ~, D) - B(-, t, D)] u(., ~) I1,,,,~(o~) <_ C I I A ( t ) U ( t , ~) - A ( r ) U ( , ' , ~)11" luoll~(~) + C ( t - ,~) I1~(', ~)11~(~) <_ C IIA(t)U(t, ~) - A(,~)U(, , ~)11 I1'~o11~(~) + C ( t - ,~)(~- ~)-' I1~o11~(~) Inequality (3.64)implies (3.77).
[]
Proof of (3. 78)" This is nothing but the adjoint form of (3.64) applied to A(t) = Ah(t) and U(t, s) = Uh(t, s). [] Proof of (3. 79): Letting u(., t) = U(t, S)Uo, we have s
s, D) (u(x, t) - u(r, x)) = [s
+ [s
x, D) - E(s, x,
s, D) - s
t, D)] u(z, t)
D)] u(z, r) + [A(t)U(t, s) - A(r)U(r, s)] to(Z)
in t2 and B(x, s, D) (u(x, t) - u(x, r)) : [B(x, s, D) - B(x, t, D)] u(x, t) + [B(r r, D) - B(x, s, D)] u(r r)
on cgf~. The elliptic estimate now gives [U(t, s) - U(r, s)] no dr
=
('u(-, t) - u(., r ) ) d r
H2(U)
<_ C(t - s)II[E(., s, D) - s
+C
f'
I[s
r, D) - s
H2(~)
t, D)] u(.,
t)JlL~(~)
s, D)] u(-, r)llL2(U ) dr
+C(t - s)I [~(., s, D) - B(., t, D)] u(-, t)ll,,/~(on) +C
[A(t)U(t, s) - A(r)U(r, s)] uo dr
II
L'~(n)
<
C ( t - s) 2
Ji'u(-, t)llu~(n) + C
(r -s)tl'u(,
r)liu~(~)
dr
~)11 ~o11~(~) + C IIU(t, ,~) - I I. I1~,o11~(~) <_ C ( t - ~) I uoll~,(~) + C I uoll~(~) ___C I1,,,o I~(,) 9 +c(t
- ~)IIA(t)U(t,
3.3. Semi-discretization
125
The proof is complete. Now the proof of (3.76) is given as follows. First, with/3 E (0, 1) we have
IIF ( t, )ll -< <
/s
- s)Ilgh(t,
9I I I - R~(~)ll.~(~).~(~)"
C~h 2
IIU(~. ~) - u ( t . ~)l ~ ( ~ ) , ~ ( ~ ) dr
8 ) ( t - T) -1
(r9{ ( t -
<
r)&(r)l
T)~(T -- 8) -~-1 + ( t -
T)~(r -- s) -1 } dr
Ch 2
and
jfs t (t - r)IIUh(t, r)Ah(r)ll
~
IIFZ(t, s)ll
U(t, ~)llL=(~),H~(~)dr
9IIRh(~) - Rh(~)l H=(~),L=(~)" IIU0", ~0 -
_< c ~ h ~ fS t (~ - ~) {(t - ~)~(~ - ~)-~-~ + (t - ~)~(~ - ~)-1} a~ <
C h 2.
Next, with 0 < "i' < ~ _
_<
r) l U h ( t , r ) A h ( r ) -- Uh(t,s)Ah(s)l]
(t-
9III -
R"(S) IIH=(~),L=(~)"IIU(t,
~) - U(~', ~) IIL=(~),H=(~) d,"
( t - T) { ( t - r ) - l - ~ ( r - 8) ~ Jr- ( t - r ) - l ( r -
C~h 2
8) ~}
{ (t - T)7(r - 8) -1-3' -Jr (t - T)m(r -- S) -1 } dr <
C h 2.
W i t h / 3 c (0, 1), we have
I[F2(t, )ll
Ilun(t,
s)&(s)ll.
9
<_
<
III-
Rh(S)IIH=(~),L~(~)
IIg(t, s) - U(r, S)[IL2(a),H:(a)" (r -- s) dr
C~(t- s)-~h 2 9
{ (t-
~)~(~ - ~ ) - ~ - 1 + ( t -
_<
IUh(t,s)&(s)
~)~(~-
~ ) - 1 } (~ _ ~) d~
C h 2.
Finally, we have [IF2( t, s)ll
9 <
C h 2.
9liRa(s) - IllH~(~),L~(~)
[u(~, ~) -
u(t,
~)] d~
L2(~"/),H2(n)
3. Evolution Equations and F E M
126 Summing up those estimates, we obtain
IlE2(t,~)ll <_ ch~(t_ ~)-, All the proof for inequality (3.68) is complete.
3.4
Full-discretization
Fully discrete approximations are obtained by discretizing the semidiscrete equation
duh dt
+ Ah(t)Uh = 0
(0 < t <_ T)
with
Uh(O) = Phuo
(3.80)
in the time variable t. In the present section, we adopt the backward difference method with the mesh length r > 0 satisfying T = N r . T h a t is,
U~h(t + r) -- U~h(t)
+ Ah(t + r)u,~(t + r) = 0
(t = n r )
(3.81)
in Xh with
~,~(o) = P~,o, where n = 0, 1,-.- , N. Thanks to (3.14), tile scheme is uniquely solvable and
e~h(t) = u h ( t ) - U,;(t) denotes the error, where t = nr. we shall derive
Id,(t)llx _< c~-t-' luol x and extend a result in w
(3.82)
Combining (3.82) with (3.67), we obtain
u(t) - u;(t)l x < C (h? + r) t-~ I 'uo Ix
(t = n r ) .
(3.83)
Let t,~ = nT and set
U[~ (tn, tj) =
{ ( / h + TAh(tn)) -1 (Ih + TAh(tn-1)) -1"'" (Ih + TAh(tj+l)) -1 Ih.
We have ~,;(t) = ui:(t, o)&~o
(t = t,,)
and
,,,,, ( t ) = u,, ( t, 0)P,,,~,o.
(~ > j) ( ~ = j).
3.4. Full-discretization
127
Drop the suffix h for simplicity of writing. Because of (3.80) and (3.81), we have for t = tn that
e ~(t + r) - e r(t)
tnt-r
f f
=
[A(t + r)u r(t + r) - A(r)u(r)] dr
at
=
t+r
[A(t + r)u(t + r ) -
A(r)u(r)] d r - r A ( t + r)e" (t + r).
,It
Hence
er(t -+- 7) = (1 + r A ( t + "r) )-l er (t) + (1 + r A ( t
+
T)) -1
f
t+r
[A(t + r)u(t + r ) -
A(r)u(r)] dr.
Jt
Because of e ~ (0) = 0 we obtain
e ~(tn)
-- E ~(tn)Puo =
(I q- 7A(tn)) -1 (I + "rA(tn_l)) - 1 . . . (I + TA(tk)) -1 k--1
-1
9[A(tk)U(tk, O) - A(r)U(r, 0)] drPuo,
(a.s4)
Er(t) being the error operator: E ,(t) = u(t, o)P - u ~(t, o ) Inequality (3.82) is reduced to those on the stability of the approximate operator U r (tn, tj+l) and the smoothness of the original one A(t)U(t, s) - A(r)U(r, s). We have proven Lemma 3.2 for the latter. As for the former, we have the following lemma, proven similarly to the continuous case: L e m m a 3.4. For each/3 E [0, 4/3), the inequality
II
tj)A(tj+l)Zll <__Cz(tn - tj) -~
(a.85)
holds true. Proof." Taking adjoint form reduces inequality (3.85) to I A(tn)ZU~(tn, tj)l] <__Cz(tn
-
tj) -~,
of which continuous version is proven in w
IIA(t)ZU(t, s)ll<_ Cz(t- s) -z All we have to do is to trace computations
in the context of discreteness.
(3.86)
3. Evolution Equations and F E M
128
In fact, inequality (3.26) holds for A(t) = Ah(t) uniformly in h as is described there. Identity (3.30) holds for A(t) = Ah(t) so that we obtain
U'(tn, tj) - (1 + rA(tj)) -(n-j) =
~
[(1 + ~-A(t~))-(~-k)U'(tk, t j ) - (1 + 7A(t~))-(n-k+l)U'(tk_x,tj)]
k=j+l
=
~
(1 +7-A(tn))-(~-k+l) [(1 +~-A(tn))- (1 +~-A(tk))]U'(tk, ty)
k:j+l
= T ~ (1 + rA(tn)) -('~-k+~) [A(tn) - A(tk)] U~(tk,tj) k=j+l = ~p=l
(1 + 7A(tn))-(n-k+UA(tn)l-PPD(tn, tk)A(tk)PPVr(tk,tj).
~
(3.87)
k=j+l
Given operator-valued functions fit = Ke(tn, tj) with t~ = 1, 2 on
D r={(t~,tj) I N>_n>_j>_O}, we define K = K1 .r /42 by n--1
(K1 *~ K2) (tn, tj) = ~- Z
Kl(tn, tk)K2(tk, tj).
k=j+l
Furthermore, we set
WT(t~,tj) Y~(rn, tj)
= Ur - (1 + TA(t~)) -(~-j) = A(t~)qPWT(t~,tj).
(3.88) (3.89)
Then, equality (3.87) reads: m
Yq = E
H q~p , ~- U + Yq,~O,
(3.90)
p=l
where
Hq~p(t~, tj)
= A(tn)I-pP+qP(1 + ~-A(t~))-(~-J+l)D(t~, tj),
(3.91)
m ,0
=
,p
,- 1
(3.92)
p=l
with
Yp~_l(t,,,tj) = A(tn)PP(1 + TA(tn)) -(''-j)
(3.93)
Note the elementary inequality (3.94). The desired inequality (3.86) can be derived in a similar way to that in w
3.4. Full-discretization
129
Details are left to the reader, but the following inequality is worth mentioning, where
O < a <wb a n d a < l :
B N(a, b) - -~1 k=l
1
- -~
< S(a, b) =-
/o (1
-
-
x)a-lxb-ldx.
(3.94)
In fact, f(x) = (1 - x)a-lx b-1 is monotonically increasing in [0, 1) when b _> 1 _> a, while f(x) is convex in (0, 1) when a, b _< 1. Those facts imply (3.94). [] We are able to give the following.
Proof of (3.82):
The operator E ~ in the right-hand side of (3.84) splits as
n~k E~(tn) =
(1 + ~-A(tn)) -1... (1 + TA(tk))-lA(tk)[e -tkA(tk)- e -rA(tk)] dr
E k=l
-1
(1 + rA(tn)) -1... (1 + rA(tk))-lA(tk)~Z~(tk, r, O) dr
+ k=l
=
-1
(1 + TA(tn))-lA(tn)
[e-tnA(t~) - e -rA(t~)] dr --1
+
Ur(tn, tk_l)A(tk)[e -tkA(tk)- e -ra(tk)] dr k=l
-1
~
+~
k=l
U'(tn, tk_l)A(tk)ZZz(tk, r, O) dr.
(3.95)
-1
Inequality (2.19) is applied to A = Ah(r) uniformly in h. We have for n > - 1 that
[IA(r)~[e -tA(~) -~-~A<~>]II C (t- s)s -n-l,
(3.96)
where 0 < s _< t < oc. Supposing n > 2, we can estimate the first term of the right-hand side of (3.95) as
[1(1 + TA(t~))-lA(tn)ll
~
n
lie -t~A(t~) - e-~A(t~)ll dr --1
CT
( t n - r)r-~dr <_ CT-1T2(tn-1
-1 --1
~ Crt-
3. Evolution Equations and F E M
130
Next, by Lemma 3.2, the third term of the right-hand side of (3.95) is estimated as
k~l
Ilg~(t~,t~-l)A(t~)z[I 9IIZz(tk, r,O)ll dr --1
< C a k1= ~
-, (tn - tk-l)-Z(tk -- r ) r - l + Z d r
n
___ c~ ~ ( ~ -
k + 1 ) - ~ - ~ ( k ~ ) ~-'
k=l n = c~
Z(~
- k + 1)-~k ~-1
k=l Inequality (3.94) now gives that
~~ IIg'(t~,
tk_l)A(tk)Zll 9]lZz(tk, r, O)l I dr <_ CT. k=l The desired inequality (3.82) has been reduced to
fi'
,11
To see this, take/3 C (0, 1/3). We get
n En-1 o~tl k U~(tn, tk_l)A(tk) k=l
[e - t k a ( t k )
- e -~A(tk)] dr
-' n-1( n = E k + 1)UT(t~, tk_l)A(tk) 1+'. ~tlk A ( t k ) - ' [e -tkA(tk) --e -rA(tk)] dr k=l -1 n-1
+ E(k
- 1)UT(tn, tk_l)A(tk) '-~
k=2 9
A(tk) ~ [e -tkA(tk) - e -rA(tk)] dr.
(3.98)
-1
The first term of the right-hand side of (3.98) is estimated by Lemma 3.4 and (3.96) as
n-1 ~-~.(n-k+ 1) llUT(t,~,tk_~)A(t~)'+Zll. ~t.lk IlA(tk) -~ [et~A(t~)--e-~A(t~)]l I dr k=l
-1
_< Cr Z ( r t - k + 1)-'T - 1 - ' ~t.lk (tk -- r)r'-idr k=l -1 n-1
rt-1
< C/3 Z ( ~ 9 , k=l
]~:-J-- I)-~qT -1-/~ 9T2(~T)/9-1
rt
:
Ca E ( n k=l
If + 1 ) - ' k '-1 _< C.
131
3.5. A l t e r n a t i v e A p p r o a c h
Similarly, the second term of the right-hand side of (3.98) is estimated as
n-1 E(kk=2
1)IIU~(t~,tk_l)A(tk)l-~ll.
/i k
IIA(t,)~
[~-~(~)- ~-~(t~)] II d~
-1 n-1
)-1+/~
_< C 0
( k - 1)(tn - t k - ,
k=2 rz-1 _< c , ~ ( k k=2 n-1 < Cr E ( k k=2
-/~-1 9
(tk - r ) r
dr
-1
- 1 ) ( ~ - k + 1) -1+~ 9~-1+~. ~ . ((k - 1)~) -~-~
- 1)-r
k -+- 1) r
< C.
The proof is complete.
3.5
Alternative
Approach
Inequality (3.83) is proven in a different way by the energy method. In this section, we describe the arguments of M. Luskin and R. Rannacher for the semidiscrete approximation and M.Y. Huang and V. Thom~e for the fulldiscrete approximation, respectively. A Priori Estimates
First, we show some estimates concerning the solution of the equation du d--7 + A ( t ) u = f (t)
with
(0 < t < T )
u(0) = ,to
(3.99)
in a Hilbert space X over R. Here, A ( t ) is an m-sectorial operator associated with a bilinear form At( , ) on V x V through a triple of Hilbert spaces V C X C V*. We suppose
I.,4t(,< ~)1
<
C1 Ilu Iv IIv Iv
and
.a~(~, ~) > 5 II~ll~-
(3.100)
for u, v E V, where C 1 > 0 and 5 > 0 are constants. It is also supposed to be smooth so that oq A (u, v)
and
02
exist and satisfy
IAt(,
and
A~(,~, ~)1 ~
c
I1~11~Ilvllv
132
3. Evolution Equations and F E M
for u, v C V. Problem (3.99) admits the weak form (Ut, ?J) + .At(u , v) ~- (f(t), v)
(3.101)
for v C V, where ( , ) denotes tile inner product in X. This implies (Utt , 72) nt- .At(ttt, v) -- - A t ( u ,
v) -nt- (It(t), v).
(3.102)
/0
(3.103)
We have the following. P r o p o s i t i o n 3.5. The inequality
Ilu(t)ll 2x +
/0
Ilu(s)llv~ ds _< C I1~oll ~x + c
IIf(s)ll~.ds
hold, fo~ th~ ,ol~t~o~ ~ = ~(t) of (S.99).
Proof: Putting v = u(t) in (3.101), we h~ve ld
2 dt Ilu(t)ll~ + 5 Ilu(t)ll~ _< Ilf(t)llv. Ilu(t)llv by (3.100). Then, Schwarz's inequality gives
-
and
d dt Ilu(t)ll2x + l u(t)llv2 < C Ilf(t)l 2v. -
_
inequality (3.103) follows.
[]
Letting .A;(u, v) = .At(v, u) and A ~ = (.At + .At)/2, we suppose that
v) + (B(t)u, v)
.At(u, v) = A~
(3.104)
holds with a bounded linear operator B(t) : V ~ X. This assumption is satisfied if .At(, ) is associated with a second-order elliptic operator with real coefficients. We have the following. Proposition
3.6. The inequality
ilu(t)ll 2v +
/0
Ilu~(s)llx2 ds _< C Iluoll 2v + C
/0
f(s)l
x
ds
holds for the solution u = u(t) of (3.99). Proof: Putting v = ut in (3.101), we have
Ilu~ I~ + A~(u, u~)
= (f,
Ut)
< --
1 2 1 ~ II/11X + 2 Ilu~ I~
"
On the other hand, equality (3.104) gives
1d
1
.At(U, Ut) = -~--~.At(U, U) -- = A t ( U , U) nrZ
1.B.t.u,( ( )
Ut)
Q
(3.105)
133
3.5. Alternative Approach
Those relations imply lutll x~ + ~dA , (u, u ) <_ C I/ll ~x + c I1~ Iv. Then, inequality (3.1o5) follows from (3.100) and (3.103). P r o p o s i t i o n 3.7. The inequality Ilu,(t)l x~ +
/o
I~,(s)llv~ ds _< C Ilu~(0)ll x2 + C l'~o I.~
(If(s)I~,. + II/~(~)11~v*) ds
+ C
(3.106)
holds for the solution u = u(t) of (3.99). Proof." Putting v = ut in (3.102), we have
ld 2 dt I1~'11~ + 61~,11~, < -<
c II~llv I1~ Iv + f~llv. I~1 v 1
-~11~ ~v + C' I1~ Iv~ + C IIf~llv* 9 2
This implies
Ilu~(t)ll ~x +
/o
Ilu~(sDllv~ds_< II~,(o)ll~ + c'
/o
II~(~)ll ~vds +
c
/o
II/t(s)ll~.
ds.
Then, inequality (3.106) follows from (3.103). P r o p o s i t i o n 3.8. The inequality
/0
s ~ Ilu,(s)ll~, ds _< C Iluoll.~ + c
ji
(ll/(s)[Iv. + II/~(s)ll 2v.) ds
holds for the solution u = u(t) of (3.99). Proof: Putting v = t2ut in (3.102), we have
ld 2 dt (t2 I]utl]~.) + 5t 2 Ilu, ll ~v < - ? A , ( ~ , u,) + t I1~11x~ + t2(ft, ut) -
-
v + c II~llv + t II~ll~ + ~t ~ II~,ll~, + c II/~ll 2V. 9
This implies d~ (t2 I[ut[ ~) +
I[ut][y _
g + t II~,llx + c IIf~ll~.
(3.107)
3. Evolution Equations and FEM
134 and the inequality
/o
s ~ Ilut(s)ll*v ds _< C
/o t (ll~(s)ll~
+ s lut(s)12x + lift(s)Iv*) ds
(3.108)
follows. Next, putting v = tut in (3.101), we have
t i1~,11~x+_did (tAt(u,
u))
--
At(u, u) -t- .At(u, u) -t- At(tut, u) -t- (f, tut)
-<
~t ~ll~t ~v + c~ II~llv + c~ IIf I~.
for c > 0. The inequality
jot
II~t(~)ll ~ d ~ <_~
/~t
~ II~t(~)llv~ d ~ + C ~
jot (11~(~)11'v +
If(~)ll~.)d~
follows. Taking e > 0 small enough and combining this with (3.108), we have
/o s21lut(s)ll~ds c /o { Ilu(s)ll~ + II/(s)I~. + IIf,(s)I~. } ds. _<
Then, inequality (3.107) follows from (3.103). Proposition t llu,(t)ll~ +
3.9. The inequality
i
X
t
II~.(s)ll ~ ds <
c ilu,(o)ll x~ + c II~ollx~ + c
/o'
(llf~(~)llx~ + IIf(~)llv.) d~ (3.109)
holds for the solution u = u(t) of (3.99). Proof." Putting v
~-- Utt
in (3.102), we have
IlUttll 2X -Jr-,At (ut, utt) : -At (u, utt) + (ft, tttt).
(3.110)
Here, note that
12ddt.At(ut, ut) = < - At (.tz, tttt)
.At (ut, uu) + ~1 At (ut, ut) + 1 (But,
r
?ztt) + C I1..,11~
1
Utt)
2
d 9 ~-- - d--iAt (11., ?zt) -~- At (.~z,"~l.t) + At (ut, ut) <-
d dt
A,(,~, .,~,) + c I1,,, } + c
ul ~v ,
135
3.5. Alternative Approach and 1
We obtain
d d &(~, ~,,) -< C I~,1 ~ + CII~I ~ + CIIf, ll~ u.ll~: + aTA,(~,, u,)+ ~7 Multiplying t now gives that
'(
)
t lluttll2x + -~t A~(u~, u~)+ At(u, ut) -< c(t +
1)(llut]l
2v + I~1 ~ +
Ift I~)-
This implies
tA~(~,~) +
/o
~ll~,(411~ <_ c
/o
(ll~,(~)llt + 11,,(~)115 + IIf~(~)ll~) d~.
Inequality (3.109)follows from (3.103)and (3.106). Semidiscrete Approximation Let fZ C R 2 be a convex polygon, and A t ( , ) a bilinear form on V x V with V = H~(f~) (for simplicity) induced from a second order elliptic differential operator. The associated m-sectorial operator in X -- L2(f2) is denoted by A(t). We take the evolution equation
du + A ( t ) u = O (O < t < T) dt
with
u(0)=u0
with
uh(0) = PhUo,
and its semidiscrete finite element approximation
duh d--T- + Ah(t)Uh = 0
(0 < t < T)
described in w and w respectively. Now we develop the error analysis by the energy method. Recall t h a t Rh = Rh(t)" V ---+Vh denotes the Ritz operator associated with At( , ):
At(Rh(t)v, Xh) = At(v, Xh)
(v e V, Xh r Vh)
(3.111)
We have the following. L e m m a 3.10. Any v = v(t) c C 1 ([0, T] --+ V) satisfies the inequality
Ila~(v- nhV)llH,(~) <_ Ch ~-j (llv g~(~)+ IIV~IIH~-,(~>) fork
1,2 and j = 0 , 1 .
(3.112)
3. Evolution Equations and F E M
136
Proof." Equality (3.111) implies
.A~(a,(v - R,,v), x~) = -A,(,, - R,,~. x~)
(3.113)
for Xh E Vh, and hence
a l a~(v - R~v)ll~
~
.At (oqt(y - R h V ) , Ot(v -- f~h'U))
--
.At (Ot(V -- R h V ) , Otv -- Xh) -- A t ( v -- R h V , Xh)
___ c (lla,(v follows. T h e results in w
Rhv)liv Ila~v
-
- xhll,,,. + IIv - Rhvllv
Ilxhllv)
are summarized as
II(Rh(t)-
1)v g,(n) <--
Chk-Jll~llH~(n)
(3.114)
for k = 1,2 and j = 0, 1. Taking Xh = Rh(t)Otv(t), we get (3.112) for j = 1 and k = 1,2. The remaining case j = 0 is obtained by the duality a r g u m e n t similarly, in use of (3.113) and the elliptic estimate of A(t). It is left to the reader. [] Suppose u0 E V, and take the solution Uh = gh(t) E Xh of duh
dt
+ Ah(t)gh = O (O <_ t <_ T)
~(0) = R~(0)~0.
with
We have the following. Lemma
3.11.
The inequality
Z
~ II~h(~)ll~-d~ ~
Ch~l ~o11~"
(3.115)
holds for -eh = u -- Uh. Pro@
We have
(9~,,, x,,) + A~(~,,, xh) = 0 for Xh E Vh. Taking Xh = Rh(t)-eh(t), we get ld
2dtll~hll~c+ A,(~h,~h)
=
(0,~h, (1 - Rh)-~h)+ At (eh, (1 -- Rh)~h)
=
(ateh, (1 - Rh)'U) + .At (-eh, (1 -- Rh)'U)
(
<_ c~ h~ I1,~11~(~) + Ila,~,~llx
) +cllehllv
for c > O. This implies
d -dt
1~,,112 x + -~hl ~ _<
Ch ~
( I1~11~,-,~(~)
+ Ila~,,llx
)
137
3.5. A l t e r n a t i v e Approach
by (3.100), and hence
/0
II~h(~)ll ~V dx
<
Ch ~
<_ c h ~
I1~(~) .:(~) + IIO~h(~)ll
ds + II~h(o)llx
II~(s)
+ ch ~ ~ol follows. The elliptic estimate gives
Ilu(s)ll/_z~(r~) < c IIA(~)~(s)Ix : c Ilu,(~)l x, while inequality (3.105) of Proposition 3.6 is valid for u = Uh uniformly in h. The righthand side is further estimated from above by ch ~
(11~ol ~ + IIRh(0)~oll~) <
Ch ~
I1~o
and the proof is complete.
[]
If one assumes the inverse assumption, L 2 orthogonal projection Ph " X ~ X h satisfies
llPhllv_v <_ c
(3.116)
by Proposition 1.5. Then, the inequality
fo ~ II~h(~)tl ~V d~ < is proven in a similar manner, where eh = u assumed in the following lemma.
Ch ~
I1~o11~,
Uh. However, inequality (3.116) is not
L e m m a 3.12. I f uo E V A H k - J ( ~ ) we have
~0 t II~h(~)ll x~ d~ <-
c t l - J h 2k
Iluo
2
,
(3.117)
where k = l , 2 and j = 0 , 1 . Proof:
Given t > 0, we take the backward problems (v, O~w) - A~(v, w) = (v, e~(s))
(v 9 V)
with w(t) = 0 in X and
(x~. 0 ~ , )
- As(X,. ~ , ) = (x~. ~,.(~))
(x~ e v~)
with wh(t) = 0 in Xn, respectively. Those problems have unique solutions w = w(t) 9 C O ([0, t), D (A(t)*)) A C O([0, t ] , X ) n C ~ ([0, t ) , X )
(3.118)
3. Evolution Equations and F E M
138 and
~
= ~(t)
~ c ' ([o.t]. x h ) .
respectively. In spite of the inhomogeneous terms in the right-hand side, the argument in the proof of Lemma 3.11 is applicable with the time variable t reversed. A quick overview reveals that
/o'
w(~) - w~(~)l ~. d~ _< C h ~
/o'
I~,(~)1 ~ d~.
Combining this with inequality (3.105) implies that
L
' (llo~(~)-<
/o' 1'
(21a, w,,(~) ~x + 2 IIO, w(~)llx~ + h -~ IIw,.(~) - w(~)ll~) d~
< c --
h-2 II~(~) - ~ , ( ~ ) l l ~ ) ds
o.~,.(~)+
Ile,(s)ll ~ ds.
(3.119)
X
Taking v = eh(s) in (3.118), we have
II~,,(s)ll%
=
(e~(s),O~w(~)) - A~(~,,(s),w(s)) {(~,,. o ~ ( ~ - ~ , , ) ) - .4s(~. ~ - w,,)} + ( ~ , , . O s ~ , , ) -
..4~(~,,.w,,).
Here, eh = ( u - RhU)+ (Rhu- uh) and
(~,,. o~(w,,- ~)) - .4~(x,,. ~ , , - ~) = o holds for Xh E Vh. It follows that
I~,(~)11~
=
{(~-
~,,~. o ~ ( ~ - ~ , ) ) - A s ( ~ - R.,,~. ~ d + ~ (~,,. ~ ) - ( o ~ . w~) - .4~(~,,. ~ ) .
~,,)}
Furthermore, the equality
(O~eh, Xh) + A~(Ch, Xh) = 0 holds for Xh E Vh. W e have
I1~(~)11 ~x = ( ~ - R,,,~, o ~ ( w - w , , ) ) - A ~ ( ~ - R ~ , ~ -
d
w,,)+ y~ (~,,, ~,,).
In use of e,,(0) = wh(t) = 0, we h a v e
/0'
II~,,(~)ll X~ ds = _
/o' ]o'
( ( ~ - ~,,~,, a~(w - w , , ) ) -
~
A ~ ( u - R,,~, w - w,,))d~
(llOs~ - a~,,,,,, ~ + t,, -~ w - w , ,
+ c~
/0'
I~)d.~
(I ~ - ~,,,~11~ + h ~ I1,~- ~,,,~11~)d~
(3.120)
139
3.5. Alternative Approach
for c > 0. We obtain
/o
I I ~ ( ~ ) l l ~ d ~ -< C
/o
(ll**-Rh~ll ~~ + h ~ I1~- R ~ I I ~ ) d ~
by (3.119)and (3.120). Inequality (3.117)follows from (3.114). As is described in w
the following theorem implies Ileh(t) x <-- Ch2t-1 Iluollx
by Helfrich's duality method. T h e o r e m 3.13. If Uo E V r~ H 2 (f~), the estimate le.(t)llx < CA ~
(3.121)
luo H~(~)
holds true. Proof." Putting = [(t)= Rh(t)eh(t)= Rh(t)u(t)-
Uh(t) E Vh,
we have ({t, ~) 2t r
~)
--
(Ot(t~hU), ~) -- (OtUh, ~) Jr- r
=
(at(Rhu), ~) - (atu, ~) -(a~(~ - R~), ~).
:
~) -- At(Uh, ~)
Multiplying t implies
1 d (t I1{11~)
2 dt
1 t~t,(~, {) = ~ I1{11~ -
+
t(at(u
-
Rhu)
~)
and hence
t ll{(t)ll~: < c --
follows. We obtain
J/o'
I1{(~)11~ d ~ + C
t Ileh(t)ll 2X -< t l l ( u - Rhu)(t)ll~: + C + c
/o
f'
~ Ila~(u- R~u)(~)lt ~ d~
Ile~(s)ll~ d~
I I ( u - R~u)(~)ll~ d~ + C
Here, Lemma 3.10 applies. We get
X
jo
~ Ila~(u- R ~ ) I I ~ d~.
t lleh(t)ll 2X <- C foot Ileh(s)llx2 ds + Cth 4
{
max [[U(S)IIH2(~) + O<s
)}
8 [[U(S)IIH2(~) + [[Ut(8)[[ u~(n) 2 ds
140
3. Evolution Equations and F E M
For the right-hand side we have
fo' I1~(~)11~ d~ X
< --
C t h 4 I1~o11~(~)
by (3.117), and also
I~(~) IH~(~) _< c II~oll.~(a) and
Jo"(
u(~)ll ~
)
~
d~ < C I ~oll ~
by Propositions 3.7 and 3.9, respectively. Inequality (3.121) now follows.
Full-discrete
Approximation
The energy method is also applicable to the error analysis for fully discretized problems. Here we study the backward difference finite element method,
~;(t + r - ~ ( t )
+ Aj~(t + T)U "(t h + ~) = 0
T
with
~;~(o)
=
Phi0,
w h e r e t = t n f o r 0 _ < n < N. We drop the suffix h, and the error is denoted by e(t) = u ' ( t ) - u ( t ) .
The inequality
L e m m a 3.14.
n--1
t 2 Ile(t)ll x2 -< c~ -2 I~011x~ + Cr ~
IIe" (tj)llv.2
(3.122)
j=l
holds for t = tn. Proof:
Given v = v(t), we put
Otv(t) = v(tj) - v ( t j _ l ) T
for t = tj with j _> 1. We have (ut, v) + At(u, v) = 0
and
(O~u~,v) + A~(u ~, v) = 0
for v E V and t - tj. It follows that
(Ote(tj), v) + At3 (e(tj), v) = (Tj, v) ,
(3.123)
3.5. Alternative Approach
141
where
1 f~~, 1 (~_ "~j = Otu(tj) - ut(tj) = -~
tj_,)~.(~) d~
(3.124)
Putting ~(ty) = tje(tj) and ~j = ty3'j, we get
(Ote(tj), v) 4- At3 (e(tj), v) = (~j 4- e(tj-1), v) . Letting v = ~(tj) now gives that
1 (~(tj) - e(tj-1), ~(tj)) 4- Atj (~(tj), ~(tj)) = (~j 4- e(tj_l), ~(tj)). T In other words, the inequality 1
:2 (ll~-(tj)ll~-II~(tj
-
12) 4- TAt, (~(tj), ~(tj)) = T (X/j 4-
- 1)112 x + r 2 IIO~(tj)
e ( t j _ l ) , e(tj))
holds and we obtain ila(tj)ll x~ - I I ~ ( t j - 1 ) l l
~ + 2 w ~ l ~ ( t j ) l l ~V X
<_ 27 (Sj 4- e(tj_l), ~(tj)) < 2~- { c (llgjll ~v . + I l e ( t j - 1 ) l l v ~. )
+ ~
-
Ila(tj)llv}
9
We have II~(tj)ll X ~ - I l e ( t j - 1 ) l l x<- 2Cw ([l@jl[~. + Ile(tj-1)[2v.)
and hence
t 2 Ile(t,)llx = II~(t,)lI2x <__2 c T
tj 1l'TjIlv. + j=l
Ile(tj)lIv 9
{3.125)
j=l
follows. Equality (3.124) implies ii,),jll2 _~ 1 T
/t;
' (8 - t j _ l ) 2 Ilutt(8)ll 2. d8 1
and hence tj II~jll..
follows because t y ( s with f = 0 implies
_
..
ftj 'j--1
tj-1) <_ Ts for s E [tj-1, tj]. On the other hand, equality (3.110) IIuttllv. < Cllutllv + CIlullv.
(3.126)
142
3. Evolution Equations and F E M
Therefore, we get
n
2
2
< ~-~
j~0tn
~ I1~.(~)111 V* d~
j=l
-< c wi
/o ' (ll~(~)ll~+s ~ll~r
_ cw 2 Ilu011~ by Propositions 3.5 and 3.8. Inequality (3.122) is now a consequence of (3.125). For the operator T(t) =_ A(t) -1, the relation Ilfllv.
~
IIZ(t)fllv
(f ~ Vh)
holds uniformly in h. In use of T'(t) = - A ( t ) - l A ' ( t ) A ( t ) - l ,
I Z'(t)fllv <_ C
Ilfllv.
we have
9
In terms of T(t), equality (3.123) is written as
T(tj)Ote(tj) + e(tj) = T(tj)3,j. Setting Fj = T(tj)~,j, we get
Ot [T(tj)e(tj)] + e(tj) = Fj + [OtT(tj)] e(tj_l). Here we take L 2 inner product with T(tj)e(tj).
We have
(T(tj)e(tj), e(tj)) ~ I e(tj)l
~.
and
(Ot [T(tj)e(ty)] , T(tj)e(ty)) 1 = 2T (llT(tJ)e(tJ) 12X - I l T ( t j - l e ( t j - 1 ) I x
+ 7-IIOt [T(tj)e(tj)]llx),
as in the proof of L e m m a 3.14. It follows that
_1 2
(llZ(ty)e(tj)ll 2x --[IT(tj -1 )e(tj_ 1 )1 x2 + ~ II0, (T(tj)e(tj))ll 2 ~) 2 x / + ~ I~(tj)llv. 7<_ Ilrjllv. IIT(tj)e(tj)llv + II[O~T(tj)] e(t~_l) x IIT(tj)e(tj)l x
with a constant # > 0. The right-hand side is estimated from above with some t, 6_ (tj_l, tj)"
I FjlIv. IIT(tj)e(tj)llv § IIT'(t,)e(tj-1)llv T ( t j ) e ( t j ) I x _< c (IIFjlIv. e(tj)llv. + Ile(tj-,)l v. IIZ(tj)e(tj)l x) 1
2
2
_< -~. (lle(tj)llv. + I e(tj_~)llv.) + C (llT(tj)e(tj)ll 2X + IIFj I~*).
(3.127)
Commentary to Chapter 3
143
We obtain 1
9
3
1
2
2
2
+ ~ - , (If~(tj)lf v. -II~(t~- ~)11~.) _< c~- (llT(tj)~(t~)fl~ + Ilrjlt~.) and hence
]]T(tn)e(tn)]] x2 + ~ s
II~(tj)ll~. -< c ~ s
j=l
Ilrjll ~v.
n + c~- ~lf(tj)e(tj)ll2x
j=l
j=l
follows. Then the discrete Gronwall's inequality implies that n
IIT(t~)e(t~)llx + f
Ile(tj)ll 2V* <- - CT ~ j=l
Ilrjll V* 2
"
(3.128)
j=l
Here, Schwarz's inequality gives JIFjll2v.
IIZ(tj)'~jll~. < 1__
-
T
~ (s - tj_~) 2 ]lZ(tj)u~(s)l[~. ds 1
similarly to the proof of Lemma 3.14. On the other hand, T(t)ut + u = 0 implies
T(t)utt = - u t - T'(t)ut. Therefore, given s E [tj-1, tj] there exists s. E (tj-1, tj) satisfying
<_ Ily(s)utt(s)llv. + f IlY'(s,)u,(s)lIv. _< Ilu~(s)llg. + IIT'(s)~,(s)ltv. + w IlT'(s,)utt(s)llv.
IIT(tj)u,(s)llv.
<__ c (ll~(~)llv. + II~,(~)llv + ~ II~,(~)llv.). In use of [lutllv. = I A(t)ullv.
< C I1~11~, w~ h~v~
IIT(t~)~.(~)llv. <
c
(ll~(*)llv
+ ~
II~,(*)llv)
by (3.126). Consequently, the inequality
~ - ~ Ilrjll ~. _< C j-1
j=l
-1
(~- tj_~ (11~(~)11~+
<-
c~ ~
<
c w 2 IIPu0112
--
(ll~(s)ll =v +
I1~(~)11~)d~
~ Ilu,(~)ll~)ds (3.129)
X
follows from Propositions 3.5 and 3.8. Combining (3.122) with (3.128) and (3.129) gives inequality (3.82):
II~(t)lt~
< CTt-1
II~ollx.
3. Evolution Equations and FEM
144 Commentary
to C h a p t e r 3
3.1. The generation theory of Tanabe-Sobolevskii was given by Tanabe [377] and Sobolevskii [353], independently. Inequality (3.17) is a consequence of the elliptic estimate of Agmon, Douglis, and Nirenberg [3]. The generation theory of Fujie-Tanabe was given by Fujie and Tanabe [132]. Inequality (3.23) follows from the coerciveness of At. The righthand side may be replaced by C~ Izl, but this form simplifies the treatment of fractional powers of A(t). The proof of (3.23) is also given in Tanabe [378]. Generation theory of Kato-Sobolevskii was given by Kato [201] when p-1 is an integer and Sobolevskii [354] for the general case. Inequality (3.27) was made use of by Heinz [169] and Kato [202], [200]. For the theories of fractional powers of m-sectorial operators, see Kato [199]. Inequality (3.26) was also proven there. Equality (3.3o) is due to P.E. Sobolevskii. The theory of Kato-Tanabe was given by Kato and Tanabe [209]. See also Suzuki [368] and ~3.3 for the fact that smoothness of bilinear forms confirm the assumptions of the last theory. Most of those generation theories are described in Tanabe [378] in details. 3.2. Inequality (3.55) was proven by Suzuki [368]. Proof of (3.64) was also given there. Note that necessary assumptions for A(t) are those on the theory of Kato-Tanabe and inequalities (3.60)-(3.62). Once they are provided, the Banach space structure induces the conclusion. 3.3. Inequality (3.67) was proven by Fujita and Suzuki [147] for boundary condition (3.a) and Suzuki [3661, [36s] for the general case. Related works were done by Helfrich [171], Fujita [140], Suzuki [365], and Sammon [335]. 3.4. For the detailed proof of (3.86), see Suzuki [368]. Inequality (3.82) was proven by Suzuki [366], [3681 Related works were done by Sammon [335]. 3.5. Error analysis on the fully discrete approximation of temporally inhomogeneous parabolic equations has been done independently by Baiocchi and Brezzi [23], Huang and Thom~e [184], [185], Luskin and Rannacher [256], [257], and Sammon [336], [337] Contrarily to Suzuki [366], [368], the Hilbert space structure of the problem were used systematically there. The works described in this section were done by Luskin and Rannacher [256] and Huang and ThomSe [185]. The error analysis for schemes with higher accuracy were done by Baiocchi and Brezzi [23], Luskin and Rannacher [257], and Sammon [336], [337].
Chapter 4 O t h e r M e t h o d s in T i m e D i s c r e t i z a t i o n We have studied fully discrete finite element approximations for parabolic equations with the forward, the backward, and the Crank-Nicolson schemes for time discretization. However, many other methods have been examined in numerical computations. The present chapter is devoted to systematic studies for them.
4.1 Let
r(z)
Rational Approximation of Semigroups {e-tA}t>_o be a uniformly bounded (Co) semigroup on a Banach space X. We take
= (1 + z) -1 and describe the idea. It is a rational function provided with the properties
r(z) ~-- r
_.it_0 (]Zl p + I )
as z ~
0
for p = 1, and
Ir(z)l _< 1
for Rez >_ 0.
They are referred to as being of order p and A-acceptable, respectively. As we have seen, the operator r'~(TA) = (1 + TA) n is regarded as an approximation of e -tA for t = nT-, where ~- > 0 denotes the time mesh size. Based on those facts, we can argue as follows. First, the relation
d--~d (r~(sA)e_n(T_s)A)
=
nr n - l ( s A ) (r'(sA) -4- r ( s A ) ) Ae -n(r-s)A
-- nrn+l(sA)sA2e-n(~-s)A holds, and therefore we have [r~(TA) - e-nrA]A -2 - n
rn+l(sA)se -n(~--s)A ds.
Next, from the semigroup theory, the uniform boundedness of the semigroup represented as
lie-tAll _< C
(0 < t < oc)
(4.1)
is equivalent to the stability of its backward difference approximation:
]rn('rA)ll <_ C 145
(4.2)
4. Other Methods in Time Discretization
146
Those relations imply
II[rn(~-n) - e-n=A]A-P-lll <_ CtT p
(4.3)
for p = 1, where t = n7. Taking this observation in mind, we shall give error estimates for the approximate o p e r a t o r rn(rA) when the semigroup {e-tA}t>_o is holomorphic and r(z) is a general rational function of order p and A-acceptable. For this purpose, we recall the notion of A-acceptability of G. Dahlquist and generalize it in connection with a sector in the complex plane, where 0 E (0, 7r):
z0 = {z e C l 0 _< larg zl _< 0}. Namely, we introduce the following notations: 1~ T h e rational function
r(z) satisfies (i)o if Ir(z)l_ 1 for any z E E0 and Ir(oc)l < 1.
2 ~ T h e rational function
r(z) satisfies (ii)o if Ir(z)l < 1 for any z C E0 \ {0}.
3 ~ T h e rational function some 5 > 0.
r(z) satisfies (iii)o if Ir(z)l < 1 if 0 < Izl < ~ and z E E0 with
If I~(z)l _ 1 for any z E E0, then r(z) is said to be A0-acceptable. Therefore, Aacceptability means A~/2-acceptability in this terminology. Any rational function r(z) of order p (>_ 1) has p r o p e r t y (iii)o for 0 E (0, 7c/2). In fact, for p E [-0,0] we have 0
Op where ~ = x/~-l. Because of r/
r(O) = e-Zlz=o = 1
and
r'(0) =
~-2-e-Z dz z=O
"- --1,
it holds t h a t c)---fi r(Ps
= - 2 R e e~ = - 2 c o s ~
_< - 2 c o s 0 < 0
(4.4)
p=0
and hence the conclusion follows. If r(z) is the Pad~ a p p r o x i m a t i o n of e -z with degrees n and m of the numerator and the d o m i n a t o r , respectively, then it is of order p = n + m. Furthermore, according to n < m, n = m, and n > m, it has properties (i)o, (ii)o, and (iii)o for some 0 E (0, 7r/2), respectively. In fact, in this case we have r(z) = R,,,m(z) = P,.m(z)/Q ......(z) with (n + m - j)!n!
P.,m(~)
=
(~ + .~)!j!(~ _ j)! (-~-Y, j=O
m
Qnm(Z)
(n+m--j)!m!
x-'z_.,(,, + ,~,)b~(,~, - .J)~J j=O
4.1. Rational Approximation of Smigroups
147
The relation
]Rnm(Z)
-
I <_ c zl n+m+l
-
(z ~ 0)
is known. Let
(n + r e - j ) ! n ! anm(j) = (n + re)!j!(n-- j)! for 0 < j _< n. If n <_ re, we have
anm(j) > 0
and
a,~m(j) <_ anm(j).
j =0
T h e inequality
j =0
m
<-- E anm(J)PJ cosjqp j=o
= Re
Qnm(pe +zv) 5r 0
follows if m~p E [0, 7r/2]. Under the same assumption we also have n
IIm
Pnm(pe+'~)[ =
anm(j)(--p) j sin(-t-jqp)
_< E j =0
j =0 m
anm(J)PJ sinjqp
m
-< E a~m(J)PJ s i n j ~ = j =0
E anm(J)PJ sin(+jqD) = IIm j =o
Q,~m(pe+~)l.
Therefore, I~(p~)l ~ 1 holds for qp E [0, 7r/(2m)]. If n < re, r ( o c ) = 0 follows and (i)o holds with 0 = 7r/(2m). If n = re, we have Ir(oc)l = 1 < + o c . T h e n the inequality r(z)l < 1 for z E E~/2m is a consequence of the classical P h r a g m 6 n - L i n d e l 6 f theorem. The m a x i m u m principle now gives Ir(z)l < 1 for z E E0 \ {0} with 0 C (0, 7r/(2re)). This means t h a t r = Rnm satisfies (ii)o in this case. Finally, r(z) satisfies (iii)o always for some 0 E (0, 7r/2) as is described. It is known t h a t the Pad6 a p p r o x i m a t i o n Rnm of e -z is An/2-acceptable if and only if r e - 2 _< n _< re. If n = re and n = r e - 1, Rum is subject to a recursive formula based on the continued fraction expansion. Namely, we have the expansion e z = Z
1Z
1+ Z 2~
-
-
3+-.-
expressed by Z
C z ---
Z
Z
Z
Z
-- i+~-5+~-
Z
Z
g+-..+~-
2j_1
This implies eZ=l+
2Z
2-~
+
Z2
-6-
+
Z2
~
Z2
+ - - - + ~
2(2j-1)
+ 9
....
4. Other Methods in Time Discretization
148
Accordingly, a rational approximation H2k+l(Z)= G2k+l(Z)/F2k+l(Z)of e z is introduced inductively with F1 = 1, Fa = 2 - z, G1 = 1, G3 = 2 + z, F2j+I
=
2 ( 2 j - 1)F~j_I + z2F2j_3,
and
Ggj+I = 2(2j - 1)G2j-1 + z2G2j-3. Similarly, H2k(z)= G2k(z)/F2k(z)is defined by Fo = 1, F2 = 1 - z, Go = 0, G2 = 1,
F2j =
{
2(2j-1)+
2 } 2j-3 z
2j-lz2F2j_4, F2j-2+2j 3
and
G2j = {2(2j - 1 ) +
2
2j- 3
"[
2j-
z f G2j-2 + 2j
1
3
G2j-4.
Then, the relations H 2 k + l ( - z ) = Rkk(z) and H 2 k ( - z ) = Rk-l,k(z) can be verified. Let A be an operator of type (00, M0) on a Banach space X, where 0o C (0, 7r/2) and Mo >_ 1. The holomorphic semigroup generated by - A is denoted by {e -tA }t>o. Suppose that A is bounded for simplicity. We can show the following. T h e o r e m 4.1. The estimate T
e-n'AII < C ( t )
P
(4.5)
holds for t = nT under one of the following assumptions, where fJ1 > 0 is a constant and 0>00" 1~ r(z) has property (i)o. 2 ~ r(z) has property (ii)o and 7 IIAI < M~ < +oo. 3 ~ r(z) has property (iii)o and T [JAIl < /l}I~ < ~.
We need several lemmas. constants, respectively.
Henceforth r > 0 and C > 0 stand for small and large
L e m m a 4.2. If a rational function r = r(z) is of order p and Ao-acceptable with p > 1 and 0 E (0, 7r/2), then there exist constants cy > 0 and fl > 0 such that [rn(z) - e -nz ] _< C n zl p+I e -''~'ee(z)
for z C Eo with
Izl
_ ~.
(4.6)
149
4.1. R a t i o n a l A p p r o x i m a t i o n o f Smigroups Proof." We have
rn(z)
--
e -nz ~-- s
r J - l ( z ) ( e - z -- r ( z ) )
e -(n-jT1)z
j=l
and
It(z)
-
e-Z[ <
C
Iz[ p+I
(Izl < ~o)
for some Oo > 0. This implies [r(z)[ _< e -Re(z) + C (Re(z)) p+I for z E E0 and Izl _< a0. T h e function f ( t ) = e - t + C t p+I satisfies f ( 0 ) = 1 and f'(0) < 0, and hence f ( t ) <_ e -~t holds on t E (0, al) with some /71 > 0 and ~ E (0, cos0). We get
for [zl <_ min(crl, a0) and z e E0. Therefore, if Izl _< (7 -- m i n ( a l , a0) and z e E0, we have
Irn(z) -- e -nz ] ~
s
e-(J-1)~Re(z)c [zip +1 e-(n-j+l)cosO'Re(z)
j=l C n ]Z[p+I e -/~nRetz) .
The proof is complete. Lemma
4.3.
Under the a s s u m p t i o n of Theorem ~.1, the estimate
II[<(~A) - e -~] A-~II ~ C < holds f o r t = h r . Proof: Take M > 0 satisfying r [IAII < M, ]~/'1 < M, and -s < M < 5 according to the cases in T h e o r e m 4.1, respectively. We may suppose t h a t a < M in Proposition 4.2. We take a p a t h of integration F0 divided into three parts, Fo = F1 U ['2 U ['3 with {90<01 <{9"
Pl
-
{)~=pg+ZOl [ O ~ p ~ g T - 1 } ,
r 2
=
{A =
r~ =
pc+Z~ I oT -1 < p < M T - 1 } ,
{M~-'~'~l
I~J < 0 1 }
Then, we have -
2m
+
1 =
I+II+III.
+
2
( r ~ ( r z ) - e-tz) z - p ( z I - A) -1 dz
3
4. Other Methods in Time Discretization
150 In use of Proposition 4.2, we have
IlZll <_ c
n.Tp )P+ae-Znwc~176 P
=
c
.~-p?+'~-~(n'o)c~176 ( )
/0
) - ~ d ~ -~ = C~-~. P
From the a s s u m p t i o n follows t h a t
I11:
__
c ]r
+
[ l ( z l - A)-'II dzl
2
C
fl_p~dp = CTP. T -1
P
Finally, the inequality
II(zI-
a)-l]l ~
7-
1
I z l - {IAII
M-
7-IIAll
holds on F3 and hence
IXIII ~_ c Jr ( ~(~z)l +
I - zl)Izl
-~' I ( A - A)-']I Idzl
3
< c -
f
o,
0,
(M~-') -~
Md~
M-~ItAII
-- CM Tp
follows. In the first two cases of the theorem, the constants C in the estimates of III]] and IIIIi] are independent of M, while CM goes to zero as M ---, +co. Therefore, the t h e o r e m follows. [] Proposition
4.4. I f r = r(z) satisfies (iii)o, then each 5' E (0, 5) admits/3 > 0 satisfying
Ir(z)l _< e -~'zj. If ~ = ~(z) satisfics (ii)o, thcn ~' > 0 can be taken arbitrarily large. Proof: As is noted, inequMity (4.4) holds. Any/30 E (0, cos0) admits a0 > 0 satisfying
I~(S~)I _< oxp (-/3op)
for p c (o, ~o) ~nd I~I-< 0. Next, let max {Ir(peW)llp E [c~0,~'], ~ C [-0,0]} = 1 - c < 1, and take /~1 > 0 satisfying exp (-/31~') = 1 - e .
Then, wc have
lr(pe'~)] _< exp (-/~,p) for p c (00, (Y) and [~p] < 0. S e t t i n g / 3 = min(/30, ~1), we get the conclusion.
[]
4.1. Rational Approximation of Smigroups
151
L e m m a 4.5. In the last two cases of Theorem ~.1, the estimate
IIA~<(wA)II < C~(n~) -~
(4.7)
holds for c~ > O. Proof: Taking a path of integration as in the proof of L e m m a 4.3, we have -
+ 27"~'~
=
1 -'~F2
(nTz)arn(Tz) (Z[ -- A) -1 dz 3
I+II.
From the assumption, we can take ~' of Proposition 4.4 in M < ~'. We have the following:
I111 IIHII
<
< -
Then
c
c
Z
(n"rz)C~e -~nrpc~
dD = Ca,
P
fie ( n T M ) ~ e - Z M n ~ d ~M< C ~ . ol
M-/1)1
-
the proof is complete.
[]
Helfrich's duality argument now gives the following.
Proof of Theorem ~.1 for the last two cases: 6, m E N a n d O _ < ~ - m _ < 1. We have
Divide N E N into n = t~ + m with
rn(TA) -- e-tA = (re(TA) -- e-e~A) A-Prm(~-A) + e-e~dA-p (r'~(TA) -- e-m~d) . Operator norm of the first term of the right-hand side is estimated as II[rl(TA) -- e-I~A]A-PlI . IIAPrm(TA)I < CT-P(mT) -p < C / n p = C(~-/t) p by Lemmas 4.3 and 4.5. T h a t of the second term is estimated similarly by the adjoint form of those lemmas. The proof is complete. [] Generally, r(c~) = 0 does not arise even in (i)o. We cannot take d' = +cx~ in Proposition 4.4. For the first case of Theorem 4.1 to prove, therefore, we require some more considerations. L e m m a 4.6. Let f l, f2 " R+ ~ R+ be continuous functions satisfying
n f l ( r ) dr < +oo r
and
dr < +oo 7"
for R > O, and let ~ be a meromorphic function provided with the properties IF(z)l < f~ (Jz[) and l~(z) - ~ ( o o ) l < f2(Izl) for lz[ <_ R and largz I = 0 ~ , where O~ > 0o. Then, the inequality
{ fl(r)~ dr + ~00Rf2(r) d~. + I~(oo)1
II~(A) < CR Jo(.R holds true.
152
4. Other Methods in Time Discretization z
Proof." Letting h(z) = qp(z)- 1 + z qp(~176we have ~(A) = h ( A ) + ~(oo)A(1 + A) -1 Here, lid(1 + A ) - l l _ C holds and we have
[I~(A)II <_ IIh(A)ll + Cl~(c~)l. To estimate IIh(A)ll, we recall the path of integration F given in w oriented boundary of E01, which produce the equality
h(A) = ~
the positively
h(z) (zI - A) -1 dz.
We have z
lh(z)l = ~ ( z ) for Izl _< R and larg z I =
01,
I + ~ (oo)
f l ( [ Z [ ) - 4 - [ z [ [~(Cx:))l
and 1
Ih(z)l= ~(z)-~(~)+
1
l+z ~(~)
A(Iz[)+~l~(~)l
for Iz[ >__R and larg z I = 01, respectively. Those inequalities imply IIh(A)ll
_< c
=
C
{/0"
( k ( r ) + rl~(oc)l) -dr- +
{/0" 'r/0" fl(r)
-r- +
) } } -dr-
A ( r ) + -1I ~ ( r ) l
r
r
'" ( R +
f 2 ( r ) -r- +
~(oc)l
The proof is complete.
r
.
[]
Now, we complete the proof of Theorem 4.1.
Proof of Theorem 4.1 for the first case: Take 01 E (0o, 0) and a > 0 as in Proposition 4.2. The function ~(z) = e . . . . . r"(z) satisfies (4.6) if z E E0, and Iz <_ a. On the other hand, making ~ > 0 smaller if necessary, we may suppose that sup{Ir(z)ll
z~E0,,
zl _ ~ } = e
-z
from the assumption. Furthermore,
~(z)- r(o~)l_
cI ~-I
(Izl >_ ~)
holds because r = r(z) is rational. Those relation imply rt--1 j=O
4.2. Multi-step Method
153
and hence ~(~) - ~(oo)1 _< ~-~,zlcos0, +
c~-nZ/Izl
(4.8)
if z C E01 and Izl >_ a. Finally, we have
I~(oo)1 ~ e -n~.
(4.9)
Inequalities (4.6), (4.9), and (4.9) give
II~(~-A)II = II<(~-A)by L e m m a 4.6. The proof is complete. Theorem 4.1 is applicable to the semidiscrete finite element approximation of the parabolic equation, dub + Ahuh = 0 dt
(0 <_ t <_ T)
a~d
uh(O) = Phuo
in Xh with Ah, Xh, and Ph as before. We recall that the spectrum of Ah lies in a parabolic region in the complex plane uniformly in h. Each 0o E (0, rr/2) admits constants M0 _> 1 and ~ independent of h such that Ah -- .X is of type (00, Mo). Taking Vh = e-)'tUh instead of Uh, we can make the exponent 0 > 0 as small as we like in applying T h e o r e m 4.1. On the other hand, the inverse assumption guarantees the inequality IIAh[I <_ 7h -2 with some "y > 0. We have the following. T h e o r e m 4.7. Let r = r(z) be a rational function of order p (> 1) satisfying one of the following conditions for some 0 > O: 1~
(~)o.
2 ~ (ii)o and r h -2 < M1 < +c<~. 3 ~ (iii)o and r h -2 <_ M1 < "~-15. Then, the estimate (~.5) holds for t = n r with a constant C > 0 independent of h. Applied to the semidiscrete finite element approximation with higher accuracy, Theorem 4.1 gives natural results from the viewpoint of the correspondence of the rate of convergence with respect to the time discretization and the space discretization. Then, theorems on backward, forward and Crank-Nicolson schemes in w167 and 2.6 arise as special cases.
4.2
Multi-step
Method
Time discretization schemes studied in the preceding section may be called the singlestep m e t h o d totally, as the value un = u(tn) is determined by t h a t of un-1 = u(tn_l)
154
4. Other Methods in Time Discretization
for tn = nT. Multi-step m e t h o d of order q(>_ 2) determines Un from the values at the preceding q-steps, Un-1, U~-2,''" , U~_q, after determining u l , . . . , Uq-1 suitably. In this section we study this kind of schemes adopted to the evolution equation
du
dt
+Au=O
(O<_t<_T)
with
u(0)=uo
(4.10)
on a Banach space X, where A is of type (0, M) with 0 E (0, 7r/2) and M >_ 1, so that - A generates a holomorphic semigroup {e -tA}t_o. Suppose that A is bounded for simplicity. In the numerical scheme which we are studying, one determines U l , ' " , Uq-1 from u0 by a single-step approximation in use of a rational function first, and then computes Un+q through the relation q
E(ai
-t- TbiA)Itn+i : 0
(4.11)
i=0
for n = 0, 1 , - - . , where ai, bi E R. W i t h o u t loss of generality, we take aq = 1 in (4.11). Setting q
P(r
q
a~r
= E
and
S(() = E
i=0
bir
i=0
we call scheme (4.11) the multi-step m e t h o d (P, S). It is said to be of order p(_> 1) if the following relations hold: q
q
q
q
q
Ea~=0, Eia,= Eb,,
EiJa~=iEiJ-lb~
i=0
i=1
i=0
i=0
(2
/=1
Letting 0 ~ = 1 and 0 . 0 -I = 0, we can write them simply as q
q
E ijai = j E iJ-lbi i=0
(4.12)
i=0
for j = 0, 1 , . . . ,p. To understand tile meaning of those equalities, take A = d/dx and compute tile Taylor expansion around r = 0 of the finite difference operator q
L~-[y] = E
[aiy(x + iT)+ fbiy'(x + iT)].
i=0
Then, equalities (4.12) follows if tile coefficients of T up to pth powers are let to be 0. If q = 1, scheme (4.11) is either one of the backward, the forward, the Crank-Nicolson, and the modified Crank-Nicolson (of order 1). Letting w((, z) = P ( ( ) + zS(() and 0 E (0, rr/2), we introduce the following notations: 1) The m e t h o d (P,S) satisfies (III)o if any root (j (1 _< j _< q) of P ( ( ) = 0 is simple and lies on the closed unit disk t(I _ 1, and moreover, if (j is in (j = 1 the inequality Re Aj/[Ajl > sin0 holds for Aj = CjS(Q)/P'(Q).
4.2. Multi-step Method
155
2) The m e t h o d (P, S) satisfies (II)o if it has property (III)o, and any root Cj(z) (1 _< j _< q) of w(r = 0 is simple and lies in the open unit disk [r < 1 for
z e x0 \ {0}. 3) The m e t h o d (P, S) satisfies (I)o if it has property (II)o, any root of S(~) = 0 is simple and lies in the open unit disk [C[ < 1, and bq > O. If (P, S) has property (III)o, then the requirements for Cy(Z) (1 < j < q) stated in (II)o holds if 0 < Iz] < ~ and z E E0, where ~ > 0 is a constant. In fact, Cj(z) is continuous in z and we may suppose t h a t Cj(0) -- Cj by re-ordering the number if necessary. We have only to show that [r
< 1 holds if 0 < [z I < ~ and
z e E0 for some n > 0, assuming I(jl = 1. Because ~j is simple, g(p, r differentiable in p at p = 0, where [r < 0: 09
8p
= [(j(pe~r
is
(o, r = 2Re (r162
The relation
r
- -s(r162
follows from P(Cj(z)) + zS(Cj(z)) = 0. We have ~p 0g (0 , Oh) = - 2 R e Aje ~r and hence
sup
{ ~~1 6 2
r
holds. This gives the assertion. We make use of the rational function r(e) of order p - 1 to construct the approximate solution of (4.10) for the value of ui = u(ti) with i = 1 , - - . , q 1 and then take the multi-step m e t h o d (P, S) of order p for the values un = u(t,~) with n > q to determine. We have the following. Theorem
4.8. The inequality T
P
holds for t = nr ~ (0, T] with n > q, if one of the following condition is satisfied with 0>0o" 1) ( P , S ) and r have properties (I)o and (ii)o, respectively. 2) (P, S) and r have properties (III)o and (iii)o, respectively, together with the condi-
tion
IIAII _< Me <
min
6, ~, ~q[
.
Here, 6 > 0 is so taken as for r(z) to have no poles in z[ < ~ and z E Eo. Furthermore, ~ > 0 is the constant described above as (II)o holds for 0 < z I < ~ and z C Eo.
4. Other Methods in Time Discretization
156
In the second case, min(5, n, 1/lbql ) can be replaced by m i n ( ~ , l / b q [ ) and min((~,n), respectively, if (P, S) satisfies (II)o and bq > O, respectively. Setting
5~(z) = a~ + b~z aq + bqz
(1 _< i _< q),
we can write relation (4.11) as q
5,(TA)un+i = 0
(n = O, 1, 2 , . . . ) .
i=0
First, we study the functions un = u,~(z) of z E C satisfying q
E
5~(z)un+i(z) = 0
(4.13)
i=0
for n = 0, 1, 2 , . . . . We have the following. Lemma
4.9. If ~i(z) (1 <_ j <_ q) are distinct, then un(z) satisfies q
I~n(z)l <_ C ~ ICj(z)l" j=l
fo, ~ ,~ >_ q, ~h~,-~ c
>
o ~ a ~on, tant d ~ t ~ , ~ n ~ d
by
"~o,"" ,~q-,, ~Plr
inf I C j ( z ) - r ~#j
Proof." From (4.13) follows q-1 i=0
while the relation q
p(r
+ zS(r
= (a~ + ~qZ) ~
(r - r
j=l
implies q-1
q
~ = - Z ~ ( z ) ~ ~ + 1-I (~ - ~ ( z ) ) i=O
j=l
with ~i = cri(z) (i = 0 , . . . , q - 1) determined by
{P(r162 (aq + bqz)
~-' i=0
J
and
4.2. Multi-step Method
157
Now we write ai and un for ~i(z) and un(z), respectively. If ui = (Q)i holds for i = 0, 1 , . . . , q - 1 with some j, then we have un = (Q)n for n = q, q + 1 , - . . In fact, we can show inductively that Un+q
-- --
~i'U'n+i
-- --
(~i ( ~ j ) i
. (O)n
--
(o)q+n
i=0
Therefore, in the case that q
(/=O,--.,q-1),
j:l
(4.14)
for some 0/j E C, we have q
(n=q,q+ 1,.-.). j=l
This implies
<_ l<_j<_q m a x I jl
q j=l
The linear transformation S"
(o/1,
..
. , 0/q)T~.+
(UO,
..
.
, U q _ 1 )T
is expressed by the matrix 9 9 9
~'1
"'"
1
~'q
9
~.-q-1
.
...
,
~'q--1
whose determinant is that of Vandermonde. Since Q's are distinct, UO,
" " " , ~q-1}
of (4.14) can represent an arbitrary element in C q. Then the l e m m a follows immediately. []
If some of Q(z) is not simple, the inequality
c(14-'/l)rn(z)--i
q
E j=l
ICj( )l
holds with m = re(z) > 2 standing for the m a x i m u m of their multiplicities.
158
4. Other Methods in Time Discretization
T u r n to the scheme in consideration. T h e a p p r o x i m a t i o n o p e r a t o r is denoted by T~(A):
un = T~ (A)uo Define the rational function sn(z) inductively as sn(z) = r(z) n for n = O, 1 , . - . , q - 1 and q
E
Si(Z)Sn+i(z) = 0
(n = O, 1, 2 , . . . ) .
j=o We have T , ~ ( A ) = s~(TA) and hence q 5
(z)
_
=
i=0 holds with q
Fj(z) = E 5~(z)e-(i+i)z"
(4.15)
i=o In the following lemma, the error o p e r a t o r T , : ( A ) - e -tA is represented by Fj('rA). Define the rational functions 7i(z) (j = 0, 4-1, 4 . 2 , . . . ) inductively as "~i(z) = 0 for j < 0, "),j(z) = 1 for j = 0, and q
=o k=O for j > 0. Lemma
4.10.
rite identity q
e -t"+~A - T~+q(A) = E 7 .... j('rA)Fj('rA) j=o q-1 j (4.16)
j=O k=0
holds for n = O, 1 , . . . Proof." E q u a l i t y (4.15) gives
j=o
j=o z=o n q j=o i=o n+q j=O
4.2. Multi-step Method
159
where J
Bj(z) -- ~
?,-k(z)(~j-k(Z)
(j : O, 1 , . . . , q)
k=O
and q
J~q+j(Z) -- E
"~/rt-J-k(Z)(~q-k(Z)
(j : O, 1, 2 , ' ' " ).
k=0
We have Bj(z) = 0 for j = q , . - . , n + q - 1 from the definition of 7j(z). We also have q
]~n+q(Z) ---
E ~/-k(Z)(~q-k(Z)
--- (~q(Z) = 1,
k=O
so that q--i
e--(n+q)z -- Sn+q(Z) --
~/n-J(Z)PJ(Z) -- E j=o
j
E
~/n-k(Z)(~J-k(Z) [e-Jz -- 3j(Z)]
j=0 k=o
follows. This implies the lemma.
[]
Estimate of
II~xp (-t~+~m) - T~+q(A)[[ is reduced to those of 7j(z) and Fj(z) in this way. For the former we have the following. 4.11. Let (P, S) have property (III)o, and suppose (II)o for 0 < Izl < and z c Eo with t~ > O. Then, each ~' e (O, min(~, 1/Ibql)) admits constants C > 0 and /3 > 0 satisfying Proposition
for Izl < ~' and z E Eo. Proof." As we have seen, the roots Q(z) (1 _< j _< q) of P ( ( ) + inequalities
ICj(~)l _< ~-.,zj
z S ( ( ) = 0 satisfy the
(1~1 _< ~', z e r~0)
for some ~ > O. This implies
IO,n(Z)l < Ce -znlzl
(Izl < ~', z e Eo)
by L e m m a 4.9. Here, the constant C > 0 depends on % , - . . , ~[q-1, which are polynomials o f / ) 0 , " " , (~q-1. The latters are bounded on Izl _< ~' < 1/Ibql, hence C > 0 also is bounded there. Recall t h a t aq 1. The proof is complete. [] :
As for Fj(z) we have the following.
4. Other Methods in Time Discretization
160
P r o p o s i t i o n 4.12. /f (P,S) is of order p(>_ 1) and ec' < 1/Ibq] , then the inequality
IFj(z)l ~ C [zlp+I e-jRez holds for Re z >_ 0 and Iz <_ ~'. Proof: Letting v(t) = e -tz, we have F J ( z ) : ;-~"Si(z)e-(J+i)Z:(aq+bqz)-l
{ ~~:o - ~ " a i v ( j + i ) - ~ - ~i:o -"biv'(j+i)}
Here, (P, S) is of order p, and tile right-hand side is equal to
{~-~ fjj+l(J"~-i--t)Pv(P+l )(t) i=, a~ p!
(aq -t- bqz)-i
~-~lJ+i(J~-i--t)P-lv(p+l) ( p - 1)!
dt - i=, bi,y
In fact, we have
Aai -
fj+~
Jj
(j + p! i - t)Pv(P+l)(t ) dt
t=j+i
=
,,
-- -
(t)
t=j
g=O
-~.v(e)(j) + v(j + i) e=O
and
B~ =
jj
j+i (j + i -- t) p-1 )! v(V+l)(t) dt (p- 1
[ ~(j+i-t)e-lv(e)(t)]e=l ( f - 1),
t=j+i _- -
ie-1
_ ~
~v(e)(j)
+ v'(j + i).
e=l ( e - i)!
t=j This implies q
q
i=1
i=0
-
e=l
C!
i ea~ -
i=0 q
= E i=0
by {4.12}.
{aiv(j + i ) - bv'(j + i)}
g
b~
i=O
dt
}
.
4.2. Multi-step Method In
use
161
of
_Fj(Z) = (aq -F bqz) -1
ai i=1
p! JJ
•
-
(t) dt bi
i=1
with
v(p+l)(t) = (-z)p+le -tz,
we
)!
JJ
v(P+l)(t) dt
(p- 1
}
get the desired inequality.
[]
Note that if bq > 0, we can take ec' < rc and a = oo in Propositions 4.11 and 4.12, respectively. Now, we can give the following.
Proof of Theorem ~.8 for the second case: Take A C
/1//1, rain
, n,
The rational function r(z) satisfies
I~j(z) - ~-Jz I < c I~" for 1 < j < q - l , z I < a, a n d z E Ee with s o m e a > 0. Recall (4.16). In the second term of the right-hand side, we have
I')'n-k(z)hj-k(z) [e -jk - rJ(z)] l <_ C Izl p e -z(~-k)lzt
(4.17)
for 0 < k _< j < q - 1, Izl < a0 = m i n ( a , ~ ' ) , and z E Ee. On the other hand, given M E (M1, ~'), we have
I'Yn-~(z)hj-~(z) [~-jz
~j(z)]l_<
C~,(~-~)lzl
forao <__ ] z ] _ < M a n d z E E o . We take a path of integration Fo = F1 U F2 U F3 with 01 E (0o, 0): pl
--
{Z---pe-t-zO,
I
0 ~ p ~ O-0T-1} ,
r~ =
{z = ~•176
F3 =
{A4T- l e zq~ I I~1 _< 01 }"
~o~-1< p_< M~-~},
It holds that
"/n-k(TA)(~J-k'(zA) [e-tjA-- rJ('rA)]
=
27c~
=
I+II+III.
1 (]; 1
+
fr 2
+
d / r3)
"Yn-k('rz)(~j-k.('rz)
[e -j~-z- rJ(Tz)] ( z I -
A) -1 dz
4. Other Methods in Time Discretization
162
Each term admits of the following estimates for n = q, q + 1,.--"
II111
<
c fo ~ ~-Z(n-k~'o(W)~-2 = C(ndp
k) -p < Cn -p
I
IZZll
<_ c
IlzIzll
< c
__ _ e-Z(n-k)wdP - c
o~--~
_pdp --<_Cn off(n-k) P e
P
-p,
e-Z(n-klMd~< Cn -p. 9
01
The second term of the right-hand side of (4.16) is estimated as
[q-1 j
Z Z ~,,_k(~-A),~j_,~(',-A)[~-,jA _ ,.j(.,_A)]
~ CTt -p.
j=0 k=0
To estimate the first term, we make use of Propositions 4.11 and 4.12. The inequality
Is j=o
<_Cne-~nlz',zP+l
follows for z I _ < M a n d z E E 0 .
In use of
j :0
1-It-I"2 =
-~-OfF ) s
(zZ-A) dz -1
3 j=0
IV+V,
we get
II/Vll
<_ c
ne_~nrP(7.p)p+l do = C r _ p P
~0 ~176
and
IlVll
< c -
_01 .
0,
~.~-~""
M -
~
M1
dw < o~-~. -
The first term of the right-hand side of (4.16) is also estimated as
[["
~_ CTz -p.
j=O
The proof is complete. W h e n (P,S) has property (III)o, we can replace min(,~., 1/Ibql) by 1 / b q in Proposition 4.11, and accordingly, min(5, 1/[bql)by min(5, 1/bql ). Similarly, min(~, K, 1/Ibql)is replaced by min(~, K) in tile case of bq > 0 from tile reason described after the proof of Proposition 4.12.
4.2. Multi-step Method
163
Now we give the following.
Proof of theorem 4.8 for the first case:
Because (P,
ICj(z)l _< ~-z~zf
S) has
property
(I)o, the
inequality
( z l _ ~', = e zo)
holds for arbitrarily large ec' > 0 with s o m e / 3 > 0. T h e conclusion of Proposition 4.11 still holds for any ec' >> 1. M a k i n g / 3 > 0 smaller, we have
~j(z)l _< ~-~ < ,
( =1_ ~', = ~ x0),
I~'~(~)1 _< c~ -n~
(z >~', z~Eo)
(4.18)
so t h a t
follows as in the proof of Proposition 4.11. To estimate the first t e r m of the right-hand side of (4.16), we take M > r IIAII and represent n
1
+ ~r ) s %_j(rz)Fj(rz)(zI-A)
E %_j(rA)Fy(rA) = j=0
2 7"i-~
i nt_l-,2
3
-1
dz
j=0
= I+II. Similarly to the case (b), we have
I-Zll _< o~-p with the constant C > 0 independent of M. If j _> 1, we get
Vn_j(~z)Fj(~z) (~Z- A) -1 & <_C
~-("-J)eM~+'~-jMc~176
3
01
of which the right-hand side goes to zero as M -+ oo. To estimate the term for j = 0, we make use of the a r g u m e n t s in the preceding section. First, we note lim Fo(z)= Izl~oo zEEo
bo bq
and b0
M a k i n g / 3 > 0 smaller if necessary, we have e-/3n
I-Tn(z)- "Tn(oo)l _< C ~
Iz
( el >_ R, z e r,o)
4. Other Methods in Time Discretization
164
for R > 0. Actually, from the proof of Lemma 4.9 this is reduced to e-~n
I~j(z)"- ~j(oo)nl ~ c ~
where ~j(z) (1 _< j _< q) denotes the root of P(r ~(z) o - ~(oo).l
:
(1=1 _> R, z ~ ~o),
Izl
+ zS(r
,~3(z) - ~ ( o o ) ,
(4.19)
= 0. Inequality (4.18) implies
n-1 . }2 ~;-~-~(z)~y(oo) k:0
<_ ~-"~ I~j(~) - ~ j ( ~ ) t , where
arises because (P, S) has property (I)o. Inequality (4.19) has been proven. On the other hand, If0(z)l _< c and
IF0(z)- F0(oo)l--
ao - bo/bq 1 +bqz
+ ~q
C -t- Ce -Izl cos01 7--7, Izl
~(z)~ -~
i=1
are obvious. We obtain b0 <_ C e - ~
%~(z)ro(~)- -~.(oO)Vqq
(1
)
~
+ e-tzl c~176 .
The first term of the right-hand side of (4.16) is estimated as
irj=0a by Lemma 4.6. We proceed to the second term. Recall (4.17) with
js ~ e_~,~Opp dp = Cn_ p P We note
+ I~._k(oc)~j_.(o~)] ]~J(z)- ~J(o~)].
(4.20)
Here, 0 _< k _< q - 1 in this term and inequality (4.19) assures e-~n
17,~-k(z)- 7n-~(oo)l < C ~
tzl
165
4.3. Product Formula for Izl ~ ~0 and z C E0. Furthermore, 3' and (~j-k satisfy C (~j-k(z)- (~j-k(oo)l _< 757 Izl
and
I t ( z ) - r(oc)l _<
C
Izl
for Izl _> a0, respectively, because they are rational functions. We also have 17n(CC)l < Ce -zn
and
I~n-k(z)l _< C.
Summing up those relations, we see that the right-hand side of (4.20) is dominated above by Ce-Zn/Izl. The second term of the right-hand side of (4.16) is estimated as q-1
j
E E 3`n-k(TA)(~J-k(TA)(e-t'A-- rJ(TA)) <_C/n".
j=0 k=0
The proof is complete.
4.3
Product
Formula
From the classical theory of Lie groups, it holds that one paraeter families of subgroups {exp ( - t A ) } t > o and {exp ( - t B ) } t > o satisfy
)in lim [r ( t ) ( o r n--,+c~ nt m
=exp(-t(A+B))
where r = exp ( - t A ) and r = exp ( - t B ) . H.F. Trotter extended this property to (Co) semigroups on Banach spaces. It has been called Trotter's product formula. Error estimates in the operator norm were known for bounded operators, but only strong convergence has been discussed for the other cases. However, D.L. Rogava has succeeded in giving them for analytic semigroups. We shall describe the simplest case. T h e o r e m 4.13. Let A1 and A2 be positive self-adjoint operators on a Hilbert space H satisfying IIAll/2e-tA2All/21 < C
(4.21)
for t e [0, J], and suppose that A = A1 + A2 is self-adjoint with D(A) = D(A1) N D(A2) and furthermore, D(Aal/2) C D(A~/2) N D ( A 3/2) holds. Then, T ( t ) = e-tA2e -tA1 satisfies the estimate
sup
tC[O,nJ]
IIe -tA -- T ( t ) n l l
with a constant C > 0 independent n = 1, 2, 3 , - - . .
< C---~ lx/'n
(422)
4. Other M e t h o d s in T i m e Discretization
166 This t h e o r e m bounded domain "10a = 0, and A2 We take a few
is applicable, for example, if A1 is the differential operator - A on a ft C R n with s m o o t h b o u n d a r y 0f~ provided with the b o u n d a r y condition = V ( z ) >_ 0 with V ( z ) sufficiently smooth. preliminaries. Let
A =
fO ~
AdE(A)
be the spectral decomposition of a positive self-adjoint A operator in H. t >_ 0 and c~ E [0, 2], we have
First, given
OG
[(I + t A ) -1 - e -tA] A -~ =
[(1 + tA) -1 - e -t~] (At) -~ d E ( A ) , t ~.
Because sups>0 I[(1 + s) -1 - e -~] s-~l < +oc, this implies
II [(I +
t A ) -1 - e-tA]
A- II
ct~.
(4.23)
(s > 0).
(4.24)
We also have tile elementary inequality (1 - e - S ) -'/2 <_ s -1/2 + 1
To see this, let f ( s ) = s - S / 2 - ( 1 - e - S ) -'/2. We have limst0 f ( s ) = + o c and l i m ~ + ~ f ( s ) = - 1 . To show i f ( s ) < 0 for s > 0, suppose the contrary. The case i f ( s ) = 0 is equivalent to
g(t) - t 2 - t -a - 31ogt = 0
fort=e
s/3 > 1.
However, this is impossible because g(1) = 0 and
g'(t) = t - 2 ( t -
1)(2t 2 + 2 t -
1) > 0
for t > 1.
Inequality (4.24) implies (1 - e - ~ ) -1 d lE(A)'u 2
=
<_
/0
<_
((tA) -1/2 + 1) 2d E ( A ) u 2 =
IIA- / ul
(t-1/2A-1/2 + I ) u l 2
+ I1~11)2
We obtain (4.25) forttCHandt>0. Now we show the following. w
167
4.3. Product Formula
L e m m a 4.14. The operator W ( t ) = e -tA - e-tA2e -tA1 satisfies
IIAT1/=W(t)A-~/=]I< Ct=
(4.26)
for t c=_[0, J]. Proof: We have
W ( t ) -- [e -tA - (I -it- tA) -1] -Jr-[(/-t-
t A ) -1 - ( I
+ tA2) -1 (I -t- tA1) -1]
+ [(I + tA2) -1 -- e -tA2] (I + tAl) -1 q- e -tA2 [(I + tA1) -1
-
-
e-tAil.
The first term of the right-hand side is estimated by (4.23) as
IIATX/2[e-tA-(I+tA)-l]A-a/2[[ IlA71/2AI/2[I II [e -tA - (I nt- ~A) -1] A-2I[ ~_~C~;2. Here, the inequality [IA1/UAT~/211 < C follows from Heinz' inequality and D(A~/2) C D(Aa/2). Similarly, the third and the fourth terms of the right-hand side are estimated as IlAT1/2[(I+tA2)-l-e-tA2](I+tA1)-lA-a/2[I
_<
IIAI~Ag'=[I II[(, + ~A~)-I - e-~=] A;=ll
and
IIAT1/2e-tA2[(I--l-t;A1)-l-e-tA1]A-3/211 IIAll/2e-tA2All/21t [[[(/-t-tA1) -1 - e -tA1] A1211 llAal/2A-3/211 ~ Ct 2, respectively. Finally, writing (I + tA) -1 - (I + tA2) -1 (I + tA1) -1 = (I + tA) -1. tA2 (1 + tA2) -1- tA1 (I + tA1) - t , we have
<_ I{ATI/=AI/]I I[(tA) The proof is complete.
(Z + tA)-lll [leA= (Z + tA=)-*ll lllllAT' A []
168
4. O t h e r M e t h o d s in T i m e Discretization
We are r e a d y to give the following. P r o o f of T h e o r e m 4.13: we p u t E(t)
-
e -tA
L e t t i n g t = n r , U ( n ) = e -nrA, U1 = e - r A ' , and U2 = e -rA2,
-
(
e -tn-lA2
)~ - -
. e -tn-lA]
g(n)
(82-gl)
-
n
and W = W(7-) = e -rA -- e - r A 2 . e -rA' T h e n , it holds t h a t
E(t)
-
8281. W . U(?z - 2) -Jv ...-Ji- (U2U1)n-1. W . U(O) 1) 4- U2U~/2. (I 81) 1/2" V. U(Tt 2) nt - ' ' " "Jr- U 2 U ~ / 2 . S n - 2 . ( I - 81) 1/2 V. U(0),
-
U1) -1/2
=
W U ( n - 1) +
=
WU(n
-
-
-
where V = (I
U~/2W
and
We o b t a i n
n-1 E(t)
=
W U ( n - 1) -I- U28~/2 ~
S k-1
(I
--
81) 1/9" V-U(Tt - 1~ -+- 1).
k=l Now we shall show
li~ 1(, ~,)1,~ I < ~
1~,,~
~4~/
for k = 1 , 2 , . . - . In fact, t a k i n g A / + 6 for A/ and m a k i n g 6 ~ 0, we may suppose t h a t A/ k 6, where i : 1, 2. In this case, we have
I1~' ~1)1'~1l< (' ~'~/1'~
'~,J <~"
~
'~'J ~ ~-'~
For s = e -ra C (0, 1), we o b t a i n
I1~~-' (, ~l)"'l < II~,ll~-1 Ill, f,~-' I1(1 ~')"~11 _~ 82(k-1)(1 Here, f ( O ) = / ( 1 )
-
-
8) 1/2 ~ 8k-1(1
= O, and m a x f (s) = f (So)
sE[O,1]
holds for So = (2k - 2 ) / ( 2 k - 1) C (0, 1). We get f ( s ) <_ (1 - So) 1/2 = ( 2 k -
1) -1/2
-
-
8) 1/2 ~ f(s).
169
C o m m e n t a r y to Chapter 4
and inequality (4.27) follows. On the other hand, we have U 1/2 (I - U1) -1/2 (TA1) 1/2 ~- ~o~176-z~- (1 -- e--TA) -1/2 (TA)I/2dE(A)
with SUps>o e -s/2 (1
-
e-S)-1/2 s 1/2 < +oc. This implies [IuI/2(I-U1)-I/2(TA1)I/2[]
<_C
and hence [IV. g ( n -
1)[[
k-
[IU1/2 ( Z - Ul) -1/2
(TA1)I/2II.T-1/2IAll/2wA-3/2II [[A3/2U(TL--k--1)[[
G C (n - k) -3/2
(4.28)
follows for k = 1, 2,--- , n - 1 if t E [0, nJ]. Inequalities (4.27) and (4.28)imply
I
2u 1
.
k=l <
.
.
. n-1
g2ll Igll1172- c . ~
i,i[ k - l / 2 ( n - k) -3/2
k=l 1-
9n -1 = C n -1/2
ft k=l because
1 ~l (k)-1/2 (
k) -3/2 ..~
n k=l
n
1
J0 1
x-1/2( 1 _ x)_a/2d z ,.0
()n
-1/2
_1
The proof is complete.
Commentary
to C h a p t e r 4
4.1. This section follows the description of [148]. For later developments, see Crouzeix, Larsson, Piskarev, and Thom6e [100], Palencia [312], and Sava% [338]. See also the monograph Thom6e [383]. The notion of A-acceptability was introduced by Dahlquist [104], and studied by Ehle [118], axelsson [16], Cryer [103], Dahlquist [106], and others. For the equivalence between (4.1) and (4.2), see Yosida [410], Kato [205], and so forth. Recall that i f - A is a generator of a contraction semigroup, then constant C in those inequalities is taken to be 1.
4. Other Methods in Time Discretization
170
General rational approximations of (Co) semigroups were studied by Hersh and Kato [174] and Brenner and Thome~ [61]. Inequality (4.3) holds if r(z) is of order p and Aacceptable. In this section, we applied the method of Baker, Bramble, and Thom~e [25] for the approximation of holomorphic semigroups. A related work was done by Bramble and Thom~e [58]. The notions (i)o, (ii)o, and (iii)o were introduced by [148]. The Runge-Kutta method is regarded as a rational approximation of semigroups. Stability and error analysis were done by Crouzeix [97], Keeling [215], Ostermann and Roche [311], Lubich and Ostermann [254], [255], and Nakaguchi and Yagi [275] including the case applied for quasilinear equations. The fact that the Pad~ approximation r = R,.m of e -~ is An/2-acceptable if and only if m - 2 < n < rn was conjectured by Ehle and proven by Wanner, Hairer, and Norsett [402]. Related works are Birkoff and Varga [36], Ehle [119], and Saff and Varga [329]. Pad~ approximation Rnn(Z) can cast the implicit Runge-Kutta method of Butcher [69] applied to du/dt + Au = 0. The use of continued fraction expansion of the exponential function was proposed by Mori [270], where A-acceptability of Hn(z) is proven. It has also the practical use. Baker, Bramble, and Thom~e [25] showed Theorem 4.1 for the case that A is self-adjoint under weaker assumptions on r(z). The first case of this theorem is due to LeRoux [234]. Large constants in stability or error estimates can cause unreliabilities to the numerical computation. For this topic, see Lenferink and Spijker [232] and the references therein. 4.2. The first case of Theorem 4.8 was proven by LeRoux [234]. We have followed [148] for the notions (I)o, (If)o, and (IlI)o. Multi-step methods for ordinary and partial differential equations were also studied by Dahlquist [105], Zl~imal [421], Nevanlinnca [293], Raviart [324], Crouzeix [98], and Palencia [313]. 4.3. Product formula for (Co) semigroups was introduced by Trotter [388]. Nelson [292] gave a different proof making use of the Feyman path integral. Chernoff [78] extended the formula to a general principle as follows: Let {F(t)}t> 0 be a family of non-expansive operators and {e-tC}t>_ 0 a (Co) semigroup on a Banach space X. Suppose that )~ > 0 admits s-lira [I + At -1 ( I tl0
F(t,))] -1 :r = (I + )~C) - 1.'/;
for each :r E X. Then, it follows that s-limn_.~ F(t/n)'~z = e-tCz locally uniformly in t E [0, oo). Later it was generalized to nonlinear semigroups by Brezis and Pazy [64]. See w for details. On the other hand, Kato [206] studied unbounded perturbations, and Feit, Fleck, and Steiger [123], [124] applied the formula to find eigenvalues numerically. Related works were done by Chernoff [79], Kato and Masuda [208], and Kuroda and Toshio Suzuki
[228]. Concerning the case of bounded operators, see M. S~zuki [362], [363], [364] and the references therein. R.D. Rogava's paper is [327]. There, O ( l o g n / v ~ ) is asserted under the assumption of D(A1) C D(A2), instead of assuming (4.21) and D(A3~/2) C D(A32/2) A D(A3/2). Related works were done by Ichinose and Tamura [186], [187], [188].
Chapter 5 Other M e t h o d s in Space D i s c r e t i z a t i o n
Continuous problems have several fine structures and it is desirable to realize them in the discretizing process. In this chapter, we describe three techniques of FEM; lumping of mass, upwind difference, and mixed elements. The first two are concerned with the dissipative structure, and the last one is related to the variational structure. Those methods are efficient particularly in nonlinear problems as we shall describe in the following chapter. In the last two sections, we describe the method of boundary elements and that of charge simulations. Both of them are associated with fundamental solutions for specific operators and reduce the quantities of computation in some cases.
5.1
L u m p i n g of M a s s
We suppose that f~ C R 2 is a polygon. In Chapter 2, we studied the finite element approximation of the initial boundary value problem
ut-Au=O
i n f ~ x ( 0 , T)
with
Ulo~ = 0
ult= o = Uo(X).
and
This problem, however, is provided with the dissipative structure such as the maximum principle and the L 1 contraction. Those properties remain valid even in nonlinearly perturbed problems and control basic features of the solutions. Numerical schemes kept with those properties, therefore, are desirable from both theoretical and practical points of view. Method of lumping of mass realizes them.
S c h e m e of
Lumping
Recall that {%} is a family of regular triangulations of gt, and Z-h the totality of vertices subject to Th locating in Ft. The set of continuous functions, linear on each T E Th and taking the value 0 on the boundary, is denoted by Vh. For a E Zh, the function Wa E Vh is given by 171
5. Other Methods in Space Discretization
172
10 (x = a)
Wa(X) =
(X e Zh \ {a}).
The family {Wa J a C 27h} forms a basis of Vh and the interpolation operator rrh" W --+ Vh is defined by
7ChV= E
v(a)Wa,
aEZh
where W = {v e C(U) Jv = 0 on Of/}. Each a C Zh takes the barycentric domain denoted by D~. It is nothing but the closed convex hull of barycenters of the triangles in rh containing a as their vertices. Let 1 0
~,(x) =
(xeDa) (xe~\D~).
and denote by Vh the vector space generated by {~a I a C 2"h}. The linear transformation Mh " Vh ---+Vh, sometimes referred to as the lumping operator, is defined through Wa ~-+ Wa. Let M;: 9Vh -+ Vh be the dual operator associated with the L 2 inner product, and
I ( h = M;: Mh" V~ --~ Vh. Given u0 E W, we can show that the scheme d -~KhUh + AhUh =
0
with
Uh(0) = 7rhuo
(5.1)
in Vh has the desired properties, where Ah indicates the finite element approximation of - A . In the weak form, it is written as d
dt (gh, wa) + (VUh, Vwa) = 0
with
(Uh.(0), Wa)
=
(7rhUO,Wa)
for a E Z-h, where 'uh = MhUh. The operator Mh is an isomorphism and Kt71 = M,71 (M,7:)* is well-defined. Problem (5.1) is equivalent to d'It h
d--7 + K~'A,,~,,,
:
0
with
u,,(O) :
~o.
There are works by T. Ushijima concerning L 2 and L ~176 convergences of approximate solutions. Here, we study L 1 and L ~ structures of (5.1) in details. In this section, Xh and Xh denote the spaces Vh, and Vh equipped with the topology induced by Ll(f~), respectively.
5.1. L u m p i n g of Mass
173
/2-stability Before doing so, we show L p stability and convergence. First, we have a constant C >_ 1 independent of h satisfying
c-'
IIM~x,~ll,.,,,(~) <
xh
I~,(~)_< c IIMhxhll.,(r)
(s.2)
for Xh E X h , T C rh, and p C [1, co]. This implies
(s.a) In fact, take the canonical reference element T of which vertices are P0(0, 0), PI(1, 0), and /:'2(0, 1). Inequality (5.2) is then reduced to the case T = T from the regularity of {%}. Linear functions on T, provided with suitable boundary conditions, form a finite (up to three) dimensional vector space and the desired estimate follows because any two norms are equivalent there. Furthermore, because the extremal cases p = 1 and p = oc are valid, that inequality is uniform in p E [1, oc] by Riesz-Thorin's interpolation theorem. This gives (5.2). It is possible to evaluate C _> 1 explicitly by elementary calculations in use of the least angle of the vertices of T for instance, and is left to the reader. As a consequence, lumping operator is shown to have the following property, where p E [1, cxD]: Uh e X h ---+U
in LP(t2)
=:::a
To see this, take g > 0 and w E W satisfying
M,~
Jlu-
MhUh ---+ U
(5.4)
in LP(t2)
wJls,(a) < e. In use of
- ~ = M~ [~, - ~ , ~ ] + ( M , ~ , ~ - ~) + (~ - ~).
we have
IIMhUh -- UlIL~(~) < c (llu,~ - ~ I,..,,(~) + c + I1'~ - ",,-~'~ll~oo(~)) + IIMu,,,,.~ - wll,r.,o~(~)
9
Here, w is uniformly continuous on f~ and we obtain lim sup I l M h u h - Ullr,(a ) _< Ca. hi0
Because e > 0 is arbitrary, we get the conclusion. Now we study the uniform boundedness of Kh " X h --+ X h with respect to t h e / 2 norm. Taking -Xh E X h and v E W, we have
_< II~hllLp'(~)" IIMh~vlILp(~)<---II~IILp'(~)IIP, vlIL~(~)
5. Other Methods in Space Discretization
174
with 1 / p ' + 1/p = 1, where Ph'L2(f~) ~ Xh denotes the L 2 projection. If {7-h} satisfies the inverse assumption, {Ph} is regarded as a family of uniformly bounded operators on LP(gt) by Theorem 1.5. In this case we have
IIPhvll~.(~) _< c IIv
(5.5)
~,(~)
for p E [1, cx~]. The inequality
I(M;N,,..'v)I _< c ~hll~,,(r~)IIv
L,(~)
follows and hence
IIM;~,.II~p,(~) _< C I ~,.11...(~) is obtained. Combining this with (5.5), we have
I K.~.II~(~) _< C l ~ l ~(n)
(xh e x~).
(5.6)
If p = 2, the inverse assumption is not necessary for this inequality to derive, because IPh r2(n),r2(n) -< 1 always holds true. The L 2 projection P h " L2(f~) --~ Xh has the same property and we have I](Kh) -1X~hIIL~(~)-< C IX~hllL~(a)
(X~h C X~)
(5.7)
if {Th} satisfies the inverse assumption, and without it if p = 2.
Maximum
Principle
In the following theorem, the inverse assumption is not necessary. 5.1. If any T C Th is an acute or right triangle, scheme (5.1) is provided with the maximum principle so that
Theorem
0 <_ fh E Xh
==~
e-tg':'n"fh >_ 0
(5.8)
holds for the associated semigroup {e-tK;i ' Ah }t>_O on Xh. To see this, let Bh be the set of vertices subject to rh locating on Cgf~. Even for a E Bh, we can define a continuous function Wa on f~, linear on each T E ~-t~by I
Wa(X) -'~
0
(x = a) (:1;, E •h U ~h \ {a}).
Then we have
E aEZhUBh
wa(a)= 1
(z E-~).
(5.9)
5.1. L u m p i n g of Mass
175
We can also take the barycentric domain Da even for a 9 •h" Taking Wa as
Wa(X) =
(xeDa) (X 9 \ Da)
1 0
similarly, we have
Z Wa(X)---1
(x 9 -~).
(5.10)
aE:ThUI3h Let a, b E Zh t2 Bh with a =fl b. It is obvious t h a t
(Wa, Wb) "--0.
(6.11)
(V~a, VWb) <_0
(5.12)
The inequality
is a consequence of the following l e m m a proven similarly to L e m m a 1.9. Lemma
5.2.
We have VWa" VWb ~ 0
(5.13)
on each T E ~-h if a, b C ffh U Bh and a sC b. Proof: I f a E Z h U B h is not a v e r t e x o f T E ~-h, then Vwa = 0 o n T and (5.13) is obvious. Let a be a vertex of T and Ta be the face of T not containing a. Because the left-hand side of (5.13) is invariant under the r o t a t i o n of the variable x, we may assume t h a t T~ is located on the line x2 = 0 and a lies in {x2 > 0}. In this case OWa/OXl = 0 and hence 0w~ 0Wb V w ~ . Vwb = Ox2 " Ox2 follows. Let {Pj [ 0 < _ i < _ 2 } be the vertices o f T with P0 = a. Because T i s a r i g h t or acute triangle, the vector e = ( 0 , - 1 ) is expressed as 2
e = ~_s aj eoj j=l
with some aj >_ 0, where e0j = P 0 ~ . j = 1, 2. It holds t h a t 0W a
Ox2
Here e 0 j . Vw~ is a negative c o n s t a n t on T for
2
= e . VWa = ~ a j e o j j=l
9VWa <_ O.
5. Other Methods in Space Discretization
176
On the other hand we have e0j 9Vwb _> 0 (j = 1, 2) if b -r a, and
Owb > 0 O3g 2
--
holds similarly on T. Therefore, inequality (5.13) follows and the proof is complete. We are now ready to give the following.
Proof of Theorem 5.1: Note that (5.8) is reduced to A > 0, fh E Xh
===> m_ax(Ih + AK,71Ah) -1 fh _< max{0, m_ax fh}. fl
(5.14)
f2
Here, the well-definedness of (Ih + AKs
-1 is also included. In fact, (5.14)implies
fh <_ o
~
(I,, + ~ K s
f. <_ 0
fh > 0
==~
(Ih + AKs
-1 fh >_ 0.
or equivalently,
Then, tile fundamental relation
e-tKs
Ah
=
lim (Ih + tn- 1Ks lAb) -n 71,'----+00
assures (5.8). L e t [ . ]+ = m a x { . , 0 } .
(5.1,5)
Property (5.14)follows if max_ Uh _< max_ rrh [fh]+
(.5.16)
is proven for Uh C= Xh and fh = (Ih + AI(s In fact, then fl + = 0 implies u h+ = 0 and hence fh = 0 gives 'uh = 0 is obtained. This means the well-definedness of (Ih + AI(,71Ah)-I " X,, ---+Xh and also (5.14). For (5.16) to prove we may suppose that the maximum of the left-hand side is attained at some a E :2h with a non-negative value, because the right-hand side is non-negative. Suppose those situations. It holds that
(a~, ~h) + A (v,,,h, Vx~) = (L, ~,~) for Xh E Xh. Let Xh = IDa"
(a~, tea)+ A (v,,,,,,, W*,a) = (/,,, r&) We have (gZh, g&) = uh(a)Isupp Wal
and
(-fs ~ )
= fh(a) supp ~ l .
(5.17)
5.1. Lumping of Mass
177
Furthermore, the equality (VUh' VWa)
---
E Uh(b) (VWb, VWa) bEZh
=
~
(~(b) - ~ ( a ) ) ( W ~ , W a ) + ~.(a) ~
bEZh
--
(W~, W a )
bEZh
~
(~th(b) -- ,/th(a))(VWb, VWa) -- "/th(a) ~
bEZh\{a}
(V~Ub, Y e a )
bEI3h
holds by (5.9). Because a E/-h attains a non-negative maximum of uh we have (Vuh, VWa) ~ 0 by (5.12). Those relations imply
m~x_ ~ = ~(~) <_A(a) <_m_~xfZ gt
or equivalently (5.16). The proof is complete.
Ll-contraction Now, we proceed to the following. T h e o r e m 5.3. Under the assumption of Theorem 5.1, scheme (5.1) has the property of
L 1 contraction expressed as O <_ fh E Xh
=:~
f Mhe-tK[lAhfhdx
(5.18)
where {e-tK: 1mh}t>_O denotes the associated semigroup on Xh. Proof: Similarly, property (5.18) is reduced to ~ M~ (Zh + )~KhlAh)-l fh dx < faM~fh dx
(5.19)
for 0 <_ fi~ E Xh and )~ > 0. We have only show it for fi~ = Wa with a E Ih. Writing
Uh = (In + AKhlLh) -1 Wa = E
~bWb~
bEIh
we have ~b ~ 0 by (5.14). Relation (5.17) with A -- Wa and )(~h -- wc is expressed as E
~b { (Wb, We) -~ /~ (VWb, VWc)} : (We, We),
(5.20)
bEIh
where c E :Z-h. We shall show that
E = ~ M h ( I h + ) ~ K h l L h ) -1 Wa d x -
= ~
~ bEIh,CEBh
c~ (w~, w ~ ) .
MhWa dx
(5.2~)
178
5. Other Methods in Space Discretization
This implies (5.19) by (5.12) and ~b _> 0. In fact, we have
b6 Ih
b6 Ih ,c6 Ih U Bh
b61 h ,c6 Bh
c6 Ih U Bh
c6 B h
b,cE I h
by (5.10) and (5.20). Applying (5.9) for the last term, we get
bE Ih
cE Bh
cE Bh
This means (5.21)by (5.11).Property (5.19) has been proven. Ll-estimate
In connection with the L 1 structure, elliptic estimate (1.69) is also worth noting. We shall study its discretization here. We emphasize that the convexity of f~ is not necessary in the following theorem. Furthermore, not all T E rh must be acute or right triangles. Remember, however, that {rh} is supposed to be regular. T h e o r e m 5.4. If {7h} satisfies the inverse assumption, then q E [1,2) admits C > 0 satisfying
IIA;~K,,PhlI~I(.),..,,~(~) <
(s.22)
c.
If any corner point of [2 does not have the angle 27r furthermore, the convergence lira hi0
IlAhlI(,~Pj~v -
A-lv[Iwl,q(n)
= 0
(5.23)
holds for v E LI(f~), where Ph denotes the L 2 projection. Proof: Because {Th} satisfies the inverse assumption, {Ph} is regarded as a family of uniformly bounded operators on L2(f~) and we have (5.5). If we apply (5.6) for p = 1, inequality (5.22) is a consequence of (1.69). Taking v E Ll(f~), we get it as
Turning to the proof of (5.23), we note that it is reduced to limhlo IA-h:I(hPhv -- A -lvllH:(a ) - 0
(v E L2(~)) .
(5.24)
5.1. Lumping of Mass
179
To see this, let us admit (5.24) for the moment. Given v E Ll(f~) and e > 0, we can take w E L2(f~) satisfying IIw - VlILI<~) < c. In use of (5.22), (1.102), and (1.69), we have
IIA-;lKhPh~- A-iv
wl,q(f~)
]A-s
_<
(v - w)[lWl,~(~) +
IIA-~ ( w - ~)1 w~,~<~)
<_ C (c + IlA~'KhPh~ -- A - i ~ l l / r ( a ) ) . Therefore, letting h I 0 and e ; 0, we get (5.23). We now prove (5.24). Taking v E L~(a), we have [ A h l K h P h - A -1] v
=
[ A h l p h - A - I ] W + A h l p h ( I ( h - y) Phv
=
( R h -- I ) A - i v
+ R h A -1 (Kh - I ) Phv,
where Rh" V --~ Xh denotes the Ritz operator and relation (1.24) is m a d e use of:
Rh A-1 = Ah 1Ph Because v E L2(f~), it holds t h a t A - i v E V = H~(f~). We have lim hlO
II(R
- i) A - i v
V
= 0
by (1.39). On the other hand we have IIRhllv, v ~ c , and therefore the relation lim IIR~A -1 (K~ - I ) P , ~ l l v hi0
= 0
follows from lim IIA -1 (Kh - I ) P ~ l l ~
= 0
hi0
(5.25)
Thus, we have only to derive (5.25) for ( 5 . 2 4 ) t o prove. We shall show w-lim (Kh -- I) Pt~v = 0 hlO
in L2 (f~).
(5.26)
T h a n k s to the assumption to f~, the operator L -1 9 L2(f~) + U~(f~) = V is compact. Relation (5.26) implies (5.25), and in this way, (5.24) is reduced to (5.26). For the proof of (5.26), we note t h a t (5.6) with p = 2 implies
II(Kh -- I)PhlIL=<~),L=<~) ~ C. Therefore, (5.26) is reduced to the case of v E W similarly. In fact, assume (5.26) for v = w E W. Given v E L2(t2) and c > 0, we take w C W satisfying
5. Other M e t h o d s in Space Discretization
180 We put Jh =
(Kh -
I) Ph for simplicity and take u E
L2(f~).
T h e n we have
I(&v,~,)l < II& ( v - ~)IIL=(~)II~llL=(a) + I(&w,,~)l _< g e Ilull~=(~) + I(Jhw,~)l. Letting h + 0 and g + 0, we get (5.26) for v E L2(f2). a similar argument now reduces (5.26) (for v E W ) to
(5.27)
lim ((Kh -- I) Phv, w) = 0 h,t O
for w C W. We have only to verify (5.27) for v, w E W. For this purpose we note t h a t
((ir
- ~) e~,,, ~)
=
(M;M,,P,,v, ~) - (e~v, w)
=
(MsPhv, MhPhw) - (Phv, w)
=
(M,,Phv, MhP,, (I - re,,)w) + (Mhre,,v, M h r e , , w ) - (Phv, w) + (MhPh (I - re,,)v, Mhrehw).
For the first term of the right-hand side, we make use of (5.3) and (5.6) as
IIM,~PhvlIL,(~) < c IIP,,v gl(~'2) ~ C II,, L1(~2) and
IIMhPh (I
-
reh)w
IL=(a) < IIPh (I -- ~h) WlIL=<~> ~ C I1(1 -- ~h)W L=(~),
respectively. It follows that
I(MhPhV, MhPh (I - reh) W)[ < C l v l l z x ( a ) [ ( I - reh) ~IIL~(~) ~ 0 from the uniform continuity of w c W . Similarly, we have
I(MhPh (I -- re,,)v, Mhrehw)l
,0.
For the rest terms we make use of (1.40): lim liP,,,,- vl e=(~) = o hlO
and also lim []Mhrehv- V[[f~o(a) = lim [[MhrehW -- W [L~(a) = 0, h~0
h~O
which is a consequence of the uniform continuity of v and w. We get that
I(A6,ro, v, MhrehW) -- (Phv, w)l - - ~ 0 as h i 0, and relation (5.27) has been proven.
[]
An analogy of the above theorem is verified for schemes without lumping more easily.
5.2. Upwind Finite Elements
5.2
181
U p w i n d Finite E l e m e n t s
Continue to suppose that Ft C R 2 is a polygon. Let b(x) be a Lipschitz continuous vector field, and f, g scalar fields on t2, respectively. The problem -Au+b.
Vu=f
inf,,
u=g
oncgf~
(5.28)
describes the phenomenon of convection-diffusion, so that u(x) is subject to the flow generated by b(x) and diffuses by itself. In this sense, the system keeps basic features of the problems in fluid dynamics. If the convection term is stronger than that of diffusion, then the standardized m e t h o d of w is not efficient. Here, one sees t h a t the value of u at x is under the influence of its values upwind with respect to b(x). Upwind finite element method is based on this observation and we describe the simplest case in this section. We make use of the following notations of the previous section. Let {Th} be a family of regular triangulations of acute type so that any T E Th is an acute or right triangle. The set of piecewise linear continuous functions on f~ is denoted by Wh, and Vh = Wh M V, where V = H0X(f~). The sets of interior and boundary vertices are denoted by 2-h and Bh, respectively. Each a E :Z-hU Bh admits We E Wh with the value 1 at a and 0 at each b C :Z-hU Bh \ {a}. Interpolation operator 7rh'C(gt) ~ Wh is defined by
rrhf= E
f(a)Wa"
aCZhUBh
We also make use of the lumping operator Mh " Wh -+ Wh and set MhXh = Xh in short. Given a C 2-h, we take T(a) E rh which meets the vector b(a) with the origin a, and set
Bauh = ~
(b(a).
VUhlT(a) ) Wa.
aEIh
We study the scheme to find Uh E Wh satisfying Uh -- r~hg E Vh and ( v u , . Vx ) +
=
(5.29)
for Xh E ~ . We can show the following Theorem
5.5. Suppose one of the following conditions:
1) Each a E Ih admits a sequence a l , . . . , am C Ih, am+t E 13h with ai+l is a vertex of T(ai) for 1 <_ i < m. 2) Each T E Th is an acute triangle.
Then, (5.29) is uniquely solvable and the discrete maximum principle f, g < 0
follows. (f, Xh).
~
uh < 0
(5.30)
The right-hand side of (5.29) may be replaced by the term without lumping,
5. Other Methods in Space Discretization
182
Discrete maximum principle (5.30) can imply the L ~176 bound of the Ritz operator (1.59) and also IIu -uhliL~(a) = O(h), although details are not described here. If p (~h, Xh) is added to the left-hand side of (5.29), the conclusion of Theorem 5.5 holds without the assumption, where p > 0.
Proof of Theorem 5.5: From the Fredholm alternative, the solvability follows from the uniqueness in (5.29). The latter is a consequence of the discrete maximum principle. We have only to derive (5.30), supposing that 'uh E Wh is a solution of (5.29). It is proven similarly to (5.14) and we shall describe the sketch. Because uh = 7rhg on oqf~, we have only to show urn(a) 5 O, supposing that m a x g u h is attained at a E Zh. Let Xh = w~ in (5.29). We have
(w,,,, v x , ) > 0
(5.31)
(BhUh, Xh) = supp Wa] (b(a). Vuh)r(a) --> 0.
(5.32)
and
Because the right-hand side is non-negative, the left-hand sides of (5.31) and (5.32) are equal to 0. If the first assumption holds, then a~ attains maxg'uh for i = 1, 2 , . . . , m + 1. This implies m a x ~ u h _< 0 by 'uh = rrhg _< 0 on oqf~. If the second assumption holds, we have (VWa, Vwb) < 0 for a : / b . Then, the inequality (5.31) implies that 'uh is a constant on f~. Then m a x g u h _< 0 follows similarly. To prove the latter part, we take A > 0 and u~, E Wh satisfying u h - rrh9 E Vh and
(~h, ~,~) + a (v,,,,h, vxh) + (~h,,,,, ~,~) = (~-Tf, ~,~)
(5.aa)
for )C~ E Vh. Without those assumptions, we can show m_ax a uh _< Inax { m_ax a rrhf, I ~ X rrh } .
(5.34)
In fact, if a E 5~ attains max~u,, we have (B,,u,,, ~,,) >_ O. Then, relation (5.33) implies (5.34). The proof is complete. []
5.3
Mixed Finite Elements
The m e t h o d in consideration is concerned with the variational problem with constraints. To describe the idea, we take the Dirichlct problem for the Poisson equation Ap = f
in [2
with
p = 0
on cgf~,
(5.35)
where f~ C R 2 is a polygon. Putting 'u = Vp, we llav(; u-
Vp = 0
and
V . ',, = .f.
(5.36)
183
5.3. Mixed Finite Elements
Let V = L2(f~) 2, W = H~(~), and B ' = V : W ~ V. The dual operator B = ( B ' ) ' : V "~ V' --, W _~ W' is defined through the representation theorem of Riesz. Similarly, f 6 L2(ft) determines 9 e W', regarded as an element of W, by (g, q)w = (f, q)L2(fl) for q 6 W. Under those preparations, (5.35) has an abstract form, to find (u,p) 6 V x W satisfying
u + B'p = 0
B u = 9.
and
(5.37)
As' we shall see, this problem is realized as an Euler equation on u for a variational problem with a constraint, where p acts as a Lagrangian multiplier. Then, it is transformed into another variational problem without constraints, where the solution (u, p) is to be a saddle point. Method of mixed (hybrid) finite elements arises naturally as a discretization of those structures. Although it had been applied to many problems in engineering related to solid mechanics, hydrodynamics, electro-magnetic theory, and so forth, the discovery of such a fine structure made it much more reliable practically. This section describe the fundamental ideas.
Abstract
Theory
We take the abstract theory first. Let V and W be real Hilbert spaces, and ,4 and B be bounded bilinear forms on V x V and V x W, respectively. We have
1"t4(u, ~)1 ~ C1 for u , v E V, where C1 > 0 is a constant. variational problem with constraints,
II~lIv IIvll~
Given g c W and F E V', consider the
(5.39)
inf u ,7, where U = {v E V l B ( v , q ) =
(5.38)
(9, q)w for any q E W} and
y(~)
= ~
A(v, ~) - F(~)
The bounded linear operator B : V ~ W is defined by
(B~, q)~ = z(~, q)
(q ~ w ) .
We have U = {v E V I B y = g} and hence U # 0 if and only if g 6 Ran B, which means the existence of Uo E V satisfying Buo = g. Then it holds that U = {u0} + Ker B. It is a closed atiine space in V and the s t a n d a r d argument described in w the condition
A(v, ~ ) ~ ~ I vll~
(v
e
Ker B)
applies. If
(5.40)
5. Other M e t h o d s in Space Discretization
184
holds with a constant a > 0, variational problem (5.39) is uniquely solvable. The solution u C V is characterized by u-
u0 E U
and
.A(u, v) = F(v)
(v C Ker B ) .
(5.41)
Let c~ : V' + V be the canonical isomorphism defined through Riesz' theorem, and A : V + V the bounded linear operator associated with A:
A ( ~ , ~) = (Au, ~ ) ~
(~, ~ 9 v ) .
Tile second equality of (5.41) is expressed as
(oF-
Au, v)v = 0
(v 9 Ker B ) ,
or equivalently, a F - A u E (Ker B) • in V. Here, the dual operator of B is realized as B' : W + V through Riesz' theorem: V' " V, W' _~ W, and then (Ker B) • = Ran B' follows. Under the assumption that Ran B' : closed
C V,
(5.42)
u solves (5.41) if and only if it admits p E W satisfying
A u + B'p = o F
B u = g.
and
(5.43)
Note that p acts as a Lagrangian multiplier here, and (5.37) obeys a form of (5.43). If one prefers to tile weak formulation, (5.43) may be replaced by
.A(u, v) + B(v, p) = F(v)
B(u, q) = (g, q)w
and
(5.44)
for a n y v C V a n d q C W . Problem (5.43) is reduced to another variational problem without constraints. In fact, (u, p) solves (5.43) if and only if it is a stationary point of 1
,]('u, q) = -~A(v, v) - F ( v ) + (q, B v - g)w
(5.45)
on V • W. Actually (u, p) is a saddle point,. To see this, take q E W and v C V. If (u, p) solves (5.43), then we have 2(,,,, p) - 2(,,,, q) = (~, - q, B,,, - 0 ) ~
= 0
and
! A ( . . , .~,)- ~A(.,,,, . . ) 2
F(.v-
u)+
(~, B ( v -
~))~ 1
= A(.,~, v - u) - r(,,, - ..,) + (p, B ( v - ..,))~ + ~ A ( v - .,,,, ,,, - u) 1 A ( , v - ,u, v - u) a F , v - "u)v + -~
=
(Au + B'p-
-
_1 2 A (v - u, v _ u) > O.
185
5.3. Mixed Finite Elements
In particular, the relation 2 ( u , q) _< 2 ( u , p) _< ST(v, p)
(v e V, q E W)
(5.46)
follows as is expected. Relation (5.42) is equivalent to the existence of k > 0 satisfying
IlB'qllv =
sup ~o~
> kllq ~li~
-
for q C (Ker B') -L. Because (Ker B') • = Ran B, this means sup
B(v, q) > k Ilq w
(q E Ran B).
(5.47)
v~.\{0~ II~ll. -
Here, the closed range theorem says that Ran B' C V is closed if and only if Ran B C W is so. Because (Ker B) -c = Ran B', similarly it is equivalent to the existence of k > 0 satisfying
sup
qeW\{0}
15(v, q) > k II~llv q Iw --
(v e R a n / 3 ' ) .
(5.48)
We have the following. T h e o r e m 5.6. Let V and W be Hilbert spaces over R, and ~4 and 13 be bounded bilinear forms on V x V and V x W , respectively. Suppose (5.38), (5.~0), and (5.47) (or equivalently (5.~8)). Let A : V ---+ V and B : V ---+ W be the bounded linear operators associated with A and 13, respectively, and take g E Ran B. Then, the solution (u, p) r V x W of (5.~3) (or ( 5 . ~ ) ) exists, and is characterized as a saddle point of 2 defined by (5.~3). Here, u C V is unique, while p E W is unique up to an additive element in K e r B ' . Actually, p is taken uniquely in Ran B, and then we have Ilullv + IIPlIw < C (IIFIIv, + Ilgllf),
(5.49)
where C = C (5, C1, k) > 0 is a constant. Proof: From the above arguments follow the equivalence between (5.46) and (5.43), the existence of the solution (u, p), and the uniqueness of u. The orthogonal decomposition u = Ul + u2 with Ul C Ker B and u2 C Ran B' gives Bu2 = g, and assumption (5.48) assures
I1~11~ _< k'-I Ilgll~. Furthermore, A u l = ~ F -
Au2 - B'p and hence
A(ul, v) = f(?3) -- A(u2, v) follows for v E Ker B. Putting v = ul, we obtain
II~lll. _< ~_1( FII., + 61 lu~ll.) _< ~-1 (llrl ., + c~k-' I gll~).
186
5. O t h e r M e t h o d s in Space Discretization
Similarly, the orthogonal decomposition p = Pl + P2 with Pl E Ker B ~ and p2 E Ran B gives B~p2 = o F - A u and Ilp211w <_ k - i (IiFlIv , -}-Ca IlulIv) by (5.47). Inequality (5.49) follows and p is taken uniquely in Ran B except for an additive element in Ker B'. T h e proof is complete. []
Poisson
Equation
Above t h e o r e m justifies ( 5 . 3 6 ) i n the following way. Let V = L2(f~) 2, W = H~(f~), A(u, v) = (u, v)c2(n)2, and B(v, q) = - (v, Vq)r2(n)2. The bounded linear operator t9' : I/V --~ V is realized as B ' = V. From Poincar6's inequality, IIB'qllv -IIVqlIL~(n)~ provides a n o r m on W. Hence d a n B = w , Ran B ' = v , (5.47), and (5.48) follow. Given f E L2(f~), we set F = 0 and define g E W ' ~_ W by ( g , q ) w = (f,q)L2(n) for q E W. The abstract theorem assures a saddle point (u, p) E V x W of 2(v,q)=
1
~11 v 2L2(n)2 -- (v, Vq)c:(a)2 -- (f, q)L:(n)
satisfying (5.43). We have (U, V)L2(f~)2 ~- ( V p , V)L2(12)2
and -
('a, Vq)L2(fl)2 : (f, q)L2(n)
for any v E L2(gt) 2 and q E H~(f~). This implies p E H~(f~), Vp = u E H I ( ~ ) , and V u : f E L2(ft) 2. In particular, p E H2(f~)VI H~(ft) is a solution to (5.35). P r o b l e m (5.35) is formulated differently in this framework. Let V : H(div, f~):
H(div, ~) : {~ c r~(~)~ I v . ~ c r~(~)} It is a Hilbert space with the inner product
(~, V)v : (u, ~)~(~) + ( v . u , v . ,v)L~(~) 9 Taking W = L2(ft), we set A(u, v) = (u, 'V)L2(fl)2 for u, v E V, and B(v, q) = ( V . v , q)r2(a) for v E V, q E W. This time tile b o u n d e d linear operator B : V + W is realized as B = V., and we have Ker B = {v E V I V . v = 0}. Condition (5.40) holds as A(v, v ) =
IIv ~/
(v E Keg B ) .
We shall show t h a t inequality (5.47) holds for any q E W = L2(~). This implies Ran B = W in particular. In fact, take (b E Hg(f~) satisfying V - V c b = q. We have [IV011L=(a) _< C IlqlIL=(a) and hence v = Vq5 E V satisfies v 2V -- I V r IL2(~) 2 _ ( C 2 + 1) Iq IL2(fl) 2 2 -[- ]lV " V 4) 2g2(fi) < 9
187
5.3. M i x e d F i n i t e E l e m e n t s
T h e n (5.47) follows as
z3(~ q) ,
=
I~lIg
llqll ~
w
1
>
IlVllv -
Ilqllw.
v/C 2 + 1
Let F = 0 and g = f 9 W = L~(f~) in (5.45). There exists a saddle point (u, p) 9 V x W of d(v,q)
1
2
: ~ Ilvll L 2 (a)2
-t-
(q, V " v - f)L2(a)
Relation (5.43) reads;
- (~, ~ ) ~ ( ~ ) : (p, v 9~ ) ~ ( ~ ) and ( V . u, q)L2(a) = (f, g)c2(a), where v C H(div, f~) and q C L2(f~) are arbitrary. This implies f = V - u , p E H2(f~), and Ploa = 0. In particular, p E H2(ft) A HI(f~) solves (5.35).
Vp = u,
The Stokes System In the above problem, f~ may be an arbitrary b o u n d e d Lipschitz domain in I~n. We have an i m p o r t a n t example formulated as above for this case; the stationary Stokes system described as -Au+Vp=f
and
in f2
V.u=0
(5.50)
with u = 0
on 0~.
(5.51)
Here, u and p are vector and scalar fields on f~, respectively. For V = HI(f~) n and W = L2(f~), its weak form is introduced; to find (u, p) E V x W satisfying
~(
V u <9 Vv) d x - (p, V . v)r2(n) : ( f , V)L2(a)n
(5.52)
and
(v.~,
q)~(~) = 0
(5.53)
for any v E V and q E W. Here, OU i z,3
OV i
5. O t h e r M e t h o d s in Space Discretization
188
for u = (u i) and v = (vi). We introduce the b o u n d e d bilinear forms
A(~, ~) = s ( w | w )
dx
and
and take F ( v ) = (f, V)L2(n). and 9 = 0. Condition (5.40)is a consequence of Poincar6's inequality. To examine the closedness of Ran B or Ran B', let q E Ker B'. This means (q, V 9V)L2(a ) = 0 for any v E V. The space Ker B' is composed of constant functions and hence Ran B = (Ker B') • = {q E L2(~) I (q, 1)L2(n) = 0} follows. If 0f~ is Lipschitz continuous, it is known t h a t any q E Ran B admits v E V satisfying V-v = q
Ilvll~ <
and
C llqlIw,
(5.54)
where C > 0 is a constant determined by f~. In this way condition (5.47) holds true. In use of
IVvll~= ~
~
g~<~)
1,3
for v = (vi), problem (5.50) with (5.51) is formulated as (5.44), or finding a saddle point of d ( v , q) -- ~1
Vvll~-
(f, V)L2(n), 4- (q, V ' V ) g 2 ( n )
oil H~(f2) n x L2(f~), where T h e o r e m 5.6 is applicable. The variational structure induces the discretization. Under the assumptions of Theorem 5.6, we prepare a family of finite dimensional subspaces of V x W denoted by {Vh x Wh}h>0. Assume the property that any (v, q) E V x W admits a sequence {(vh, Wh)} in (Vh, Wh) E ~ X Wh satisfying lira (ll v - vhlIv 4-IIq - qhlIw) = o. td.O
The linear operator Bh : Vh---* Wh is defined by
13(vh, qh) = (S,,vh, qh)w for vh E Vh and qh E Wh.
5.3. Mixed Finite Elements
189
It is reasonable to assume (5.40) and (5.47) with V, W, and B replaced by Vh, Wh, and gh, respectively, with constants ~ > 0 and k > 0 independent of h: inf ~or~\~o~
r
(5.55)
~h) ~___5,
I1~11~
inf
sup
B(Vh, qh)
> k.
(5.56)
Then, Ritz-Galerkin method (5.44) is realized as to find (Uh, Ph) E Vh • Wh satisfying
.A(~h, vh)+ Z(vh,ph) = P(vh)
(vh ~ Vh)
(5.57)
and
s ( ~ , q~) : (g, q ~ ) ~
(q~ 9 w ~ ) .
(5.s8)
Those relations are equivalent to
AhUh + B~hPh = PhcrF
and
BhUh = Qh9,
respectively, where Ah : Vh --~ Vh denotes the linear operator associated with Alyhxvh, B~ " Wh ~ Ws --~ l/h ~ V/~ the dual operator of Bh " Vh ~ l/Vh, and Ph" V ~ Vh and Qh : W ~ Wh the orthogonal projections. This means that Qhg E Ran Bh is necessary for this discretized problem to have a solution. Then Theorem 5.6 applies and there exists a unique (uh, Ph) E Vh x Ran Bh satisfying (5.57) and (5.58). This solution is also a saddle point of ,7 restricted to Vh x Wh, with ,~ defined by (5.45). Based on those considerations, it is natural to assign the requirement QhRan B c Ran Bh. Taking orthogonal complements in Wh, this is equivalent to assuming Ker B~ C Ker B'. Under those circumstances the following theorem holds, where (u, p) denotes the unique saddle point of J on V x W belonging to V x Ran B. T h e o r e m 5.7. We have Uh]ly + l i P - Phllw < C ( inf Ilu - Vhl]y + inf liP-- qh]lw~ \ VhE Vh qh E Vh / with a constant C > 0 independent of h. Proof: Equalities (5.43) and (5.57)-(5.58)imply A ( u - Uh, Wh) + B(Wh, p - Ph) = 0
and
13(u - Uh, rh) = 0
for any wh E V~ and rh E Wh. Taking vt~ E Vh and qh C Wh arbitrarily, we have A(v~ - uh, wh) + S(w,~, q~ - p~) = A(vh -- u, wh) + S(w~, q~ -- p) and B(~
-
~,. <~) : s ( ~ . - ~. <~).
(5.59)
5. Other M e t h o d s in Space Discretization
190 or equivalently,
(5.6o)
Ah(Vh -- Uh) + B'h(qh -- Ph) = PhA(vh - u) + PhB'(qh - p)
and s,,(v,
- u,,) = Q,S(v,,-
(5.61)
u).
Theorem 5.6 is applicable to (5.60) and (5.61) with V, W, A, and B replaced by Vh, Wh, Ah, and Bh, respectively. We have I~h - ~hl v +
Iqh - p , llw
<_ c tlPhA(v,,. - ~)llv + C ( PhB'(qh -- P)I v +
QhB(v,,.- ~) ~) _< C ((tlmll + IIBII)Ilvh- **llv + IIBII IIq,~ - PlIw),
and hence II?t
~
'/t h v + I p
-
PhlIw -----
I1~, - v,,. v + I p
-
qh
_< c ' (11,,- vhllw + l i p -
w
+
Iv,,. - ",,,,. Iv + Ilqh
-
p,,.
w
qhllw)
follows. Thus, inequality (5.59) has been proven. Although it is not easy to construct such a family {Vh x Wh}h>0 with the required properties in T h e o r e m 5.7, many ways are known now.
5.4
Boundary
Element
Method
(BEM)
If one makes use of the fundamental solution, then tim boundary value problem is reduced to an integral equation on the boundary. Boundary clernertt method arises, associated with that structure. Let f~ C R n (n = 2, 3) b c a bounded domain with Lipschitz t)oundary 0f~, and consider tile boundary value problem -A'~J = f
in [2,
'u-- g
on ~)f~.
The fundamental solution
1 F(:r) =
~
1
(rz-- 2)
log T~
1
47r [:,:[
(rz = 3)
satisfies - A F = ~(0), where d (lcnot(.'s Dira("s delta function. If on(: takes w(:r) =
.[~ F(:r-
:,j).f(y) dy,
(5.62)
191
5.4. Boundaly Element Method problem (5.62) is reduced to the case f = 0: -Av=0
int2,
v=g
on0t2
(5.63)
The fundamental solution induces also single and double layer potentials
F(x) = foa V(y)F(x - y) dS v and
o
~ r ( ~ - v) ds~
respectively. For ~ E 0t2, we put A+(~)=
lira
xCgt+~
A(x),
where ~+ = ~c and ~ _ = ~. If V is continuous, we have [F] + =_ F + -
F_ = 0
on 0~.
(5.64)
Given x c 0t2, we can take the principal value for F(x) to define. On the other hand (O/Onv)F(x- .) is regarded as an Ll(0t2)-valued continuous function of x E 0t2. Letting
Ho(~) = foa V(Y) ~---~vF(~- Y) ClSy for ~ E 0t2, we have 2H+ -- i V + 2H0
on 0~.
(5.65)
In the Fredholm theory, property (5.65) is applied to reduce (5.63) to an integral equation on the boundary. In fact, we have v(x)=-2foa#(y)o~v['(x-y)
dSy
for x E ~, with # satisfying
#(~) - 2 / o # ( y ) o ~ v F ( ~ - y) dS v = 9(~) for ~ C 0t2. Linear operator K :
C(OFt) ~ C(Ot2) defined by f
K,
(5.66)
=
0
jo ,(v)-~ r(. - v) dS v
is compact by Ascoli-Arz~la's theorem. Unique solvability of (5.66) then follows from the Fredholm alternative; in fact 1/2 is not an eigenvalue of K.
5. Other Methods in Space Discretization
192 However, the simple iteration scheme #n-I-1 -- g -~- 2 K p n
(n = 0, 1, 2 , ' ' " )
does not always converge because - 1 / 2 is now an eigenvalue of K. In particular, }--~n(2K)nr does not converge for the corresponding eigenfunction r - 1. To see this, let us apply Green's formula to -Av=0
inf,,
Ov On =g
on Oft.
(5.67)
We get
~(*)=fo (-~(v)~ r(*-y)+v(v)r(*-.~J))ds~ for x C f~. Therefore, from (5.65), it follows that
/o
/o
o
for ~ E 0f~. Because v = 1 is a solution to (5.67) for g = 0, this implies the assertion that - 1 / 2 is an eigenvalue of K. The boundary element method avoids the difficulty in the following way. First, one derives from (5.63) that v(x)=fo n
O--~F(z - Y) + ~---~v(y) 9F(x _ (-g(Y) On
y))
dS~
(5.68)
for x E ~. Then, (5.65) gives 1
-~v(~) + f o n v ( y ) ~ F ( ~ -
~.])dS~ = fon J~v(y) . F(~- y) dS~.
If the integral equation ~o q(~)F(.(-:t j ) d S y =
~f(~)+
Jo.f(:~j)~F(~-y)dSy
(5.69)
in ~ C oqf~ is solved for q = (O/On)v, then formula (5.68) gives the solution v(x). This section is devoted to the wellposedness of the scheme; we show the unique solvability of (5.69). It was founded by several authors with the efficiency of the application of Ritz-Galerkin's or the collocation method. Let
and
A(p,q)= fo~.s ,, v(e)q(e)r(e- ,),~,s~,s,, The following theorem indicates that the bilinear form A( , ) is realized as a bounded coercive bilinear form on H-1/2(0f~) x H-l/'2(Of~). Then, the unique solvability of (5.69) is a direct consequence.
5.4. Boundaw Element Method Theorem
193
5.8. There exist constants C1, C2 > 0 determined by gt such that
c, Ilqll~H-1/2(0~"~) __< A(q, q) _< 02
Ilqll 2u_,,,~(o~)
(5.7o)
forqEX. Proof: Fourier transformation of a rapidly decreasing function f C S ( R ~) is given by .T" = f (~) = f
e-'Z~ f (x) dx,
n
where z = v/L~. Given q c X, we take Tq E D ' ( R ~) defined by (r
Tq) ---
fOa qr dS
for r E D ( R n) = C~~ Its support is contained in O~. It is also regarded as a compact measure on R ~ and Tq is taken as a C ~ function on R n. Now we show
A(v, q)= (Z~)-nJ~ 1___~(~)tq(~) d~
(5.7].)
for p, q E X, with the right-hand side converging absolutely. In fact, if n = 2, we have f'({) = p.v. ~
1
+ c(~({)
(5.72)
with a constant c. From the assumption, we have Tq(0) = 0. Regarding P C 8'(R2), we can define P 9Tq C S'(R2). By (5.64), it is regarded as a continuous function on R2:
r 9T~(.) = f
JO
r ( ~ - y)q(y)
dSy .
We also have
f (r 9T~) = t . ~q =
p.v. ~
1
+ ~a(r
}
Tq(~)
9t~(r = ir
Let ~b E $ ( R 2) be real-valued. The inverse Fourier transformation is given as
~(~) = (2~)-~ a~f~ ~'~ ~(~) d~ and hence
f~ f0r(~-y)q(y)dS~r
= (r,
Tq,r (5.73)
5. Other Methods in Space Discretization
194 follows. We take p C C ~ ( R 2) satisfying 0_
supppc{lxl<_l},
and
[ JR
2
p dx = l.
Let p~(x) = e-2p(x/e) for e > 0. Given p c X, we put G = P~ * Tp. The left-hand side of (5.73) for r = ~ converges to (F 9Tq, Tp) = A(p, q) as e I 0. To handle with the right-hand side, we shall show that 72 1 2
B~c~us~ Ir
IT, 9I~1 -IT,I, th~n the right-hand sid~ of (5.73) convcr~r to
i~l= q(~)" ~-~p(~) d~ as e $ 0 by the dominated convergence theorem, and hence equality (5.71) follows. It is easy to see that (5.74) is reduced to the case p = q. Then we have
=
d~ + I_<~
I~l ~
1 ,>
12
d~ = I + II.
1~
The second term of the right-hand side is estimated as
II<_2
~1+1~12 d~-CllYqll2_,(~=).
Here, we have
[[Tqll/_/_I(IR2) -- sup { <2/2,Tq) I w ~ HI(I~2), [Iwtti/l(iR2) ~__1 } .-~ s u p { ~ n q v d S =
v E H'/2(Of~), ,.wl H,/2(on) <_ 1} (5.75)
]qlln-'/~(on)"
On the other hand, we have ]7~q(()- ]bq(0)] -< ].~a (e .... ~ - 1 ) q ( x ) d S x _< [e -'x~ - 1[ ,,,/2(oa)[IqllH-1/:(on) <- Cl~l Those relations imply
<-
ll ll "
Ilqll.-,/~(0n).
195
5.5. Charge Simulation Method
Because ] = I for n = 2, the right-hand side of (5.70) is proven. The left-hand side of (5.70) follows similarly from (5.75)as
I~
dg > ~f -
i5 dg ~
Ilq
I~-~/=(o~) 9
~ 1+1~
If n = 3, we have 1
f'(~) = ~-~ E L~oc(IR3).
(5.76)
It also holds t h a t [Tq(O)[- 1(1, Tq)l
~ O q [H_1/2(0~).
Noting those relations, we get (5.70) and (5.71) similarly. The proof is complete.
5.5
Charge
Simulation
Method
(CSM)
The m e t h o d in consideration is also concerned with (5.63) and make use of the fundamental solution F(x). We take N points y ~ , - . . , YN outside f~ and the solution v(x) is approximated by
N
(5.rr) j=l
Here, the coefficients Q 1 , " " , QN are determined by
v N(xj) = f(xy)
(5.78)
for j = 1 , . . . , N , where X l , ' ' " ,XN are taken on 0f~. Sometimes {yl," "" ,YN} and { x l , . . - , X N } are called the charge points and the collocation points, respectively. In this section we study the case that gt is a two dimensional disc, ft = {x E R 2 I Ixl < P} with p > 0, and the charge and the collocation points are so located as yj = Re ~(j-1)~ and xj = pe ~(j-1)~ respectively, where z = x/~-l, R > p and 0 = 2rc/N.
Wellposedness Equation (5.78) is expressed as G Q = f , where G = (gij) with gij = F ( x i - yj), Q = T ( Q 1 , ' " , QN), and f = T ( f ( x l ) , . . . , f(XN)). In the above case, it follows t h a t 1
gij = -~--~ log p - Rez(J-i)~ I ~ Lj-i.
5. Other Methods in Space Discretization
196 Therefore, G is a cyclic m a t r i x given as
Lo LN-1 G
L1 Lo
9
.
L1
L2
999 LN-1 999 LN-2 .
9..
Lo
and hence :
p=O \ k=O
follows for w = e *~ L e t t i n g N-1
~;(z) = ~
~r
(z- P~),
k=O
we o b t a i n N-1
det G = I - I ~P(P)"
(5.79)
p=0
We can show the following. 5.9. It holds that
Lemma
(p =- 0 mod N)
- ~ log Iz~ - ~ 1 ~p (z) = I 1 N
for
Izl
1 ( ~z~lR)
Iml
Z TM
(otherwise)
(5.80)
< R.
This implies cpp(p) r 0 for p = 1 , . . . , N holds by (5.79). Theorem
5.10.
1, in particular. Hence tile following t h e o r e m
The matrix G is non-singular if and only if RN
--
pN
?s 1,
(5.81)
in which case scheme (5.78) is uniquely solvable. Proof of Lemma 5.9: In the following, the equivalence class associated with m o d u l o N is d e n o t e d by "-". If p = 0, then w p = 1. In this case, we have N-1
qpp(z) -
1 27r Z
k=O
1
=
N-1
1 log Iz - Rwkl -- --~-~ log 17I Iz -- Rwkl
-2--; log I~~ - ~ 1 .
k=O
5.5. Charge Simulation M e t h o d
197
In the other cases, p = 1 , . . . , N -
r(~-R~
1, we have 1
~) =
-2---~ log ]z - Re,,,k ]
1 log ]Ra,,k] + log 1---R-2~1 (logR+Re ]og(l w - k z ) )
2~ 1
- -R-7-
log R -
Re
27r
-
n--1 n
Writing z = rP ~ we obtain
F (z-
R w k)
=
n=l n
1 27r
log R - Re
1 27r
log/7-E~nn n=l
1
w - n k e mO
~
R
(~,~o~-~
+
~-,~0~)
'
and hence
~(~) = - ~ 1 ~ ~
~1
log R - ~
k=0
~r
(e,,Ow_nk + e_~nOwnk )
n=l
follows. Because of N-1
Z J~
{ N (e - o) =
k=0
0
(5.s2)
(otherwise)
we get
1
~(~)
=
4~
_--
1 Z 47r
n=l
_1
r
~
-~ _1 n
n>l
n
(eznOod(p_n) k +
e_mOcd(p+n)k )
k=0
r
n P n ~N + T~ 1
1 -n
r
e-'~~
n>l
p-n=_O
p+n=O
(5),m, ,too 4 7r n GZ
-~l
e
,
m----p
and the proof is complete.
[]
5. Other Methods in Space Discretization
198 Convergence
Now we study the problem of convergence. Let f~ (n C Z) be Fourier coeMcients of f so that
f(x) = E fnem~ ncZ
I,fnl
holds for x = pe ~~E O~t. We suppose ~
+cr
<
(5.83) First, we note tim following.
nEZ
L e m m a 5.11.
We have vN(x) = ~ f,~n(p)-l~,~(x) n,EZ
(5.84)
.for x E f~ under the assumption of (5.81). Proof." We take 1 N-~
gij = "-~ ~
~)p(P) -102p(i-j)
p=0
and set d = (gij). Then, the (i, j) component of G G is given as N
E
1
Lk-i-N
k=l
~gP(P)-lcup(k-J)
=
p=0
=
1 N-1
N
-- E ~gP(D)-I~ N p=0 1
~ k=l
N-1
N E cdP(i-J) : 5ij p=O
by (5.82). We have (~ = G -1 and hence
D(I QJ
=
~=
=
ff
1/
N ~=0
~?p(p)-la2p(j-1) p=0
~-p(k-i)f
(t00)k-1)
k=l
follows from Q = G - i f . Plugging this into (5.77), we get
~p(p)-lcoP(J-1) ~"~ (.x.)-P(k-1)f (/9(,,0k-l)
uN(x) = ~ -~ j=l
p=0
1 ~
=
N
N
N z_., ~pp(p)-l ~--~ ~p0-,) F ( x - R J - ' ) E p=O
j=l
k=l
1 g-1
--" N E
p=O
r (x-/~(.,o j-l)
k=l
~-pef (owe)~p(X)
(,pp(p)-i e=O
~-p(k-i)f
(pcdk-1)
199
5.5. Charge Simulation Method by the definition of ~p(z). This implies
(;)
:
vN (x)
p,j =0
1 N
N-1
N-1
E (~gP(R)-I(cgP(X)E fn E CO(n-p)J p=0
nEZ
j =0
N-1
: E ~gP(P)-199p(X)E fn
(5.85)
n=p
p:0
by (5.83) and (5.82). The function Cp(Z) is harmonic and hence
I~p(p) -l~p(x) l _~ 1 holds. The right-hand side of (5.85) converges absolutely and we obtain (5.11) by (~gq= (~gp for q _= p. The proof is complete. [] Equality (5.83) implies r ) bl znO e nEZ
for x = rP ~ C Ft. We obtain eznO
~(x) } ~(p)
The function eN(X) is also harmonic and hence
II~N .o(~) _
max
Ixl=p
I~,(~)1 ~ ~ IAI ~
(5.86)
nEZ
follows with an=
I
sup Iezn~ oe[o,2~-) I
~,~(pe ~e) ~n(p)
Now we show the following. L e m m a 5.12. We have an = a _ n < m a x { 2 ,
1+
log log
(j~N __ fiN)
(587)
for n E Z and
NllogRI
(n=0) (5.88)
an ~_ 8n ( p ) N - 2 n N-n-R .for N >> 1, respectively.
(n= 1,
,N-~)
5. Other Methods in Space Discretization
200
Proof:
T h e relation ~n(z) = ~_n(z) implies a - n = N
1
an.
If n ~ O, we have
Iml
m~n
and hence / an _< sup | 1 + oe[o,2~-) \
~..
(pe~~
<2
~,,(p)
follows. If n - 0, we have 1 I~,~(p~'~ <- ~I log lp~e'N~ RN] I < ~1 flog (pN + RN) I.
This implies log
(R N -k- pN)
log
(~N
an_<1+
__
pN)
and inequality (5.87) is proven. To show (5.88), we note
a~ _< I~.(p)1-1 sup ]e'~~
~ (pe'~
oe[o,,) If n = 1 , . . . , N, we have N
~lml
N~ ~ 47rn '
where/3 =
p/R.
We also have
l~,,,o~,,(p) _ ~,, (p~,o) [
=
N _
7;7
(~ .. ~ -
Here, the r i g h t - h a n d side is e s t i m a t e d from above by
_ ~,(~+.)0) + 9~N-" (e ....o _ ~_ ,(~N-,~)o} )
N
~{
47r
e=l gN + n < N k(flgN+n
S ~§
(~ ....0
--
2rre= 1
=
27r(N-n)
gN+n +gN-n =
N/3N-n -< 2 ~ - ( N - n )
gN- n fleN-n)
g N + n t3(e~2n
(1+
oo
)Z/3(eg=l
+g.N-n
1)N __ N[3N-n - 27r(N-n)
1 +/3 2n 1 - / 3 N"
5.5. Charge Simulation Method Because
/~N <
201
1/2 for N large, then we get 2N
~N-n
~(N-~)
"
Inequality (5.88) for the cases of n = 1, 2 , . . - , N - 1 is proven in this way. The case n = 0 follows similarly in use of the first equality of (5.80). The proof is complete. [] We are ready to prove the results on convergence. In the first theorem, the b o u n d a r y value f is supposed to be real-analytic. Therefore, the solution v of (5.63) has a harmonic extension to {x C R~ I Ixl < to}, where r0 > p. 5.13. Suppose R r 1 and (5.81). Then, there exists a constant C = C~,n,ro >
Theorem
0 such that
I vN -,11~(~)
-~ c,-~ sup I~(~)1, Ixl =To
wh ere
~-=
{
~T;;o (~o < R ~) plR
(pro > t~~)
so that r C (0, 1). If R = 1, then v N(O) = 0 always holds, regardress with the values yy and Qj. Therefore, assumption R 7~ 1 cannot be eliminated. The following theorem deals with the case of more rough data. 5.14. Under the assumptions R 5r 1 and (5.81), the following results holds, where fn (n C Z) denote the Fourier coefficients of f indicated by (5.83):
Theorem
(i) If (ii)
~-~n IAI < §
If fn =
O
t h ~ limN_o~ II~ - ~11~(~) = 0.
(1~1-~),
then
I ~ - ~11~(~) = O (N-~+'),
wh~
~ > 1.
Proof of Theorem 5.13: The function f is supposed to be real-analytic, and therefore, Cauchy's inequality indicated as
p) i~i sup ~(~)1
IAI _< ~0
(5.s9)
~,=~o
holds. Letting m = [N/2], we split the right-hand side of (5.86) as
I1~11,_~(~)
m
_~ ISola0 +
~( n=l
SI -~- S2--~- S 3 .
oo
AI + IS-~l)an
+
E
(IAI + If-hi)an
n=rn+l
(5.90)
5. Other Methods in Space Discretization
202
T h e first a n d t h e s e c o n d t e r m s of t h e r i g h t - h a n d side are e s t i m a t e d by (5.89) a n d (5.88), w h e r e a = R ~/(pro): $1
$2
8
<
( P ) N sup Iv(x)l ,
-
NJlogRI
<
Z
--
2
R
P
Ixi=ro 9
8n
N -
n=l sup
16 ~
p N-2n n
v(z)l
Ixl=ro
sup Iv(~)l Ixl=ro
ct n n=l
N
<
ol- 1 \ro/ 16 1 -- oz
Izl=~o
--(P)x sup I~(~)1 ~
(~
Izl=ro
< 1).
O n t h e o t h e r h a n d , t h e last t e r m is e s t i m a t e d by (5.87):
s3 _< 2M~ ~
n=m+l
=
TO
sup I~(*)1
Ixl=ro
2Mn <~00) P m+l 1 1 - -~ sup I~(~)1 ro Izl=ro
<
-
1 - -p-
sup
ro
Ixl=ro
v(x)l.
If a > 1, we have p2/R2 < p/ro a n d hence
II<~ll~o~(r~) < _
~ c
~ --'
Vo
N/2 +o
s
TO
+ c
p
sup Iv(*)l Ixl=ro
sup Iv(~,)l
Izl=,'o
follows. If a < 1, on t h e o t h e r h a n d , we have p2/R2 > p/ro a n d hence o b t a i n
Tile p r o o f is c o m p l e t e .
Proof of Theorem 5.14: We have I.f-I - IlfllL~o(o~)-
[] T a k i n g m = [N/3], we
estimate
Commentaly to Chapter 5
203
each term of the right-hand side of (5.90) as follows: S1 < -
(fl)N
8 NllogR
R
s~ _<s--~v (~p)~/3 ,
Ilflc~(Oa),
(9O
S3_<(I+MR) ~
(I/~l+fAn).
n=[N/3]
The last term is o(1) and O ( N -a+l) according to the first and the second assumptions on fn, respectively, and the proof is complete. []
Commentary to Chapter 5
5.1. An important feature of hyperbolic or dispersive systems is the law of conservation. M. Mori and others tried to realize them in discretized schemes. The precise definition of the barycentric domain is as follows. Let T be the twodimensional simplex with the vertices P1, P2, P3. Then, its barycentric coordinate
{A(x", T, Pi)} 3i = l of a point P = P(x) C R 2 is defined in the following way: Put Pi = (x~, x~, x~) and set
(~) 1 A(x;T,P~)=~det
1 1 x I xl
1 x~
for x = (Xl, X2) and
A = det
1 1 1 x I x21 x31 x I x~ x~
Now, given a E Zh [.J Bh, we put Tta = {T E 7 h l a ~ T} and
rer~ Then, the barycentric domain Da with respect to a is defined as
Da = U {z E T I A(x;T,b)_< A(x;T,a) TeT~
(b E I~ \ {a})}.
5. Other Methods in Space Discretization
204
T. Ushijima's works on scheme (5.1) were published in Ushijima [396] and [395]. RieszThorin's interpolation theorem is described in [127] and so forth. As is described in the commentary to Chapter 1, see Brezis and Strauss [65] for the L ~ estimate of elliptic operators. For the fact that A -~ : L2(t2) ~ H(~(t2) is compact if t2 has no corner with angle 2re, see Ne~as [286] and Grisvard [164]. Results obtained in this section keeps to hold for the general dimensional case under the following condition for the triangulation. Given T E rh, let V(T) be the set of vertices of T. Furthermore, for a E V(T), let Ta be the n - 1 dimensional simplex (i.e. face) of which vertices are composed of the elements in V(T) \ {a}. Write f(a,T) for the line containing a and perpendicular to Ta. Then the condition is described as
If n - 1, this assumption always holds. If n - 2, it is equivalent to saying that any T E % is an acute or right triangle. Here, n denotes the space dimension of the domain in consideration. See w for related arguments. Relation (5.15) is obvious because dim Xh < +o~. But it provides a key identity for E. Hille to construct (C0) semigroups on Banach spaces. K. Yosida made use of another aI)proximation for t h a t purpose, now referred to as the Yosida approximation. Some variants are made use of in w167 and 6.3. See also Hille and Phillips [179], Yosida [410], Kato [205], and Tanabe [378]. 5.2. The properties of L ~ stability and convergence were established by Tabata [373], [374] for scheme (5.29) with # (Eh, Xh) added to the left-hand side and without lumping in the right-hand side , where # > #o = C(7)Ilbll 2L ~ ( n ) " Here, C(7) > 0 is a constant determined by the parameter 7 > 0 arising in connection with the regularity of {%}. See also [375]. 5.3. The inverse operator of the canonical injection a : V' --, V is sometimes referred to as the duality map. Note that the bounded linear operator A : V --, V is different from that of the previous sections defined through the Gel'fand triple V C H C V'. Variational problems of saddle point type arise in many areas. See Aubin and Ekeland [15], Rabinowitz [320], and Suzuki [369]. In connection with H(div, f~), the function space H(rot, f~) is also introduced when f~ C IR3. It is composed of the vector field 'v E L2(t2, R 3) satisfying V x v E L2(t2, Ra). See Girault and Raviart [160] for those spaces and applications. The existence of v satisfying (5.54) is also proven there. See also L e m m m a 7.32 in w Instead of (5.52) with (5.53), the Stokes system may be formulated as to find (u, p) E H(~(f~) n x L2(t2) satisfying 2~
(e~j('u), e,j(v))L~(n ) - (p, V 9v)L:(n ) = (.f, v),:(n),~
i,j
for ~ c H ] ( ~ ) n ~nd ( V . ~,, q)~(~) = 0 for q c L : ( ~ ) , w h ~
1 (O'u ~ Ou~) r
= ~ \o.~, + ~
Colnmelatary to Chapter 5
205
This way is more natural from the physical point of view. Then, Korn's inequality takes place for Poincar~'s one to confirm the assumptions of Theorem 5.6. The continuous Stokes system and its finite element discretization were studied by many people, including Cattabriga [70], Kellogg and Osborn [217], and Crouzeix and Raviart [101], Bercovier and Pironneau [30], Glowinski and Pironneau [163], LeTallec [237], Stenberg [357], respectively. Mixed finite elements for magnetostatic and electrostatic problems were studied by Kikuchi [219], [220], [221], [222], [223]. Concerning the scalar equation, the Stokes system, and the equation of linear elasticity, see Brezzi, Douglas, Jr., and Marini [67], Raviart and Thomas [325], Crouzeix and Falk [99], Crouzeix and Raviart [101], Falk [122], and the references therein. Inequality (5.56) is referred to as Babu~ka-Brezzi-Kikuchi's inf-sup condition. It was introduced by Babu~ka [18], Kikuchi [218], and Brezzi [66] independently. Concerning actual examples of Vh and Wh satisfying the assumptions of Theorem 5.7 in use of finite elements, see Giraut and Raviart [160] and Brezzi and Fortin [68]. Under this condition, the discretized nonstationary Stokes system is treated similarly by the semigroup theory. See Okamoto [307] for details. The Navier-Stokes system is the fundamental equation in fluid mechanics. Operator theoretical approach for nonstationary problems was initiated by Kato and Fujita [207] and Fujita and Kato [143]. For other references, see the commentary to w Its finite element discretization was studied by Okamoto [308] and Heywood and Rannacher [175], independently. Error estimates given by the former are a priori; they hold only under reasonable assumptions on the initial value. The latter's are, on the other hand, a posteriori so that are valid under some additional estimates of the solution. Up to now, fundamental theory for this system is unsatisfactory, especially for the case of three space dimensions. A posteriori approaches are intended to compensate such a situation by numerical computations. See also Bernardi and Raugel [34], Heywood and Rannacher [176], [177], [178], and Rannacher [322]. 5.4. For layer potentials, see Courant and Hilbert [94], Kellog [216], and Garabedian [155]. Theorem 5.8 was proven by LeRoux [233] for n = 2 and Nedelec and Planchard [288] for n = 3. We have followed the method of Okamoto [309] for the proof, where q E X is taken in L2(cgf~). In the actual computation, the method of collocation is more realistic than that of Galerkin's. Iso [191] and Hayakawa and Iso [168] made error analysis for this method applied to the Neumann boundary value problem. Related works were done by Arnold and Wendland [13], [14]. Several monographs and surveys were published concerning the boundary element method. See Sloan [350] and the references therein. For the theory of distributions, particularly equalities (5.72) and (5.76), see Yosida [410] and Schwartz [348]. 5.5 CSM is also called the fundamental solutions method. Since the pioneering work by Bogomolny [41], the efficiency of the scheme has been clarified rigorously. Theorems 5.10 and 5.13 were obtained by Katsurada and Okamoto [213]. Theorem 5.14 was proven by Katsurada [210]. For other charge and collocation points, it can happen that the
206
5. O t h e r M e t h o d s in Space Discretization
coefficient matrix G becomes singular. Some examples are given in [210]. Problem (5.63) is invariant under the following transformation: (1) x H c~x, yj H c~yj, where c~ is a constant (2) f (x) ~ f (x) + c and v ( x ) ~ v ( x ) + c, where c is a constant. Scheme (5.77) has lack of such properties. K. Murota proposed N
j=l
where Q o , ' " ,
Q N are determined by VN(Xj) = f ( x j )
(j=
1,...,N)
and N
k=l
This scheme is provided with such properties. In this case, tile conditions R g - oN ~ 1 and R ~ 1 are not necessary for the wellposedness and the error estimate to hold. Katsurada [211], [212] studied the case that c%2 is a Jordan curve in use of the conformal transformation. An application to the free boundary problem was made by Shoji [351]. Furthermore CSM is a powerful method to obtain numerical conformal mappings. See, for this topic, Amano [6], [7], and a m a n o et. al [8].
Chapter Nonlinear
6 Problems
Many problems in applied sciences and engineering are formulated as nonlinear partial differential equations. Fortunately, in accordance with the progress of analytical theories, reliability on numerical computations has been advanced extensively. In this area we have several monographs such as Girault and Raviart [159], [160], Kf{~ek and Neittaanmgki [227], and Zen{gek [417]. This chapter selects the following topics: (1) iterative method for solving unstable solutions for elliptic boundary value problems, (2) qualitative features of finite difference solutions for semilinear parabolic equations, (3) finite element scheme for degenerate parabolic equations based on tile L 1 structure.
6.1
Semilinear Elliptic Equations
A typical example of the nonlinear elliptic boundary value problem is
-Au-
lul p-1 u
in ~
(6.1)
with u = 0
on 0f~
(6.2)
for p E (1, ec). In this problem, u = 0 is the trivial solution and non-triviM solutions are linearized unstable so that the simple iteration scheme U k + 1 ~- ( - - Z ~ ) - 1
I~1 ,-~ ~
(6.3)
hardly converges to them. Furthermore, the existence of non-trivial solutions depends sensitively to the domain: dimension, topology, and geometry. To fix the idea, let ~ C R 2 be a convex polygon. Then, it is known that problem (6.1) with (6.2) has infinitely many solutions. The functional
J(v)-
3f~(1 ~IWl ~-
1 p+l
I~
,p+) '
d~
takes place of the energy, and u is the solution if and only if it is a critical point of J on
v = H0~(~). A least energy solution denotes the one which attains the minimum of J on the set of non-trivial solutions. It is positive in g2 and attains d = inf J > 0, H 207
(6.4)
6. Nonlinear Problems
208 where
c~(a) Note that J(v) =
(1 1) ~-v-4-r
--
Ilvl
Lp+l(f~)
IlWllL2(n) for v 9 H .
>0
"
We call (6.4) Nehari's variational
formulation and N" the Nehari manifold. By Sobolev's imbedding theorem we have inf I WilL2 > O, yEAr (fl)
(6.5)
and this implies d > 0. It is known that d is equal to the mountain pass critical value so that d is attained by a solution with the Morse index 1 if it is not degenerate. Subject to the underlying variational structure, the iterative sequence {uk}k=0,1,2,... in consideration is constructed on iV" in use of the scalar multiplication operator Q" V\{0} ---. Af. Actually, it is not hard to see that any v 9 V \ {0} takes unique t = t(v) > 0 satisfying Q(v) - t(v)v 9 N'; t h a t is,
t,_l = II Ilvl ,+1 Lp+l(gt)
Then the sequence is defined as Uk+l = Tuk for T v = Q ( - A ) -1 Ivlp-lv and u0 9 V \ {0}. More precisely, Uk+l
-~
tkwk 9 Af
with
tk > 0
and
wk = ( - A ) -~ ]'uk p - 1
Uk.
(6.6)
Here, the positivity is preserved so that if Uo _> 0, then 'uk > 0 for k = 1 , 2 , . . . This sequence is provided with several fine properties. First observation is the following. L e m m a 6.1.
We have t(v) <_ 1 and J ( T v ) <_ J(v)
(6.7)
for v 9 N'. Here the equality holds if and only if v 9 Af is a solution to (6.1) with (6.2). Proof: Take v 9 N" and set u = T v = tw, where -Aw=],vV-lv
inFt
with
w=0
inO~.
We shall show that t _< 1 and J(u) _< J(v). In fact, because of v 9 A/" and (6.8) we have
iwll~ ,+1 L~(~) = I1~1 L~, l(~) = ( v w , w )
< IlVw L~<~)IlWllL~(~)
or
IlVvllL2(~) _< IIW,,llL2(~) 9 Therefore, again by v E H and (6.8) we have
IIv~ll ~L~(~) = (1~1 ~-' ~, w) _<1~1 ~ + , (~) w i l L . , (~) = IIw, IIh=(~) ~+' ~11..+,<~) _<
v~
" a=(~) Ilwll~. 1(a)-
(6.8)
6.1. Semilinear Elliptic Equations
209
This implies o < IVwll ~
w
p+l
and tP--1 __
IlWll~(~)
Ilwl'+l I LP+
<
1 (f~)
-
1.
1 Isl p+l is a convex function, the inequality Because s H F(s) = K+-7 1
p + 1 ( r/
p+l
-
[~" p+l
1
) ~ 2 (?]2 __ ~2) ~[p-1
holds for r/, ~ E R. We have
J(tw)
=
t2
~-IlVwll ~L2(a)
tp+ 1
p+ 1
IIw''p+l IILp+~(n)
< --t 2 {ll~Twl122(f~)-(W2, IvIP--1)} 'JF( 12
-
Here we have IlVwll ~~(~) -
1 ) ivlp+l p+l i ZL~+~(a)-
2
(1~1~-~ ~, w) <
Qj~
v ~ Ivl ~ - '
)1/2 (/~)1/2
w ~ Iv ~_1
= IVvll~(~) (w ~, I~1~-') 1/~ <_ IlVwll~(~) (w ~, I~ ,,-1)1/2
(6.9)
and hence
IlVwll 2L2(f~)<-- (W2, I'Vp-l) l follows. We have (6.7) as j(u) < (1 2 --
1 p+l
)
tI~i~+l Lp+I(~) = j ( ~ ) .
Assuming the equality in (6.7), we get the equality in the second inequality of (6.9). This implies w -- Av with a constant A. Then A - 1 follows from v C Af, and therefore, v solves (6.1) with (6.2). The proof is complete. [] To describe the dynamics of the sequence {uk} defined by (6.6), we introduce its w-limit set as
in w
for s o m e
c
},
where W = {v E C ( - ~ ) l v = 0 on c9t2}. The following theorem provides the reliability on this iterative method in the numerical computation.
210
6. Nonlinear Problems
6.2. The set W(Uo) is non-empty, compact, connected, invariant under T, and contained in the set of non-trivial solutions of (6.1) with (6.2). The sequence {[ Vuk[[L2(a)} is monotone decreasing and bounded from below by a positive constant. We have tk T 1 and distw (Uk, W(Uo)) ~ 0 as k --, oo. Finally, the sequence {uk} is compact in W.
Theorem
Proof." The fact t h a t Tw(uo) C W(Uo) is obvious. Because of J(uk+l) _< J(uk) and uk C N" for k >_ 1, the monotonicity of [[Vuk c2(n) follows, while inequality (6.5) shows that it is bounded from below by a positive constant. We have Wk :
(_/~)-1
and
l?2k p-1 Uk
uk+l = tkwk
(6.10)
IIW~IIL~<~)
with 0 < tk _< 1 and = o(1). Then, the standard bootstrap argument gu~r~nt~s II~kllH~(~/ = o(1). Therefore, {uk} c WNH~(f~) is compact and in particular, w(uo) is a non-void compact set provided with the property that d (uk, w(u0)) ~ 0. II Any {u;} C {uk} admits a subsequence {u~} C {u;} and u E H~(f~) satisfying u k ~ u in H~(f~). This implies u E iV" and - i Uk,, ~-i u"k
zk - ( - A )
~
(--A)
lul ~-' u -
-1
w
in H 2(f~). We obtain w 7f 0 and (t~)p_~
"
IVz~ll~(~/
=
---,
ilzkl p+, gp+,(n)
I Vwll ~L2(fl)
p-1
= t,
> 0
(6.11)
IIw p+l Lp+,
and klim ~oo
IIt k"z k
--
t,w
[H (a) = O.
This means Tu~
~
Tu
in H 1(f~).
Here, we have I!
J(Uk+l) ~_ J(Tu'k) by L e m m a 6.1. Sending k ~ oc, we get
J(u) <_ J(Tu). Then the latter part of L e m m a 6.1 assures that (6.1) with (6.2). In particular, w(u0) in contained furthermore, Tu = u. The last relation gives t, = It remains to show that w(u0) is connected in W. exists because of the monotonicity of IlVukllL~(a).
u E N" is a (non-trivial) solution for in the set of non-trivial solutions, and 1 in (6.11) and hence tk ---* 1 follows. First, we note that l i m k _ ~ Vuk IL2(a) Next, we have tk T 1 and hence
I1~7 (Uk-t-1 - - uk)ll ~L~(~ --- I l V u k + ~ l l ~L2 (gt) - 2 ( W k + l --
,
Vu~) + IlVu~ll ~L 2 ( f ~ )
I v u ~ + , l l ~~(~> - 2 ( V w ~ , v u ~ ) +
IlW~ll =L2(a) + o(1)
211
6.2. Semilinear Parabolic E q u a t i o n s
holds. Here, we have
(Vwk, Vuk)
---
I uk p+~1
---
V?~k
~ 2(a) L
so that l i m [IX7 ( U k + 1 - k---,oo
Uk)llL2(f~) -- 0
follows. In use of the bootstrap argument for (6.10), this implies lim Iluk+l - uk w -- O.
k----~oG
Now, suppose that a~(u0) C W is not connected. Because it is compact, then we have non-empty disjoint compact sets A and B satisfying cJ(u0) = A U B. If dist (A, B) = 35 > 0, we have N such that I]Uk+l- ukll~y < 5 for k > N. From the definition, there exists kj ~ oc and k~ > kj satisfying dist (uk~,A) < 5 and dist (uk~, B) < 5. Therefore, there exists some kj" in k i < kj" <_ k~'i satisfying distw(uk~,, A t2 B) > 5. This contradicts the assumption, w(u0) = A U B. The proof is complete.
6.2
[]
Semilinear Parabolic Equations
Vast references are devoted to the qualitative theory on semilinear parabolic equations. Among t h e m are the studies on the behavior of the solution globally in time. In this section, we take their finite difference analogues. We study the blowup of the solution, taking in f~ x (0, T)
ut = uxx + u 2
(6.12)
with Ulo a = 0
and
ult= 0 = Uo(X),
(6.13)
where f~ = (0, 1) and uo(x) is a non-negative continuous function on f~. For this system the classical solution exists locally in time. H. Fujita showed that it cannot be continued globally in time if the initial value Uo(X) is large in a certain sense. This case is referred to as the blowing-up of the solution, because then limtTTm~x I1'~(., t)llL~(a) = + o c follows, where Tm~x denotes the maximal time for the existence of the classical solution. F . B . Weissler showed that if the initial value is symmetric with one peak, then the solution blows up at one-pint, x = 1/2. Finite difference analogues for those facts are studied by T. Nakagawa and Y.-G. Chen. We provide the uniform mesh for f~ denoted by /~ " X 0 =
0 ~
Xl
~
X2
~
"'"
~
XN
--
1
6. Nonlinear Problems
212
with xj = jh for j = 0, 1, 2 , . . . , N. Then, a piecewise linear function u n is identified with its values on the nodal points, u] for j = 0, 1, 2 , . . . , N. On the other hand, the time discretization is subject to Tn=T-min(l
,
]]un]]LP(FI) -1 )
'
where T = Ah 2 with )~ > 0 and p E [1, oc]. Here, it should be noted that all (1 _< p _< ec) are equivalent for h fixed. Actually we have
[lun[IL~(a)
1
[lu"llL,(a) < Ilu"llLOO(a) < h - ? ]lU n I]L,(a)"
(6.14)
Then, the scheme in consideration is the finite difference m e t h o d (FDM), implicit in linear parts, n
. n+l
=
+ (u;) 2
he
Tn
(6.15)
for j = 1, 2 , . . . , N - 1 and n = 0, 1 , 2 , . - . with
u~=u}=0
(n>0)
(6.16)
and
uj0 =Uo(Xj)
(l_<j < N-l).
(6.17)
Scheme (6.15) with (6.16) and (6.17) approximates the solution and the blowup time appropriately. We describe the qualitative features of the approximate solution in details. For this purpose we suppose that Uo(X) is symmetric with respect to x = 1/2 and N = 2m so that u 0 ( 1 - x ) = uo(x) and Xm = 1/2. We also suppose that Uo(X) is not constant and monotone increasing in [0, 1/2]. Then it follows that
u~_j = u2
(j=l,2,...,m;
n=0,1,2,...)
and n
(j=
1,2,.-.,m-i;
n=
1,2,..-).
In particular, we have and
n ?~m-
l --- I n a x
n uj.
j~=m
We suppose that the solution of system (6.15) with (6.16) and (6.I7) blows up so that oo
lira
n----* (x)
II'u'~IIL~(~)-- -t-c~
and
~
w,, < +o<~.
Then we have the following theorem. Note that'unm_2 = maxjr . . . . +1 u~.
213
6.2. Semilinear Parabolic Equations Theorem that
6.3.
Under those circumstances, three points blowup occurs to the solution so
lim urn_ ,n
~---+OO
--
+oc
(6.18)
and limsupu~_ 2 < +oc
(6.19)
n--+(X)
hold. Proof: Let "~n ~- % h 2. Then equality (6.15) is written as 9
--
Uj
k'u,j_ 1
"71- ' u , j + 1
) "JF Tn
(Uj)
,
or
(1 + 2An)ujn + l
__ /~n \U,j_ {~ n + l1 Jr- 'C~j+ . n+l~ n n 1 ) + ?.l,j Jr- Tn (Uj) 2
(6.20)
In particular, we have un+~ = /~n (?jn+_l + u n + l ) _~ --
[1 + Tnun_i] un_I
(6.21)
1 + 2A,~
and n+l n Unm+l - 2/~n'U,m_ 1 -I- [1 H- Tn,l~n] ~trn 1 + 2)~n
(6.22)
Those equalities imply u,~+ 11 > -
-
n "~nun+l -It- Urn-1
n+l
and
l+2)~n
um
> -
n Urn l+2A,~
,
respectively, and hence
~u~/I ~IIL~(~) + (1 + 2.Xn)u~_ 1 (1 + 2:~.) ~
~:+1 >_ ~ . ~ + (1 + 2 ~ . ) ~ _ ~ (1 + 2)~n)2 follows. Because
b~ ~
, n ~m u~[ LP(fl)
>_1,
we have n
~21 >
,~ + u r n _ 1
(1 + 2s
2
6. Nonlinear Problems
214 so t h a t relation (6.18) follows as II
lira inf U mn _ n~oo
1
= lim inf u~ +11 _n 1. _ > ~ lira inf A + u,~_l ~_ A + lira inf um n-.o~ n-~oo (1 + 2An) 2 n-~oo
In use of (6.21) and (6.22) we have
=
An (U~n+'l + U~n+') + [1 + %U~n_,] U~n_, 1 +2An (1 + 4An) Anun+21 -Jr An [1 + "rnu~] u,~ (1 + 2A,~)[1 +
(1 -4- 2An) 2
+
TnU,nn_l] tt,nn_l
(1 + 2A,~)2
This implies l t n t 11 <
-
A~ [1 + %%~,] u ~ + (1 + 2An)[1 + 7-,~un_l] u~_ 1
-
(6.23)
1 + 3An
From the assumption, we have I1..-I1~.(~) > 1 for n large, and then T,, = r/I1~11~(~) follows. Letting an --
, n /tm_ 1
n
UTn
'
we have by (6.23) that u~+ll an+ 1
--
(0.24)
7tn+ 1
(1 + 2~,,)
2A,, + [1 + T,,'u~] l t mn ,/ l t mn+l - 1
A. [I + ~-,,,~] u~ + (1 + 2;~,,)[I + ~',,~,m_,] '~ Urn,
-
Irt
i,~
n
i
n -t- 2A,, [1 -t- T ,,.-~m~ln 1 ] "ltm.(1 + A,,)[1 § T,u~] u m " 1
A (1 + rb,,) + (1 4- 2A,,) [1 + rb,,a,,.] a,,
(1
+
~..) (1
+
~b..)
+
2~.. [1
+
(~.25)
~..~,...] a..
In other words, we have
*"('+'~") + (1 + 2A,,)[1 + rb,~a,~] a,,
an+ 1 ~
(6.26)
a,,. - (1 + A,,.) (1 + rbn) + 2A,, [1 + rb,~a,r a,. for n large. To get a reverse inequality we reduce (6.21) and (6.22) as -)',n'~'n+ 1 . . m . -{-
1 -Jr-TnH,,nn_ 1 ILm_ 1
1 + 2A,,. 2 -.%u.,,._, -2, n+l
, n 'u;',,. § (1 + + A,,~ [1 + r,,um]
2,~,u)
[1 §
'" 1 Tn ll'Tn.- 13"II'm-
(1 + 2A,,.) 2
> -
A.. [1 + ~..,,........]. u .... " +(1+
2A,.) [ l + % u ' "..... , ] ' 'bl'm- 1
1 4- 4A,, + 4A~,
....
215
6.2. Semilinear Parabolic Equations Similarly, we obtain an+l
n Um-1 un+l
=
(1 + 2/~n) un+21
~+~1 2,~ a m_
+
[1 + r
n um
a~ [1 + ~n~ n] ~ + (1 + 2a~) ~ [1 + ~ n ~ _ , ] ~ _ ,
>
-
(1 + 2A~)[1 + ~ , ~ ] ~m + 2A~ [1 + ~ ; ~ - 1 ] ~ - 1 an (1 + ~b~) + (1 + 2a~)[1 + ~b~a~] a~ (1 + 2An)(1 + 7bn) + 2An [1 + Tbnan] an
for n large. This means t h a t an + l ~
an
ar~
- (1 + 2An)(1 + Tbn) + 2An [1 + Tbnan] an
(6.27)
We can deduce t h a t {an} is m o n o t o n e decreasing for n large by (6.26). Actually, t h e n 2 {)~n (1 + 7bn)+ (1 + 2/~,~)[1 + 7-bnan]an}-{(1 + )~n)(1 + 7bn) an + 2~n[1 -t- 7bnan]an}
= )~n (1 -- an)[1 + Tbb -~- 2an (1 + Tbnan)] -~- Tanbn (an -- 1) is negative. Because
0 < an < 1
and
1 1 _< b~ _< b = h 5,
(6.28)
this t e r m is d o m i n a t e d from above by
7n (1 -- an) [--Un_l -~- 3h -2 (1 nt- Tb)]. We have proven (6.18), so t h a t there exists k satisfying k > Urn-1 -- 3h-2(1 + Tb) > 2h -2. Relation (6.21) implies
un4-1m'-i> -- --
1+
n
TnUm-1 n Urn--I"
l+2)~n
Therefore, we conclude Unmt21 ~ U n _ l ~ 3 h - 2 ( 1 + Tb)
for n >_ k by an induction. T h u s {an}n>k is m o n o t o n e decreasing. Let a = l i m n _ ~ an C
[0,1). By (6.28), there is a subsequence {bin} C {bn} converging to some constant c _> 1. Then, (6.25) gives t h a t
a < 1 +7-ca I+TC
.a.
6. Nonlinear Problems
216 This implies a = 0 and hence lim,~__.~ b,~ = b. Furthermore, lim An = lim "rbnh -2 ~--~ a~ ~--~o~
( [lt~m, _ l )
-1
~-- 0
by (6.18). In use of (6.26) and (6.27) we obtain lim bn -- lim a~+l -1 E (0, 1). n---<x~ an 1 + ~-b
(6.29)
n---*cx~
In particular we have oo
oo
Ea,~
< +ec
E(l+Can)<+oc.
and
n:O
n=O
Equality (6.20) gives ~
A,~(u,~223 +
=
urn_l) n+
1
+
(1 +
Tu~_2)urn_ 2 n
1 + 2A,~
~n ( < % + < % ) + (1 + ~,~_~)~_~ 1 +2A,~ and hence n+l .n+l
, n
~ 2
'tt, m,_
~m-2
l+ln
--
In use of (6.23) we get ~ n n + l 2 < Anum_ 2 + J~n, 'tim_ -
-
where
fin - 1 + "rnu,'~_2 < 1 + Tbn unn-2 < 1 + Tba,~ -- A,~ 1 + An
'u~
-
and
B.
_ <
n + A. (1 + 2An)(1 + ~ - ~ nm - , ) ' / t i"n A~ (1 + ~,~<~)~'m ( 1 + A,.) (1 + 3A~) Ab(1 + Tb)[h-2T,~ + ( 1 + 2A)a,~] - Bn.
1
This leads to (1.54) as n -
u,Z_2
_<
i=0
The proof is complete.
1
n -
o
Um-2 E Ai + E i=0
2
n -
1
Bi I-i Aj -+- Bn_I <_ AI. 3=i+1
[]
6.3. Degenerate Parabolic Equations
6.3
217
D e g e n e r a t e Parabolic E q u a t i o n s
In this section, t2 denotes a flat torus for simplicity, where a, b > 0:t2 = R2/(a.Z x bZ). Let f 9 R --~ R be a non-decreasing continuous function satisfying f(0) = 0. The problem
ut- Af(u) = 0
in f2 x (0, T)
~1,=0 =
with
~0(x)
(6.30)
describes several p h e n o m e n a including free boundaries inside; diffusion in porous media for
ul'~-lu
f(u) =
(6.31)
with "7 > 1, very fast diffusion for the same nonlinearity with "V E (0, 1), two phase Stefan problem in the enthalpy formulation for
f(u):
oz(u+l) 0 fl(u-1)
(u<-l) (-l 1)
1)
with constants c~,/3 > 0, and so forth. Wellposedness in X = LI(Q) of (6.30) has been established in the early 1970's via the nonlinear semigroup theory. For the differentiation L = - A acting on X with the domain D(L) = {v E Wl'l(ft) ] Lv E X } , the nonlinear operator A in X is defined by Av = L f ( v ) for v C D ( A ) with
D(A) = {v e X lf(v) ~ D(L)}. Problem (6.30) is reduced to the evolution equation
du --+ dt
Au = 0
with
u(0) = u0
(6.32)
in X. Here, the operator - A is hyper-dispersive in X. This means that any constant )~ > 0 admits an order preserving contraction (I + AA) -1 in X. Abstract theory of M.G. Crandall and T. Liggett now guarantees the convergence
S(t) = s-lim m--,oo
(
I +
)m
t_ A T~
(6.33)
and thus the nonlinear semigroup {S(t)}t>_o is defined on D(A) = X . Then, u(t) = S(t)u0 is regarded as a solution of (6.30). Semigroup {S(t)}t>o has the properties of order preserving and L 1 contraction, described as (6.34)
218
6. Nonlinear Problems
for Uo, fto C X and t >__ 0, where v+ = max{v,0}. (I + AA)(X) = X, and
This is a consequence of (6.33),
where v,~ 9 D ( A ) and A > 0. Time discretization of (6.30) was studied by A.E. Berger, H. Brezis, and J.C.W. Rogers. Based on the nonlinear Chernoff formula, they developed the L 1 theory. Here, we take into account of the space discretization, and consider the finite element analogue for (6.32). Decomposing ~ into simplexes with the size parameter h > 0, we write Th for their totality. Furthermore, Xa denotes the set of continuous functions on f/, linear on each T r rh, provided with the topology induced by LI(~). From the L 2 theoretical point of view, it is natural to take d d~(Uh, ~h) + (Vf(u,,), V~a) = 0
with
as a semidiscrete approximation, where r
(uh(0), f#,) = (Uo, Ch)
(6.35)
r Xh. The solution
uh 9 C I ( [ O , T ) , X , ) may be regarded as a finite element approximation of u. In terms of the operator theory developed in the previous chapters, scheme (6.35) is represented in the following way. First, the finite element approximation Lh" Xh ~ Xh of L = - A and the Ritz operator Rh" V ---+ Xh for V = HI(U) are defined as follows, where A(u, v) = (Vu, Vv)" (i) L,,vh = fh r (ii) Rhv = v,, r
A(V,,, ~ h ) = (fh, fa,,)
for fa,, e Xh ;
A(v, fh) = A(v,,, gab)
for fa,, 9 X h .
Then, (6.35) is written as
duh
d----t-+ LhR,,.f('u,,)= 0
with
"u,,(0)= P,,u0,
(6.36)
where Pt, " L2(f ~) ---+ Xh denotes the orthogonal projection. If f 9 IR --+ IR is locally Lipschitz continuous, scheme (6.36) is well-defined, because then vh E X,, implies f(v,,) E V. Scheme (6.35) is conforming only in this case; otherwise it is not so. We propose another scheme, provided with the properties of L 1 contraction and order preserving. Such a viewpoint is not always regarded so significantly in the linear case f ( u ) = u, but becomes important in the degenerate case as we shall see. It is actually realized by replacing the Ritz operator by that of interpolation, and adopting the technique of lumping of mass for the linear part. Let Fh be the set of vcrtices subject to rh. For a E Yh, the function wa C Xh is defined by
W a
1
ata
0
at b E F h \ {a}.
~---
6.3. Degenerate Parabolic Equations
219
Then, {Wa I a ~ v . } forms a basis of Xh and the interpolation operator 7rh: W = C(f~) Xh is defined by
7ChV= E v(a)Wa. a6"Ph Remember that ~ C R 2 is a fiat torus so that is compact by itself. We adopt the m e t h o d of lumping of mass described in w Actually, each a 9 )2h takes the barycentric domain Da. From the periodic extension, it is identified with a subset of ~. Let
(x 9 Da) (x 9
1
Wa(X)--
0
,
and denote by X h the vector space generated by {~a [ a C ~2h}. The lumping operator Mh " Xh --~ X h is defined through We H ~a. The adjoint operator M~ 9Xj~ ~ Xh is associated with the L 2 inner product, and we set /(h = M ~ M h " Xh ~ Xh. Let u0 E W. The scheme studied in this section is described as d dtKhUh + LhZChf(Uh) = 0
with
Uh(0) = 7rhUo
with
(Uh(0), Wa) = (TChUo,Wa)
(6.37)
in Xh. In the weak form, it is written as d d--t (Uh, Wa) + (VWhf (Uh) , Vwa) = 0
for a E 12h, where Uh = MhUh. Because 7chf(Uh) E Xh, it is conforming for any continuous non-decreasing function f satisfying f(0) = 0. Furthermore, the linear operator 7rh " W = C(f~) ~ Xh is order preserving and so is the mass lumping Mh 9Xh ~ Xh. This is desirable for our purpose. We suppose that any T E Th is a right or acute triangle. This implies m a x i m u m principle and L 1 contraction for the discretized linear part. In fact, any result stated in w is concerned with the homogeneous Dirichlet boundary condition, but similar arguments assure them for the periodic b o u n d a r y condition. Similarly to the proof of Theorem 5.1, we can show the properties
0 <_ f~
6
Xh, ,X > 0
==*.
(Ih + I K / 1 L h ) - l f h
_> 0
(6.38)
and
with the well-definedness of (Ih + AIrhlLh) -1 " Xh ~ Xh. On the other hand, Lhl~h is well-defined for X:h 9 Xh with [X:h] - If~-1 fa Xh = 0 by Poincar~-Wirtinger's inequality, similarly to the continuous case. Letting E v = v - [v], we have
IILhlEKhPhlBLI([.~),WI,q(~-~)~ Cq, lim IGL-j1EKsP~v ii i~ h~0
-
L-1Ev
W 1 , q (~"~)
=0
(6.40) (6.4~) (6.42)
220
6. Nonlinear Problems
for q E [1, 2) and v E Ll(f~), corresponding to the results obtained in Theorems 1.12 and 5.4. Inequality (1.48) is also replaced as [[LhTrhL-1E[[L~(a),L~(a) < C, and hence
I Kff I LhzrhL-lzll.(~)...(~)
_< c
(6.43)
follows from (5.7) with p = 2. Property (5.4) also keeps as Uh C X h ~
u
in
LP(f~)
MhUh ~ U in
===:=v
LV(f~).
(6.44)
Wellposedness We set
AhV = I(h 1LhTrhf (v) for v E W = C(ft), which can be regard as an operator in Xh. We have the following. Theorem
6.4. It holds that
IIMh~h[~h -- ~h]+ll~'(~) -- IIM ,. Iv,.-
+ AAhvh- aA,~'Oh]+ll~,(~),
(6.45)
where Vh, Vh E Xh and A > O. Furthermore, (Ih + AAh)Xh = Xh follows. Similarly to the continuous case, the nonlinear semigroup theory guarantees the unique solvability of (6.37) globally in time by Theorem 6.4, because IIMhTrhl'l Ll(ft) provides a norm to Xh. The solution is given as uh(t) = Sh(t)Trhuo for Sh(t) = lira m---,oo
(
Ih + t A b
(6.46)
7Yt
Inequality (6.45) now gives for Vh, 77h C Xh and t _> 0 that
I M,.zrh[S,.(t)v,.- Sh(t)~,,]+llL,(~) < IlMhZr,,[Vh- '&]+IIL'(~)
9
Therefore, u0 >_ 'u0 implies 'uh(t) _> 'Sh(t) for u0, '~),0 E W, where Uh = uh(t) denotes the solution of (6.37) and ~h = ~h(t) that of d d~I(hfih + Lhrchf(ith) = 0
with
/th(O) = 7rhftO.
In particular, u0 > 0 implies uh(t) > O. Furthermore, L 1 stability of the approximate solution holds in the following sense:
IIMh~'h '~'h(t)lllL,(a) 4 Mh~',~luolll,,,(a)" Because {rh} is regular, we have II~,~IXhIIILI(T) ~ Inequality (5.2) with p = 1 can be replaced by
IlxhlIL,(T)
for T E Th and Xh C Xh.
c-' I xh lI L, (T) _< IIMh~h xhlIIs,(T)< c X,,IIL,(T)
(s.47)
6.3. Degenerate Parabolic Equations
221
and hence
II[s,,(t)vh - sh(t)~]+ IIL,(~ ) _< c II[~ - ,~]+ll~,(~)
(6.48)
follows for Vh, 9h C Xh and t _> 0. Those properties are preserved in the time discretized equation, the backward difference finite element approximation
I(h [u;(t + T)~_- uT,(t)] + LhTrhf (u~(t + T ) ) = 0
(6.49)
with u~,(0) = 7rhuo, where t = rnT with m = 0, 1, 2 , . - . . In fact, its solution is given as u[,(t) : ( ~ + ~ A , , ) - m ~ u o for t = m~-, and formula (6.46) reads as
Uh(t) = lira u'h(t ). ~I0
We have
(6.50)
IlM~h bT,(t) - ~T,(t)]+ IIL,(~) _< IIM~,~,~ [~o - ~,o]+1}~,(~) for u0, ~0 E W, where U~h(t) denotes the solution of (6.49) with u~(0) = 7chuo, and 5T,(t) that with ~!~(0) = 7Ch50. We proceed to the proof of Theorem 6.4. Variation and lattice are key structures. Talking about the variational structure, we note that both the solution and test functions are taken from Xh. The lattice structure is also non-trivial, as Uh E Xh does not necessarily imply [Uh]+ E Xh. Given Vh C Xh, we take F~: = {a E Fh I +Vh(a) >_ 0}. It holds that
'V=~~ 7rh [Vh]:l== -nt- E Vh(a)Wa, aeV~h where [ - ] + = max{0, + . }. Then, 0 <_ V~h e Xh and Vh = v~ -- Vh follows. This implies (Ih + A K h l L h ) - l v ~ >_ 0 by (6.38). Therefore,
[(Ih Jr-,~KhlLh) -1 Vh]+ <_ (i~ + a K : I L ~ ) - ' ~ : and hence
~h [(Ih + "~KhlLh)-I Vh]+ ~ (Ih + /~KhlLh)-l?2~
is obtained. Because Mh : Xh ~ X h is also order preserving we have
-1
~ Mh (i h -~-/~KhlLh)-i Vh_/_.
Therefore,
f~ MhTrh [(]h-~ /~I(hlLh)-lvh]+ ~ ~A/lhV-h= ~MhTrh[Vh]+ holds by (6.39). We have the following fact comparable to Kato's inequality for L = - A .
(6.51)
222
6. Nonlinear Problems
L e m m a 6.5. The inequality
(~.52)
s MhrCh [(KhXLhrrhV) 9sgn+v] > 0 holds for v C W = C(f~), where
sgn+ v = ~1 to
(v>O) (v < o).
Proof." Given u, v E W, we have
7rh (u. sgn+v)
~ u(a)wa aeyhn{v>O}
= =
~
u(a)wa = rrh (u. sgn+Trhv).
ae Vhn{ rrhV>_O}
Therefore, f
f
Ja Mhrrh [(K171Lhrrhv) 9sgn+v] : Jn Mh~rh [(Ir
9sgn+vh] ,
where v~ = rrhv E Xh. Writing uh = (Ih + AKt71Lh) -1 vh for A > 0, we have Yh
Uh
-
=
)~ ( I h 2t- / ~ / ( / 7 1 L h ) -1 IV- -h1
Llz~tJh"
This implies A j~ Mhrrh [(Ih+ kKt[1Lh) -1K[[1Lhvh " sgn+vh]
= s Mh~,,[(~,,- ~,,,). ~n+~,,] = s ~,,,~,, [.,,,,]+- s A,,,~,,[.,,,,-~n+~,~] >_ s ~',,~,, [v,,]+- s Mh~,, [~,,,]+ >_0 by (6.51). Letting A I 0, we obtain
.s M,,~,, [(<7'L,,~,,,) . ~g~+,,,,] > o, or equivalently, (6.52). We now give the following.
6.3. Degenerate Parabolic Equations
223
Proof of Theorem 6.~: First, we show t h a t
I I M ~ [v - O]+IIL~r --< IlMh~h [v -- ~ +
~A~v
-
AA~O]+ IL~(~)
for v, ~ r W and ,k > O. We may suppose t h a t f : R ~ IR is strictly increasing by taking f~(u) = f ( u ) + su and later making s I 0. Letting u = v + and ~ = ~ + )~I([~lLhTChf(v), we have
AK~iLh~hf(v)
[IMh~h [V -- ~]+ll.(~)
= s Mh~h [(~ - ~)sgn+(~ - ~)]
= /o M ~
[(~- ~). sgn+(~ - ~)]
- ~ f a Mh:rh [KhlLhTrh [f(v) -- f( 9 Because f is strictly increasing, for w = f ( v ) -
sgn+(v -- ~5)].
f(~) we have
sgn+w = sgn+(v - ~?). Therefore, L e m m a 6.5 assures
J~ MhTrh [t(hlnhTrh I f ( f ) - f(?))]. SgIl+(V -- ?))] = ]~ Mh7rh [I(hlLh7rh w.
sgn+w]
>0
so t h a t we obtain
I Mh:rh[ v - ~?]+lln,(fl)
<
.~o Mh:rh [(u -- ~ ) . sgn+(v -- ~)]
_< /~ M.Tr. [u
-
/~]+
as is desired. Now we proceed to the proof of (Ih + AAh)Xh = Xh. Given Uh E Xh, we show the existence of Vh C Xh satisfying Vh + AAhVh = Uh. In fact, Ta = Ih + AAh is a continuous m a p p i n g on Xh, a finite dimensional vector space. Note t h a t IIMhTrh I 9Illnl(~) provides a norm to Xh. In use of (6.45), we can take an open ball (9 C Xh sufficiently large, satisfying uh q~ Ta(OO)for any A > 0. We may suppose t h a t uh C (9. T h e n the topological degree d(T~, Uh, (9) is well-defined. Its h o m o t o p y invariance implies
d(Ta, Uh, (9) = d(Id, Uh, (9) = 1. Therefore, Uh C Th ((9) follows. The proof is complete.
L~176 Scheme (6.37) has the property of L ~ - s t a b i l i t y furthermore, stated as follows.
[]
6. Nonlinear Problems
224 Theorem
6.6. The semigroup {Sh(t) } in Xh defined by (6.~6) satisfies
IIS,,(t)~h~011~o~(n) <_ II~-h~011~oo(n)
(6.53)
for t >_ O and uo E W. Proof: Thanks to (6.33), inequality (6.53) is reduced to
II',~,~llLoo(~)_ II(Zh + AAh),~hll~oo(~) for uh E Xh and A > 0. Let vh = (I + AAh)uh and b =
(6.54)
I1~,~11~(~).The relation
(~h, Xh) + A (VTrhf (Uh), VXh) = (Vh, Xh)
(Xh 6 Xh)
(6.55)
holds for Uh = Vh = -+-b E Xh. This means + + ~ h - (Z,~ + ~m,~)-' (+b) = +b and inequality (6.54) is reduced to the following lemma of comparison. consequence of Theorem 6.4 and the proof is complete.
It is a direct []
L e m m a 6.7. Given vh, 9h 9 Xh, let Uh, ith 9 Xh be the solutions of (6.55) and
(~h, Xh) 4- A (VTrhf(~h), VXh) - (-~h,Xh)
(Xh 9 Xh),
respectively. Then, Vh <<"vh implies Uh <_ iZh in f~.
Time Discretization We have proven convergence (6.50) for the backward difference approximate solution u~ of scheme (6.49). Here we study another scheme based on the nonlinear Chernoff formula of Brezis and Pazy. We suppose that the nonlinearity f is locally LiI)schitz continuous. Let # > 0 be the Lipschitz constant of f on
[-II~-h~,0ll~oo(~),I ~h~011LOO(~)], and take the regularizing parameter c = c , --, 0 as 7- $ 0 satisfying #~-/a~ _< 1. Under such a preparation, the schcme is given as
~(t
+ ~) - ,,,,;(t)
+
1 - e-~K,Y1LhI
7Chf (",,,,,(t)) T
=
0
(6.56)
with u,~(O) = 7rhuO
(6.57)
6.3. Degenerate Parabolic Equations
225
for t = m r (m = 0, 1, 2 , - . - ) . Here, {e-~ denotes the linear semigroup in Xh generated by K~ 1Lh. We can show tile convergence lira sup IlMhTrhlu;(t) -- uh(t)lllLi(a) -----0 Ts
(6.58)
0
for T > 0, where u~(t) and uh(t) denote the solutions of (6.56) with (6.57) and (6.37), respectively. In fact, it holds that
ttt:(t )
=
[fh
(tm-l)] m (TrhttO)
for t = m r with the nonlinear operator Fh(r) " Xh ~ Xh defined by
[F~(~-)] ( ~ ) = x~ + L [L~ - ~ ' / q ~ , h f ( x ~ ) - ~ f ( ~ ) ] J O"r
9
(6.59)
Applying Theorem 6.6 for f(u) = u, we have
e-aKhlLh7rhY+ IILOC(f.t) _< I1~-~+ I~(~)
(6.60)
for a > 0 and v C W. A quick overview now guarantees the estimate
II~(t)ll~(~)
_< II~-h~oll~(~)
inductively by #r/o.,- _< 1 and (6.60). We may assume t h a t f = f(u) is Lipschitz continuous with the Lipschitz constant # in R by replacing f(u) for lul >_ 117rhuoIL~(n)if necessary and then
r If(r)-
GT
f(s)l + ( r - s) - L ( f ( r ) GT
f(s)) = I r - sl
(6.61)
follows for r, s E R. Theorem 5.1 assures L1 (f~)
for v C W. Equality (6.61) gives for Xh, Oh C Xh t h a t
IIMhTrh Ifh(T)Xh -- fh(T)~JhlllLl(t2) T < =lMh~hIf (Xh) -- f (~,,)lllL~(~) (7 T
O"m
-
IIMh~,~
Ix~ - r
L 1 (Q)
(6.62)
Convergence (6.58) is a consequence of the nonlinear Chernoff formula. Namely, it suffices to show the convergence of the resolvent in the sense that lim TiO
(
Ih + - ( I h - Fh(r)) 7-
)1
Xh = (Ih +/~Ah) -1Xh,
(6.63)
226
6. Nonlinear Problems
where A > 0 and Xh C X h. Letting Ch = (Ih + )~Ah) -1 Xh, we set ~
r
=
)-1
~,~ + -
(~-
r,~(~))
Xh
T
and
X~=r T
In use of xh = r
+ - (h, - Fh(~))Cj:, T
we have
(
IIMh~h ICh
1+
--
Chlllc,(.) --< IIMhTrh IXh -- x;~lllL,(~) + -IIMhTrh I ~ -- ~hlllLl(a) T
by (6.62). This means
IIMh~hICZ - ~hlllL,(~) ~ IIMh~hI ~ h - ~,SIIIL,<~> 9 Here we have x~r
=
C h _ A-[-
e-aT
K,: I L h T r h f (r
-- zrhf (~Ph)]
Crr
--~ r
+ AK[[1LhTrhf (r
= r
+ AAh~h = Xh
by (6.59). Hence limTl0 ~Pt: = Ch and (6.63) follows. The proof is complete.
C o n v e r g e n c e of R e s o l v e n t Trotter-Kato's theorem assures that convergence of semigroups is a consequence of that of resolvents, and under that spirit, we study the convergence of resolvents now. Here, we take the simplest case that {rh} is uniform" each vertex is shared with 6 triangles and the following property holds in rh, where oz(z) = x + h~ei for a basis {el,e2} of R 2 and uh < h i , h 2 <_ h with a constant u > 0: T E rh
==*
ai(T) C rh
(i = 1, 2)
Under such a situation, we can show tile following. T h e o r e m 6.8. Under the assumption stated above, it holds t/tat lim II(Ih + AAh) -17rh'v- (1 + AA) -l VllL~(m = 0 hlO
f o r A > O and v E W .
(6.64)
227
6.3. Degenerate Parabolic E q u a t i o n s Proof." Given I > 0 and v E W, we set vh = 7rhv and Uh : (Ih + / ~ A h ) -1 7rhV.
We have I(hUh -+- ALhTChf (Uh) = I(hVh
(6.6s)
II~ll~oo(~) ~ IIv~ll~oo(~)~ II~llLoo(~)-
(6.66)
and (6.54):
Therefore,
III~t~*hllL~(~)< C II~IIL~(~)fonows from (5.6). Similarly, IIKhvhllLoo(a) < C Ilvll~(~)
holds, and we get IILh~hf(uh)llL,(~) <_ Ca -~
117rhf(u,~)llw~,~(~) -
II~IIL~(~) by O(1)
(6.65). The relation
(q ~ [1, 2))
now follows from (6.40), because 117rhf(u,~)llL~i~) < c is obvious. Later, we shall show the following. L e m m a 6.9. The f a r o @ {uh} c L~(f~) is relatively compact as h I O. This lemma assures that any hk ~ 0 takes a subsequence { h; } satisfying 7rh;f(Uh;) ~ W weakly in w ~ ' q ( a ) and Uh,k --~ u strongly in Ll(f~) with some w E w l ' q ( a ) and u E L l ( a ) . Those convergences can be also almost everywhere in f~. We have w = f(u)
(a.e. in f~).
(6.67)
In fact, given s > 0, we have a measurable set f~ C f~ satisfying If~ \ f~l < s and uh; ~ u uniformly on f~ by Egorov's theorem, where I" I denotes the Lebesgue measure. This implies that 7rhf(Uh,k) ~ Z(U) uniformly on a~ and w = f ( u ) holds in f~. Therefore, (6.67) follows by making s I 0. In this way, 7rh,kf(Uh,k) ~ I(U) weakly in w l ' q ( a ) and Uh,k --~ u strongly in Ll(f~) with some u C L l ( a ) . It holds that
IlL; 1EZ~hUh -- L -1 Eullwl,q(f~) ]IL;1EKhPh (U h -<_ C l l u h -- u
by (6.41). Similarly,
u)llwl,q(f~ ) -t-IlL;1EI~hrhu-
L~(n) + [ l L h l E I ( h P h
u -- L-IEullw~,q(n)
L-1Eullwl,q(f~) "
Letting h = h~ ~ 0, we see that the right-hand side converges 0 by (6.42).
6. Nonlinear Problems
228
follows from Ilvh- vllL~(~> ~ O. We have [Lh:rhf(Uh)] = 0. This implies L h l K h E u h + A:rhf(Uh) = L h l K h E v h and hence
L-1Eu + Af(u) = L-1Ev follows. On the other hand, [Kh,kUh,k] : (Mh,kUh,k, Mh,kl ) = (Mh,kUh,k, 1) ---* [U] by (6.44) and similarly, [KhVh] --* IV]. This implies [u] = [v] by [KhUh] = [I(hVh] and hence u + A L I ( u ) = v is obtained. We have u = (I + A A ) - l v and convergence (6.64) now follows. Theorem 6.8 is reduced to Lemma 6.9. []
Proof of L e m m a 6.9: We show that any c > 0 admits ~ > 0 such that
o < h < e~, lyl d~ < ~
===:~
~ luh(x + y ) - uh(x)l < e.
(6.68)
Then the assertion is proven. In fact, we have the following properties by (6.66):
1) sup II~hllg,<~) < + ~ h
2) Any e > 0 admits # > 0 such that A C f~ with [A[ < ~ implies suPh
~ luhl< c
Let hk I 0 a n d r > 0. We have k. such that k_> k. implieshk < 5. On the other hand uh,, u h 2 , ' " , Uhk._, are uniformly continuous on f~. Making ~ > 0 smaller if necessary, lyl < ~ implies
f l ~ ( x + y) - ~(~)1
dx <
(k= 1,2,.-.)
by (6.68). Frechdt-Kolmogorov's criterion now holds and {uhk} C Ll(f~)is proven to be compact. We recall the translation operator ai " x H x + hei. Let 2 = (or1, a2} be the transformation group generated by cri (i = 1, 2), and
II~ll = sup ~(~) - xl xEIR 2
f o r c r E 2 . We s e t a z ( x ) = x + z f o r z E R 2. G i v e n y E R 2 , w e h a v e a E E a n d z E R I~11 -< lyl and Izl _< h such that T + {y} = az (a(T)) for any T E rh. We shall write cr*ut~ for uh o or. Tile first term of the right-hand side of
2in
229
6.3. Degenerate Parabolic Equations f
is represented as I,, [Uh(X + z) -- ~(~)1 d~ with I~l _< h by a transformation of variables. We shall show that this term is estimated from above by a constant times
sup~ep~~ I~*~(~) - ~(~)1 &. IIo-II_
This implies
s
~,(~ § y) - ~h(~)l d~ ~ C sup s
I~*~,(~) - ~,~(~)1 d~
aEE
IIo-II_
sup~ s
(6.69)
I~*uh(x) - Uh(x)l & .
IIo-I1_<1~1
To see this, we take T E % and let its vertices be {al, a2, a3}. Suppose a3 = o 1 ( a 2 ) and al = a2(a3), without loss of generality. The inverse assumption (actually uniformity of %) guarantees uh(a3) - ~h(a~.)l _< I ~ ,
Similarly, w~ h~ve I~h(al) - ~ ( a 3 ) l
IlWhll~(~) ~
C h -~
- ~ 1 1 ~ ( ~ ) ~ c h -~ I1~1~ - ~,11,(~) ~
Ch -~
9
II~r~h - ~hll~x(~) ~nd hence
sup IIo-*~- UhIILI(T) ~ Ch -3 ~ aEE
IIo*~- ~11.(~)
(6.70)
aEE
Iloll_
IIoll___h
follows. Let {T/}~I C rh be the set of triangles which share one or two vertices of T E rh. Letting T = Ui1_21Ti, we have
I~(~ + ~) - ~.(~)1 _< h IIW.ll~(% for I~1 __ h and x E T. Applying (6.70), we have IIW~ll~oo(%
=
sup IIW,~ll>o(~,)
1
<_ Ch -3 sup 1
E
II~*u~- ~llL'(a)
aEE
Ilal[_
<- ch -~ ~
I1~*~-~11~).
aCE
II~ll__h
This implies
aEE
Iloll___h
6. Nonlinear Problems
230 and hence
~ luh(x + z)- Uh(X)ldx < 12C' ~
a*Uh(X)- Uh(X)IIL,(n)
o'C Y?,
II~ll
a*uh
--
( Ih + )~Ah )- I 0" , Y h
with Vh = 7rhv and (6.45) gives
<
I~1. I ~ * ~ - ~1 ~ ( , ) .
Inequality (6.69) is reduced to
s I~,,(~ + y) - ~,,(~)1 d~ _ C' sup la*v II~ll
-
~11~o~(~) ace IIGII
and p r o p e r t y (6.68) is a consequence of tile uniform continuity of v C W. T h e proof is complete. []
Convergence of Semigroup Convergence of the semigroup is a consequence of tile resolvents. Uniformity of essential for the following t h e o r e m to hold. Theorem
Th
is not
6.10. If f is strictly increasing and rh is uniform, we have lim sup IISh(t)~rhuo -- S(t)uo ILI(~) = 0
(6.71)
hi0 0 < t < T
for T > O and uo E W . Similarly to the s t a n d a r d theory, we make use of the Yosida approximation to derive the convergence of semigroups from that of resolvents. Here, we recall the following facts. Given A > 0, let J~ = (I + AA) -1 and Aa = ( I - ,J~)/A. We take d d~UA + Aaua = 0
with
'ua(0) = u0
(6.72)
6.3. Degenerate Parabolic Equations
231
in X = Ll(f~). We can verify t h a t Aa is an m-accretive operator and hence - A a generates a contraction semigroup. Any T > 0 admits a contraction (I + TA,x) -1 on X:
I1(1 +
_< IIv -
7A,x) - i v - (I -F TA,~) -1~1
~?llLl(a)
(v, ~? E X)
(6.73)
Obviously, Aa is (2/A)-Lipschitz continuous in X because J~ is a contraction. Therefore, (6.72) is uniquely solvable; we get the solution UA e c l ( [ o , oo), X). It is known t h a t u~(t) is regarded as an approximation of u(t) = S(t)uo. In fact, the inequalities --
--<
2("~2+ / ~ t )
1/2
IIA'~ollc~(~)
(6.74)
and ~ ( x / ~ + A)IIAuollLl(~)
(6.75)
IlS(t)uo- sa(t)uo Ll(f]) __~ 3(v/,~t + )~) Auollc,(~)
(6.7s)
-
are known. In particular,
holds for u0 E D ( A ) . It is natural to introduce the corresponding operators on Xh as Jh,~ = (Ih + AAh) -1 and Ah,~ = (Ih -- Jh,~) /)~. The problem d -~uh,~ + A~,~uh,a = 0
with
uh,a(0) = 7rhuo.
(6.77)
has a unique solution Uh,~ E C1([0, oo), Xh). Our first task is to derive lim sup Iluh,a(t) - ua(t)llLl(a ) = 0 h~0 0
(6.78)
for ,~ > 0. However, in the framework of nonlinear semigroup theory, ua(t) E W does not follow from u0 E W in spite that ua(t) E L~176 actually follows from u0 E L~176 Because the interpolation operator 7rh works only to continuous functions, this causes a technical difficulty. To avoid it, we take time discretizations and deduce an analogous result first. We prove the following. L e m m a 6.11. of
Given )~ > 0 and ~- > O, let u~(t) E X and U~h,~(t) E Xh be the solutions
~X(t + ~) - ~ (t)
+ A:~u~ (t + T) = 0
with
+ Ah,:~u~h,~(t + T) = 0
with
u~(O) = "Uo
and
~L~( t + ~) - ~L~(t)
U~h,:~(0) = 7ChUo,
232
6. Nonlinear Problems
respectively, where t = m~- with m = O, 1, 2 , . . . . the continuous time t >_ O. Then, it holds that
Combining linearly, we extend them to
lim sup l u ~ , ~ ( t ) - u~(t)ilL,(n) = 0
(6.79)
hi0 0
forT>O. Proof: Because f is monotone increasing, we can show that v E W implies u = (I + w A ~ ) - l v E W . In fact, this relation is equivalent to + 1 u = - v + J~u
-
T
(6.80)
T
and hence 1 +
-
IlullL~(a)
T
IIv Ig~(a) -
u + -(u-
W
v)
T
= I&ullg~(a) <_ IlullL~(a) L~(f2)
follows from (6.54). This implies II,ulino~(fl) _< IIv ILo~(fl) < +oe. Now (6.80) means A u = u + -(u-
T
A v)+ ALf(u + -(u-
T
v)),
~ d c f (~ + ~(~,- v)) c c ~ ( ~ ) fonows from ~, ~ z c ~ ( ~ ) . This implies f
u+-(u-v) "r
E W=C(f2)
from the elliptic regularity. Because f C C (R,R) u+A(u-v)/wE W and h e n c e u C W b y v C W . From the property stated above, we get u[(t) C W (I + "rA,x)-~Uo for t = mr. As a consequence of Theorem 6.4, the operator on Xh equipped with the norm ]MhTCh 1" ]]]L'(n); it
is strictly increasing, this implies for t > 0 by u0 C W and u[(t) = Ah,~ has the following properties is (2/)~)-Lipschitz continuous and
(I + rAh,~) -1" Xh ---* Xh is a contraction with respect to ]]MhTrh 1" ]]]gl(n)" If t = m r , we have ,u rh,~ (t) --u~(t)
=
(Ih -[- TAh~) , -m 7rh'Uo - ( I + ~-A ~ ) -m Uo
=
(I + TAX)-mTrhUO- (I + "rA,x)-muo
m + E[(I
+ rA~) -(m-e) (Ih + rAh,~)-'
~=1
- - ( I + TA,x) -(m-e+l) (Ih -+- TAh,~)-(e-1)]TrhU0 by the associative law of operators. In use of (6.73), Ll-norm of the second term of the right-hand side is estimated from above by m
Z II[(1 + T A ~ ) ( I h e=l
+ TAh,,x) - e -
(Ih + TAb,A) -(f-l)] 7[hUOIILI(f2)
6.3. Degenerate Parabolic Equations
233
This is equal to
m g=l m g=l
We obtain from (6.73) that
][u~,~(t) -
u~(t)llLl(a) ~
[[(Trh --/h) U0[ILI(~)
m g=l m
(6.81) g=l
In use of the Lipschitz continuity of Aa, the second term of the right-hand side of (6.81) is estimated from above by
2~ m
)-~
-~]
g=l
-< A Ee:l (Ih + TAh,~)-eTrhUo- (I + TA~)-euo +7 ~
(I + ~A~
~o- (I +
Ll(a)
~A~) -~
g--1
m -<
2"]5-Z
g--1
2t
T (~f) Iluh,~ - u~,T ( ~ T ) l l L 1 (~) "-JI--i- I1(I-
~h) UollL,
(~)
6. Nonlinear Problems
234
To handle with the third term of the right-hand side of (6.81), we note that
[-
3
[Aa (I + TAa) - e - Ah,a (Ih + tAb,a) -e] 7rhUo = A~ (I + rA,x)-errhuo- A,x (I + rA,x)-euo +Ax (I + TA~)-euo- Ah,~ (Ih + rAh,~)-eTrhuo = [Aa (I + T A A ) -g rch- A,x (I + rA~)-ej] UO 1
+ ~ [(1 + rAa
)-e
-e - ( I h + TAb,a) 7r,,]no
1
[Ja (I + rAa) - e - Jha (Ih + rA,.a)-eTrh] Uo A = [Aa (I + rAa)-err.- Aa (I + rAa) -el UO + ~1 [(I + rAa )-e - ( I h + tAb,a) -err,,.] Uo 1 )~
[& _
Jh'~1
(I + ~A~) -~
1[ A Jh,aTrh(I + TAa) - e -
UO
]
Jh,,~(Ih + rAh,a)-eTrh Uo.
Therefore, in use of the Lit)schitz continuity of A~ and (6.73), tile third term of the right-hand side of (6.81) is estimated from above by
2 7"?Tt
,~
T
m
II(~-h- I) UollLl(~) + ; ~
T
r
I[""'~'(e"-)- ~"~(e"-)llL'(~)
g=l m
+ A
e:l
CT
gl(~)
m. g=l
2t _<
A
m I(r:h
1) uoll~,
(~) +
(C + 1) ;T
Z I1'" (e~) - ,4(e~)II~,(~) g=l
m
+ ~-
Uo
Ll(f~)
g=l
CT
m g=l
6.3. Degenerate Parabolic Equations
235
We can summarize them as
[l(Trh- I)uo[lLl(~)
CT,~ E [[Tj,~7. h,,, (g,.]_) __ U~(gT_)[ILI(~) _31_ 1 + ~ g=l CT
m
7.
T
g=l
m
7.
g=l
for t = m r . Because of the discrete version of Gronwall's lemma, this implies
~uv II~L(t) - ~i(t) I .(~) < ~ / ~ ( 1 + _c~_)II(~. - I)~0 I~(~)
0
m _FeCr/,xC7 --~- Zm [ (~h - I ) u 7.(eT) x ][Ll(a ) _31_e c r / x ~a ~--~.I[[Jxg=l g=l
Y.x~h]uX(g7)llCl(n) ' '
where T = m r . As is noted, u0 E W implies u~(lr) E W. Therefore, the right-hand side of the above inequality converges to zero as h I 0 by (6.64). The proof is complete. [] Now, we show the following. L e m m a 6.12. Given A > 0 and uo E W , let u~ = u~(t) C Cl([0, o c ) , X ) and Uh,~ = Uh,~(t) e Cl([O, o c ) , X h ) be the solutions to (6.72) and (6.77), respectively. Then the convergence (6. 78) follows. Proof: The semigroup {T~(t)} generated by A~ is a contraction in X = Ll(gt). This implies [[u~(t)I]Ll(a ) < [lU0[ILl(a), and hence
IlAAux(t) - Axux(s)lln,(~)
2
<_ -A ] l ~ ( t ) - ~(S)IILI(~>
< V4 It- ~l II~ol LI(~) (6.82)
-,~ follows. Because of
u~(t + T) - u~(t) + TA~u~(t + T) = 0 and uA(t + T) - u~(t) +
l
t+T
AAu~(s)ds - O,
,It
the error function eT.(t) - u~(t) - u~(t) defined for t = m T (m = O, 1,--. ) satisfies (I + ~-A~)u~(t + "r) - (I + TA~) 'u:~(t + 7-) = e~(t) +
f
,It
t+T
[n~u~(s) - A~u~(t + "r)]ds.
236
6. Nonlinear P r o b l e m s
This implies
_< I e'(t)llc~(a) +
f
t+T
Aau~(t + ~) - A~ua(s)llLi(a) ds
dt
m[.
by (6.73), and hence
l e'(t)llLl(n) < ~
-1)7
f=l
follows for t = mr. In use of (6.82) we obtain m 4
II~-(t)llLi(~)----- ~
T2
2tT
V ~ 2 II~0
i
ii
/=1 or
2TT
sup
0
IluX(t)- u~(t)l L,(n) < -V--Ilu011L,(~)
for T > 0. Similarly, we have 2T7
sup
I~;,x(t) - ~h,x(t)[[]~(~) ___~
0
IIMh~h I~h~olll~(~)
and hence CTT
sup 0
I uh,~(t) - u,~(t)ILI(f~ ) ~ ~
~ I'U,0 L~(Ft) -Jr- s u p I l t h , A ( t ) 0
--
ttA(t) llLl(a),,
follows. Now, we send h I 0 and then r I 0. We get (6.78) by (6.79). The proof is complete. [] Now we complete the following. Proof of T h e o r e m 6. i0:
We shall show that (6.83)
lim sup [ uh(t) -- u(t)llLl(gt) -- O. hJ.O
0
Consider, first, the case that Uo E W satisfies f (u0) E C2(f~). Inequality (6.76) is a consequence of the general theory. We also have sup o
IIMh~h lu~,~(t) - uk(t)lllL,<~) < 3 (\ ~
+ A)]lMhTCh AkTrh'uO /
by Theorem 6.4. Here, in use of LhZrh [f(u0)] = 0 we have
IIMh~h IAh~h~ofll~l(~)=
IIM,~,~ IK;' fh~hE f
(~o)lll~,(~)
< c I I K ; ' L ~ L - ~ E . Lf (~o)11~(~) < C IILf (~o)11~(~)
Commentaw
to Chapter 6
237
by [L -I, E] = 0 and (6.43). We have
<_ <
sup 0
Ilu~(t)-
sup O
I ~th(t) -- Uh&(t)l[Ll(f~) + sup []Uh,l(t) -- ~tA(t)l Ll(f~) -I- sup
sup
0
u(t)llL,(~)
O
I u~,a(t)
- ua(t)llLl(a) + C ( x / ~
O
+
ua(t) -- u(t)llLl(ft)
A) IILI (Uo)IL=(~) 9
This implies
uk(t)-
limsup sup hi0
u(t)
Af_ ,~)
LI(~--~)_~ C ( ~ - T
ILf(u0)ILk(n)
0
by (6.78), and conclusion (6.83) follows by making A t 0. It remains to extend it for the general case of u0 E W. We again make use of the strictly monotonicity of f for this purpose. P u t t i n g v0 = f(uo), we take a sequence {vk} C C2(f~) N W satisfying lim Ilvk - v0llc~(a) = 0 and set uk = f - l ( v k ) . Then it holds that f (uk) < W N Cg(f~) and lira ][uk - u0 c~(a) = 0. We have proven lira sup
h$0 0
II&(t)Trhuk --
S(t)uklIL~(~)
: 0
for k = 1, 2 , . - - . In use of (6.34) and (6.48), on the other hand, we obtain
IIS(t)uoIIS(t)uo-
Sh(t)TChUOIILI(~) s(t)u~llc,(~) + I I s ( t ) u ~ - Sh(t)TrhUkllLl(~) + II&(t)Trhuk- Sh(t)TrhUOIIc,(~)
_< Iluo -- uklIL,(~) + IIS(t)uk -- &(t)~h~kllL,(a) § C II~hUk -- ~OIIL~(~) < C I1~ - ~OIILoo(~) + IIS(t)Uk -- &(t)TrhUkllL~(a) 9 This implies sup 0
II&(t)~-huo
- S(t)uol
LI(~) !
If we make h ~ 0 and then k ~ complete.
Commentary
to Chapter
C 1]'~/r - Uo] Lee([-]) -1L- s u p
0
II&(t)Trhuk
-
S(t)ukllLl(n).
oo, convergence (6.71) is obtained and the proof is []
6
6.1. The monograph Suzuki [369] is devoted to the theory of semilinear elliptic boundary value problem. There, the existence (unique, multiple, and non-) of solutions and their qualitative features are described. To see that any non-trivial solution u of (6.1) with (6.2) is unstable, note that the linearized operator is given as - A - p lul p-1. Then, its first eigenvalue is negative by
llVul[~(n)
P
U IP+I
'L~+ (n)
(p
I)
2
<
238
6. Nonlinear Problems
and Rayleigh's principle. It indicates the linearized instability. See Henry [173] for details of this notion. In many cases, the simple iterative sequence defined by (6.3) becomes monotone, which makes its dynamics simpler. For this topic, see Amann [5], P.L. Lions [248], and the references therein. However, the following fact essentially due to Fujita [136] is worth noting: Let u. is a non-trivial solution, positive in f2. Then, I'uk L~(a) --+ +oc and IlukllL~(a) ---+ 0 according to u0 > u. and 0 _< u0 < u., respectively. Therefore, scheme (6.3) is not suitable to get unstable solutions. See also Suzuki [369] for the proof of this fact. Ambrosetti and Rabinowitz [9] obtained infinitely many solutions for (6.1) with (6.2) by tile variational method of mini-max type. See Berger [32] and Rabinowitz [320] for various methods of variation and applications to nonlinear partial differential equations. Concerning the Morse indices of tile mini-max solutions, see Bahri and Lions [22] and the references therein. Uniqueness of tile least energy solution for (6.1) with (6.2) is proven by C.S. Lin [240] when [2 C R 2 is a convex smooth domain. Nehari's variational principle was introduced by [289], [290], [291] for the two-point boundary value t)roblem. Applications to partial differential equations were done by Coffman [90], [91], Struwe [361], and others. Its equivalence to the mountain pass lemma was noted by Ni [294]. However, to our knowledge, Nehari's iterative sequence was introduced by Coffman and Ziemer [92] in the study of quasilinear elliptic boundary problems, although a related idea can bee seen in Ding and Ni [113]. Mizutani and Suzuki [268], [269] gave a systematic study on that method of iteration, generalizing nonlinearities, unifying the method of inverse powers, and so forth. There, tile numerical computation was done by the use of finite element method. See also Suzuki [369] for some theoretical aspects of this sequence. An example of numerical experiments is presented in Figure 6.1. The Nehari manifold is closely related to the stable and unstable sets in the theory of double wells. For this topic, see Ikehata and Suzuki [189], [190] and the references therein. The standard bootstrap argument valid for the subcritical nonlinearity in elliptic boundary value problems is presented in Suzuki [369]. Actually the elliptic estimate utilized in this section is indicated as (-A)-IL2(f2) C H2(f2). It holds because f2 is a convex polygon. Recently, several trials have been made to provide a proof of the existence of the solution with the aid of numerical computations. Among others, Nakao's method works to the fixed point equation u = r ( u ) on a Banach space X. For instance, if T is a (nonlinear) compact operator, then Schauder's fixed point theorem is applicable. If one can construct a bounded convex closed set U satisfying T(U) C U, then there is a fixed point of T in U. For this purpose, we construct a finite dimensional set Uh and its complement U ~. If T is split as T = Nh + M~, satisfying Nh(g) c Uh and Mrs(U) C U 1, then T(U) C U follows for U = Uh | U 1. Here, numerical verifications of Nh(U) CUh and Mh(U) c U 1 are done differently, Newton's method and norm estimate, respectively, for instance. For more details, see Nakao [276], [277], [278], [279], Watanabe and Nakao [404], Yamamoto and Nakao [408], Nakao and Yamamoto [280], Yamamoto [407], Nagatou, Yamamoto, and Nakao [272], Nakao, Yamamoto, and Nagatou [281], Nagatou [271], and the references therein. 6.2.
Non-existence of the classical solution globally in tittle for some kind of parabolic
C o m m e n t a r y to C h a p t e r 6
239
Figure 6.1: Shapes of approximate functions Uk,h of minimizing sequences to --Au = u 2 with ulon = O. The detail of the scheme is described in [269]. The domain f~ is assumed to be L-shaped with vertices (0,0), (1,0), (1,3/16), (3/16,3/16), (3/16,1), (0,1)and is divided into 2784 elements (equal right triangles).
equations was noticed by Kaplan [196], Fujita [135], [136], Tsutsumi [392], [393], and others. A lot of remarkable works have been done up to now. See Levin [238], Suzuki [369], Samarski, Galaktionov, Kurdyumov and Mikhailov [334], Deng and Levine [109] and the references therein. Blowup phenomenon for (6.12) was first observed by Fujita [137]. See Kohda and Suzuki [225] and the references therein for later developments. One point blowup is first noted by Weissler [405]. Later, Friedman and McLeod [131] and X.-Y. Chen and Matano [74] refined the study. Theorem 6.3 is due to Y.-G. Chen [75]. It also studied more general nonlinearities and showed that the exponent p = 2 in the nonlinear term u p , is critical for the three points blowup described here. Related studies are done by Nakagawa [273], Nakagawa and T. Ushijima [274] and Chen [76]. See also Berger and Kohn [33]. The validity of scheme (6.15), especially approximating the blowup time, was recently studied by T.K. Ushijima [399]. Another important application of FDM to the theoretical study on the semilinear parabolic equations was done by Tabata [376]. It studied the Euler approximation to
240
6. N o n l i n e a r P r o b l e m s
Figure 6.2: Behavior of blowup solution to ut = uzx + u 2 with u[x=o,1 = 0. The time discretization is subject to rn = r . min (1, IlunllL ~ ) with r > 0. We take as the initial value uo(x) = #x(1 - x)(x 2 - x + 3/10) with a constant p. We note that uo(x) has two peaks. These figures show the behaviors of blowup solutions when (a) # = 1000 and (b) # = 5000. In the farmer, the blowup takes place at only one point, but in the latter, it occurs at two points.
one space dimensional problem under tile Neumann boundary condition for general nonlinearity, and showed that if r / h 2 < 1/4 then the property of tile monotone decreasing of the number of peaks of the solution arises, where h and r denotes tile space and the time mesh size, respectively. Its continuous version was provided by Matano [260], and Angenent [11] showed sharp profiles of the movement of peaks. Stability of the FDM to the reaction-diffusion system was studied by Mimura, Kametaka, and Yamaguti [265]. Then, many works have been devoted to the discretized semilinear parabolic equations. See Thomde and Wahlbin [385], Thomde [383], and the references therein. 6.3. Concerning physical backgrounds of (6.30), see Friedman [130]. The hyperdispersivity of operator - A associated with the degenerate parabolic equation was proven by Br~sis and Strauss [65]. It was Crandall and Liggett [96] that proved the fact that a hyper-dispersive operator in a Banach space generates a nonlinear semigroup by formula (6.33). For this abstract theory and also the regularities of the solution, see Barbu [26] and Miyadera [266]. The work by A.E. Berger, H. Brdzis, and J.C.W. Rogers was published in [31]. Figure 6.3 shows an example of numerical computation by their scheme. Nonliner Chernoff formula was proven by Brezis and Pazy [64]. Recently, Evans [120] and Barles and Georgelin [27] observed that a scheme by Bence, Merriman, and Osher [28] for constructing mean curvature flows is justified by that formula in the framework of the level-set approach due to Osher and Sethian [310], Evans and Spruck [121], and Chen, Gig& and Goto [77]. See Crandall, Ishii, and Lions [95] for basic notions of the level-set approach. Scheme (6.35) was studied by Rose [328], Nochetto and Verdi [304] and Kacur, Handlovicovg, and Kacurovg [195] for the porous media and tile Stefan nonlinearities, ~ d by L ao x [2351 and Lesaint and Pousin [236] for (6.31) with ~, E (0, 1). It is known that nonlinear semigroup associated with the degenerate parabolic equation (6.30) is generated also in H-~(t2). See Nochetto, Savard, and Verdi [305] and the references therein
C o m l n e n t a l y to C h a p t e r 6
241
for numerical analysis based on that point of view.
Figure 6.3: Shapes of approximate solutions to ut = A f ( u ) with u[oa = 0, where f ( u ) = u + 1 (u < -1), = 0 (-1 < u < 1), and = 0 . 8 ( u - 1) (u > 1). The time discretization is done by the scheme (6.56).
Provided with the properties of order preserving and L 1 contraction, scheme (6.37) gives reliable numerical solutions very rapidly. It is also simple enough to perform actual computations, while theoretical studies are performed in the framework of the theory of nonlinear semigroups. See [63] and so forth for Poincar~-Wirtinger's inequality. Kato's inequality for - A was given by [204]. Concerning the topological degree and its properties see the standard monographs on nonlinear functional analysis, Nirenberg [295], Berger [32], Zeidler [416], Deimling [108], and so forth. The nonlinear Chernoff formula was presented by Brezis and Pazy [64]. Scheme is a finite element analogue of the one by [31]. Several versions of nonlinear Trotter-Kato's theorem are known since the original paper Trotter [387]. A typical form, making use of the Yosida approximation, is described in Brezis [62]. For the linear theory, see Kato [205] and the references therein. Ushijima [394] applied it to the finite element approximations. We adopt the method of B~nilan, Crandall, and Sacks [29] for the proof of Theorem
242
Z Domain Decomposition Method
6.8. See Yosida [410] or Brezis [63] for Frechdt-Kolmogorov's theorem. If t2 is a polygon, Theorem 6.10 follows if f is strictly increasing, convergence (6.64) holds for ~ > 0 and v C W, and t2 has no corner with angle 2re. For the fundamental properties of the Yosida approximation for m-accretive operators such as (6.76), see Miyadera [266]. There, inequalities (6.74) and (6.75) are indicated in the formulas below (4.5) and above (3.49), respectively. If t2 is a polygon with any vertices not to have angle 2re, the elliptic regularity necessary to the proof of Lemma 6.11 is assured.
Chapter 7 Domain Decomposition
Method
Basic idea of the domain decomposition method (DDM) is found by the work of H.A. Schwarz in the 1860's. The domain in consideration is divided into several parts, and the approximate solutions are computed separately and succecively on them, provided with suitable interaction conditions on artificial boundaries. This method fits parallel computations, domains with complex shapes, and coefficients with strong inhomogeneties. Target problems in this chapter are the Poisson and the stationary Stokes system. Tools necessary for our program are; square root of self-adjoint operator, Dirichlet to Neumann map, variational principles, reflection principles, inf-sup constant, and so on. Several monographs have been devoted to the mathematical theory of DDM such as Chan and Mathew [73], Smith, BjOrstadt and Gropp [352], Quarteroni and Valli [319] and Gu [165]. Especially, the problems which we are going to consider are discussed in [319] in more generalities. Howerver, operator theoretical analysis enables us to gain sensitive relationships between the rate of convergence of iterations and the shape of decomposed domains, as we are now describing. We mainly describe the continuous case, because the essence of the problem does not depend on the way of discretization.
7.1
Dirichlet to Neumann (DN) Map
This section is devoted to the Dirichlet to Neumann (DN) map, which plays a fundamental role in the analysis of DDM. We take the case that f~ C R 2 is a bounded domain, that OFt is composed of two smooth curves 7 and F, and that ~ and F are intersect transversally at two points. Those intersections make corners of f~. The Dirichlet to Neumann map S is defined formally as follows. First, given a function defined on 7, let h solve Ah=0
inf~
and
h=
Then, S is defined by
s ~ = ~~ 243
I,
{~ [0
on4/ on F.
(7.1)
7. Domain Decomposition Method
244
where O/On stands for the differentiation along the outer unit normal vector n, and "iv the trace to 7. This definition, however, has a usual sense only when h is appropriately regular, say, h E H2(f~). One obstruction is the corners of ft. The condition h E H2(f~) does not follow even if ~ is sufficiently smooth. Another is the requirement of later sections to regard S as an operator in L2(7). Those difficulties are overcome by tile representation theorem concerning bilinear forms on Hilbert spaces. Examining the wellposedness of (7.1), we take more generally that
-A,u=f
inf~
with
u=
[~ 0
l
on7 on F,
(7.2)
where f E L2(f~). We introduce a closed subspace I l l ( a ) of HI(f~) by
- {v e H ' ( ~ ) I
Ir
v r - o}.
Poincar~'s inequality is still valid for functions in KI(Ft), and the Dirichlet norm is equivalent to the H l ( f i ) - n o r m in I(l(ft) 9
IIV.llg=(~)
~
I1 I.,(~)
The underlying Hilbert space in our consideration is X = L2(7). The standard L2(7) inner product and norm are denoted by (., ")x and I I Ix, respectively. We set
Holo/2 ( ~ ) = { ~ e H1/2 (~) ~ p- 1~2 ds < + ~ } with
II~llglg:<+ =
I1~11~,,~(+
+
p-'~
ds
,
(7.3)
where p = dist,(., 07) denotes the distance along 7 from tile end points of 7, and ds the r4~/2 (7) becomes a Hilbert space equipped with the norm line element of 7. The space -:oo 1/9
(7.3). P u t t i n g V = H00 (7) and[ -lie - II .11.~o~<~), we thus obtain the triple V C X C V' of Hilbert spaces. In the present chapter, symbols C and C' are used to denote positive constants depending only on the domain in consideration. The values of these constants may differ from each other in a single context. When we want to specify the dependence of such constants on a parameter p, which may9 not be a number, we write Cp or C'p" In spite of the corners of [~, trace theorem keeps to hold in the following way. Proposition
7.1.
We have the following:
(i) Every ~ E Kl(f~) admits ~ = "~h~ E V satisfying I~llv <- C I ] ~ ) l l [ [ l ( ~ ) . (ii) Every ~ E V admits ~# E t(~(f~) satisf~ling ~ ~ = ~ and I1~1 +,,(~) _< c ' I~llv.
245
7.1. Dirichlet to Neumann Map
Consequently, problem (7.2) takes a unique solution for given ~ E V and f E L2(f~); we have a unique u E Kl(f~) satisfying u], = { and
(Vu, VV)L~(a) = ( f , V)L~(a)
(7.4)
for any v 9 H~(a). In fact, if u is a solution for ~ = 0 and f = o, then u E H~(f~) follows. The relation IIVullc2(n) = 0 holds by (7.4) with v = u. This implies u = 0 from Poinca%'s inequality, and thus the uniqueness follows. On the other hand, given ~ E V, we take its extension ~( E Kl(f~) satisfying [~)~[[H'(f~) ~ C [[~[[v" Because the linear functional
v E H~(f~)
~
(f, V)L2(a) - (V{;~, VV)c2(a)
is bounded, we have w E Hl(f~) satisfying (Vw, Vv)c:(a) = (f, V)c2(a) -- (V~e, VV)L:(~)
(7.5)
for v E H~(a). T h e n u = w + r solves (7.4). If f = 0, u is nothing but the harmonic extension of ~ into f~. This case is referred to as u = ~ . Then we have ]lVw La(a) < IlVCelln~(a) by (7.5). The following inequality holds by Proposition 7.1 and Poinca%'s inequality, where ~ E V:
c' IIg Iv -< IV~gll~(~)_< c I1~11~
(7.6)
In fact, we have
c ' I~llv <
I ~ll,-,~(r~)~ IlWll~(~)= I I V ~ l L~(~) < IIW, ~(~) + IIV~,~ll~(~) < 2 IIV~l ~(~) < 2 I1r HI(~) --~C II~llv.
We define a bilinear form ,.7 9V x V ~ R through the following relation, and call it the J - f o r m pertaining to (f~, 7)"
j ( ~ , .) = ( v ~ ,
v~,)~.(~)
Because of (7.6), it is bounded and coercive; we have
J(~, ~)>__ c '2 II~ll~,
[fl(~, ~)l _< c 2 I~ Iv II~llv
and
(7.7)
for ~, r/E V. Therefore, a positive self-adjoint operator S 9D ( S ) C X --. X is introduced, associated with J :
J ( ( , r~)= (S[, r])x
(~ E D ( S ) C 17, r]E V)
General theory now guarantees the following. P r o p o s i t i o n 7.2. We have D ( S 1/2) = V and J ( ~ , ~) = ( $ 1 / ~ , S1/~r])x for ~, r]E V. The mapping S 1/2" V ---->X i8 an isomorphism. In particular, we have
c' I ~llv < Ilsl/~Jl~ _< c II,~llv for~e
V.
(7.8)
7. Domain Decomposition Method
246 If h = 7-{~ is suitably regular, we get
f
ob
for r/ E V. This means S'~ = (Oh~On)Iv in the sense of distributions, and in this way the Dirichlet to N e u m a n n map is realized as a self-adjoint operator in L2(',/). We write S = S(t2, 7) and call it the D N map pertaining to (t2, 7), to express this correspondence. The o p e r a t o r S 1/9 can be characterized by the variational principle. In fact, the variational principle for the harmonic equation claims t h a t
IIV~ L=(~)~ IlVv L~(~) for ~ E V, where v E Kl(f~) is so taken as v[~ = ~. This, together with Proposition 7.2, yields the following.
Proposition 7.3. Given~ E V, we have
Is,/=
IIV,, L=(~), o~ equivalently J ( ~ , ~ ) <_
[IVv[l~2(~), for v E I('(t2) satisfying vl~ = ~.
7.2
Dirichlet
to Neumann
(DN)
Iteration
Studying the typical domain decomt)osition algorithm, we assume t h a t fi C R ~ is a b o u n d e d d o m a i n with the Lipschitz b o u n d a r y F = 0g~ and take the problem -Au=f where f E L2(f~).
~=~
in t2
with
u=0
onF,
(7.9)
The exact solution of this taryct problem is henceforth denoted by
H~(a).
We divide the target domain t2 into two disjoint subdomains t21 and t22 by an artificial boundary, a smooth simI)le curve denoted 1)y 7: n
f~ = f~l U [22,
fil N ~22 = O.
We assume t h a t 7 connects transversally two points oil F. The outer unit normal vector is generally denoted by n, but we specify the one to 7 outgoing from t21 by u if necessary. We put F1 = 0t21 \ 7 and F~ = 0t22\"/. Dirichlet to N e u m a n n (DN) iteration is as follows. Take a function it (~ E V as the initial guess. Then, for k = 0, 1, 2 , . - . , we sllccessivcly generate:
,tt(1k) :
the k-th a I)proxiInation to 'u,[~2~;
u~k ) : p(k+l) .
the k-th api)roximation to 'u,[~2" the (k + 1)-th aI)I)roxima,tion to "~.]~
247
7.2. Dirichlet to N e u m a n n Iteration
by solving
~'-'~1
-/Xu~ k)-- f
in
--Au~k) = f
in f12
with
u~k) __ l 0
with
{
on 1-'1
#(k)
on 7
on F2
u~k/= 0
Ou~k) On -
Ou~k) cgv
on 3'-
The value of p(k) is adapted by #(k+l)
_._
(1 - 0)# (k) +
o~)1~,
(7.10)
where 0 is the relaxation parameter subject to 0 < 0 _< I. This procedure is rigorously understood as follows: I) Take p(0) E V and put k = O.
2) Find u~k) e /(1(~-~1) satisfying
~")1-~=
,(~) and
(~7"U'k)' ~ VVl)L2(f~I)
-
-
"
( fial , Vl)L2(al)
for vl E Hl(f~l). 3) Find u~k) C Kl(f~2)satisfying
(vu~ k), VV2)L~.(~) --
V2)L~(n:) --
(fln~,
(Vu~ k), V~)I)L2(f~I) -]-
for v2 e K'l(a2), where 7./)1 E Kl(f~l) stands for an extension of
(flf~l' ~)1)L2(al)
~l~.
4) Renew p(k) as (7.10). Set k = k + 1. Go back to 2) until convergent.
Amplification Operator Taking attention to the convergence, we introduce error functions ~k) = u~k)_ Ulal, ~k) = u2(k) - u]a2, and ~(k) = #(k) - '51~. We obtain A~p~k) -- 0
in ~1
with
Ar k) = 0
in f~2
with
@[k) = { 0 ~(k)
on F1 on 7, on F2
0
On and =
(1 -
&,
on 7,
7. Domain Decomposition Method
248
Letting ,5'1 = E(~I, "y) and S2 = S(f~2, 7), we have Sly(k) -_ 0~)~k)_. J _ 0%, "r
01/)~k)
__ --$2 (~z~:k)[7)
On
and hence ~b~k)l~ = - S [ ' & ~
(k) follows. The (k + 1)-th error ~(k+l) can be expressed, at least in the formal manner, as {(k+i) = (1 - 0){ (k) - 0S;-~&~ (~/ This formula is justified by the following. L e m m a 7.4. The operator Ho = 321S1 with its domain D(Ho) = D(S1) admits of a bounded extension H into V, explicitly written as
H-
S21/2(S~/2S21/2) *.q'l/2 J,a 1 ,
(7 l l )
where * denotes the adjoint in X . Proof: It is easy to verify H0( = H ( for ~ E D(Ho). T h e boundness of H follows from ql/2 ql/2 S~ 1/2 E s V), (S~/2S~1/9) * E s and ~'1 E s In fact, since ~1 S21/2 C L;(X), the adjoint operator (S~/2S21/2) * is also b o u n d e d in X. [] Replacing $21El by H, we get the recursive expression
~(o) e V,
~(k+l) _ Ao~(k)
(k - O, 1 , 2 , . . . )
for Ao = (1 - 0)I - OH. We call it the amplification operator for the error of the DN iteration. A l t h o u g h H is not sclf-adjoint with the usual X - i n n e r product, we can treat it as a self-adjoint o p e r a t o r in V. For this purpose, we introduce a special inner product in V in terms of the DN m a p as follows:
((~ , 77))= (S~/2e , s~,/'e ~)~
(~, ~ c v )
The space V again forms a Hilbert st)ace with this new inner product. corresponding n o r m II1" Ill = ((', .))1/2 is equivalent to I1" IIv in V by (7.6).
(712)
In fact, the
L e m m a 7.5. Under the new inner product (7.12), the operator H defined by (7.11) is self-adjoint in V.
Proof." Since H is b o u n d e d in V, it suffices to verify t h a t H is symmetric. This is seen from
((H~, 7])) -- ((X~/2E21/2)* S~/2~, ~-b2r~l/27])XX= (X~/2~, Ell/2T]) x, ((~, HI])) -- (S~/2~, (S~/2S21/2)*S~/2I])x -- (Sl/'2~, S~/2I])x. The proof is complete.
[]
7.2. Dirichlet to Neumann Iteration
249
Since H is bounded and self-adjoint in V with the inner product (7.12), Ao has the same properties. The recursive formula for ~(k) implies (7.13) for k = 0, 1, 2 , . - - , where ro(Ao) denotes the spectral radius of Ao. The spectral mapping theorem now guarantees the following. L e m m a 7.6. We have
ro(Ao)=
sup [ 1 - 0 - 0 s l , sEa(H)
where or(H) denotes the spectrum of H. Proof." Because Ao is expressed as a linear function of H, we have
~(Ao) :
sup ~6a(Ao)
Ir
=
sup I1 - 0 - Osl, sEa(H)
where a(Ao) stands for the spectrum of Ao. The proof is complete.
C o n v e r g e n c e w h e n 7 is a L i n e S e g m e n t The spectral radius of Ao is evaluated in accordance with the geometry of f~i (i = 1, 2). First, we shall deal with the case that 7 is a line segment on the z2-axis. Furthermore, we say that the condition (I) is satisfied if 7~Lf~2 C_ f~l, where 7~c denotes reflection with respect to the z2-axis: 7~r: ( x l , x 2 ) ~ ( - x l , xg). L e m m a 7.7. If condition (I) is satisfied, then we have 0 <_ H <_ 1, so that 0 <_ s < 1 holds for any s E or(H).
Proof: We take J-forms ,.71, J2, and ,-72* pertaining to (f~l, ~'), (f~2, 3') , and ( ~ * 2, ~), _I1/2 respectively, where f~29 = 7~rf~2. Let ~ E V = /--00 (~/). We have ,72(~ , ~) = ~ , *2 (~ , ~) from the symmetry. We show J l ( ~ , ~ ) _< J2*(~,~)- In fact, let h~ be the harmonic extension of ~r into f~, and v its zero extension into f~l" 1,
Ah; = 0 = ~h~ V
[0
inf,;
with
h;=
~~ 0
[
in f~ in f~l\f~
Then the variational principle Proposition 7.3 implies
on-7 on0f~;\~,
250
z Domain Decomposition Method
Therefore, we get
o _< ((S~, ~)) <_ II1~111=
by Oq'l(~, ~) -- ((H~, ~)) and f12({, {) = II1~111~. The proof is com)lete.
[]
Lemmas 7.6 and 7.7 imply r,(Ao)<_ sup I i - O - O s l = ? ( O ) .
0<s
This, together with (7.13), gives
II ~(~)111___~(0)lll~(~-')lll _< ~(0)~111~(~ Because we have positive constants c~) and c~ determined by t22 satisfying
Iv -< (O) co II (~
c0
by (7.8), we get the following theorem with Co = cg/c'o. T h e o r e m 7.8. If condition (I) is satisfied, then there exists a constant Co > 0 determined only by f~2 such that
I1 ( )11 _< co#(0) I1(~
~
(7.14)
for k = 1 , 2 , 3 , . . . .
Here, we have
f(0)=
1 -0 20-1
(0 < 0 _< 2/3) ( 2 / 3 _ < 0 < 1)
and hence 0 < f(0) < 1 follows if 0 C (0, 1]. Furthermore, (7.14) is equivalent to
I1~~ -
~1~1I~ _< ~o~(0)~1I# ~~ ~1~I1~.
This implies the convergence of the approximate solutions {u}k)}. Theorem
7.9. Under the same assumption of Theorem 7.8, we have
[lUlk ) - ?~ll2,IIHl
C1r
k [U~0 ) - ?~[f~lI[Hl(f~l),
11'4~>- ~[~211n1(~2) _< c~(0)~11~5 ~
~ ~2llHltm),
(7.15)
(7.16)
where Cl and c2 are domain constants. Proof:
Inequality (7.15) is a consequence of Proposition 7.1 and (7.14). Let '/3 be the I
harmonic extension of fi = 'v~k)[ into f~l" 1
7
L 0 on (~1 \'~.
251
7.2. Dirichlet to Neumann Iteration In use of
we obtain
IlVv~k)~
vv~ ~)
Wll~
This implies
In a similar manner, we get
I Vv[ ~) IL~(~)<_ CliVv~)llg~(~). Inequality (7.16) follows from (7.15).
[]
We proceed to a natural modification. Given m > 0, let
We say that condition (I,~) is satisfied if 7~LTmf~2 C_ ~1 holds, where Tmf~2 denotes the image of f~2 by Tm. L e m m a 7.10. If condition (I,~) is satisfied, then we have 0 <_ H < m. Proof: The J-form pertaining to (fY, 30 is denoted by J ' , where fY = TmQ2. Similarly to the proof of Lemma 7.7, we have Jl(~,~) < J ' ( ~ , ~ ) for ~ E V. Taking the harmonic ~xt~n~ion h~ of ~ into a', w~ p~t ~ ( ~ , ~ ) = h ~ ( ~ l , ~ ) for ( ~ , ~ ) e a'. We h~v~ v E KI(Q ') and
_< m & ( ~ , ~ ) . Y'(~ , ~) _< IIVvll~(a,) ~ This implies
Oq"l(~, ~) ~ mJ'2(~, ~), and the proof is complete.
[]
Similarly to Theorems 7.8 and 7.9, we have the following. T h e o r e m 7.11. If condition (Ira) is satisfied, then we have (7.14) with ~(0) defined by 1-0
( 0 < 0 < 2) - m + 2
(7.~7) (m+1)0-1 and a constant co > 0 determined by Q2.
m+2
<0< 1 -
7. Domain Decomposition Method
252 Theorem
7.12. Under the same assumption, we have (7.15) and (7.16) with f(O) defined
in (Z1"7). This time, ~:(0) C (0, 1) follows from 0 < e _< 1 and 0 < e < 2/(m + 1). Exchanging the roles of ~ and ~2, we define the condition (Ie) in accordance with (Ira), where g > 0: 7ELTe~I C_ ~2. Then, conditions (Ira) and (I e) imply J l ( { , { ) < rnJ2({,{) and J2({,{) _< gJl({,{), respectively, for { C V. We get the following, where the case g = oo is permitted under the agreement of g-1 = 0. L e m m a 7.13. /f both conditions (fl) and (In) are satisfied, then we have g-1 ~ H < m. 7.14. Under the same assumptions, inequalities (7.14), (7.15), and (7.16) hold with f (0) defined by
Theorem
1_(1_+_e_1)0(0<0 (m+l)O-
Optimal
~-1
2
-- m + ~ -1 + 2
1
)
<0<1 77l--[-~
If 0 < 0 < 1 and 0 < 0 < 2 / ( m +
<
-1
--{-2
--
--
"
+ 1), this ~(0) is in (0, 1).
Rate of Convergence
Theorems 7.8, 7.11, and 7.14 imply tile following. Theorem
7.15. As an optimal value ropt of f(O), we have the following.
1~ /f condition (I) is satisfied, then ropt
--
1/3 is assumed by 0 = 2/3.
2 ~ If condition (Ira) is satisfied, then fopt = m / ( m + 2) is assumed by 0 = 2/(m + 2). 3 ~ If both conditions (fl) and (In) are satisfied, then ~opt = (m + g-~)(m + g-1 + 2) is assumed by 0 = 2 / ( r n + g-1 + 2). One can show that the choices of 0 in the above theorem are really optimal in the following sense. Namely, let ~ be a rectangle and 7 a line segment parallel to the lateral sides of the rectangle with 0 < a1 _< a2 and b > 0:
~1
=
{(x,,x2) [ - a l < :I:l < O, 0 < .T2 "~ b } ,
a~
=
{(*l,~)10
w h e r e T = { ( 0 , x2)[ 0 < : r 2 < b } . A~
\
b ]
< * ~ < a~, 0 < ~ < b},
We note that and
qZ,,(x2)=
sin x/~,,x2
253
7.2. Dirichlet to Neumann Iteration are the eigenvalues and the eigenfunctions, respectively, of
- d 2 ~ / d x ~ = A~
with
~ ( 0 ) = ~ ( b ) = 0,
where n = 1, 2 , - - . . Given { E V, we have O(3
-'- E Cn~n n=l
for cn = Cn(~) = (~, ~Pn)x. The harmonic extensions hi of { into f~l is written explicitly. It holds that oo
Ohi
= ~
~ X 1 Xl--0
x / ~ c , coth \
b /~,(x2)
n=l
where coth s = (e s + e-S)(e ~ - e-S) -1. Therefore,
( na l Tr"~
holds by cj = (qPn,~j)x = an,j, and A(1) = x/~coth(na17r/b) (n = 1 , 2 , - . - ) are the eigenvalues of SI. Similarly, A(2) = ~ c o t h ( n a 2 7 r / b ) (n = 1, 2 , . - . ) are the eigenvalues of $2, and 9n is an eigenfunction of $1 and $2 corresponding to A(1) and A(n2), respectively. It can be shown that H is a compact operator in X. The spectrum of H coincides with the set of its eigenvalues, {/~(nl)//~(n2)}n~176 1. B e c a u s e /~(ni)//~(n2) is non-increasing in n, we have ((H~, ~))
sup ~ II1~111 - -~ ~
)~I)
A~2)
tanh(Tra2/b) _ tanh(Trai/b) = r(al, a2, b)
and inf ((H~' ~)) _ inf
II1111
= 1
This implies 1 < H < 7-(al, a2, b). Both conditions (I e) and (Ira) are satisfied with f = 1 and m = a2/al. assures
1 < H < a__~2.
L e m m a 7.13
(7.18)
al
Because T(al, a2, b) ---, a 2 / a 1 as b ~ +oc, estimate (7.18) is really optimal. This (f~, 7) satisfies conditions (I z) and (Ira), and the really fastest speed of convergence is arbitrarily close to ro~t. Similar tendencies are observed for other (f~, 7)'s numerically; the universal optimality is realized by the choices of 0 in Theorems 7.15-7.15 if (f~, 7) satisfies (I ~) and (Ira) . We call them the general optimal choices of 0 in this meaning.
254
7. Domain Decomposition Method
Convergence
w h e n 7 is a C i r c u l a r A r c
We proceed to the case that 3' is a part of a circle with radius R and with its center at the origin of the coordinate plane. We take the circular reflection with respect to the circle, where (r, ~) denotes the polar coordinates"
We say that the condition (R) is satisfied if 77.cf~2 c f~l. Let f~ = 7~cf~2 and take Jforms ,.72 and J2* pertaining to (f~2, "7) and (f~, "7), respectively. Then f12(~, [) = J2*([, [) holds for [ E V from the reflection principle, and the following facts are proven similarly. L e m m a 7.16. If condition (R) is satisfied, then we have 0 < H < 1. Theorem
7.17. Theorems 7.8 and 7.9 remain valid with the condition (I) replaced by
(R). The condition comparable to (Ira) is given in terms of
T~,R" (r,
~)
~
/~ + ~ ,
~
m
We say that the conditions (Rm) and (R e) are satisfied if 7r f~2, respectively.
C_ f~l and TCcTe,Rf~I c
L e m m a 7.18. If condition (Rm) is satisfied with m > 1, then we have 0 < H < m. If
(R e) is satisfied with g >_ 1 furthermore, then f-1 < H < m follows. Proof: First, we take the case that f~2 is adjacent to 7 from outside, and hence r > R holds on f~2. Let J ' be the J-form pertaining to (~', 7), where s = T,,,,Rfl2. Then we have o"7"1(~,~) J ' ( { , { ) for ~ C V. We take the harmonic extension h.2 C I('(f~2) of ( into s and put v(r', qp) = h2(r, qp) with r' = r + (r - R ) / m . We have v E I ( ' ( ~ ' ) and ,
-
-
L'2(fl,) = .
~
,
k, Or'J
+ 777 ~
(Oh.2 2r' ) -- 9rdrdqp + L
1
(
r'dr'd~
c3h2
)
2
--r 9 l 9rdrdqp
_< mS~(<,,~), because 1/(m 2) _< r'/r _< 1 on ~22 from the assumption. This implies J~((, <) _< m,.7:,2((, () and the first assertion follows. The second assertion is a consequence of the reflection principle. Let, J~* and J2* be the J-form pertaining to (s V) and (f~,~,~/), respectively, where s = Tr163 and ~ = TCcFZ2.
7.3. Dirid]let 2 to Neumann 2 Iteration
255
We have J l ( ~ , ~ ) = J~*(~,() and J 2 ( ( , ~ ) : J2*((,() for ~ e V and J2*(() - eJl*({) similarly. This implies the conclusion in this case. The other case that r < R on 91 is treated similarly by the reflection transformation described above. [] We can argue similarly to the case that 7 is a line segment by L e m m a 7.18. Theorem
We have the following.
7.19.
1~ Theorems 7.11 and 7.12 remain valid with condition (Ira) replaced by (Rm). 2 ~ Theorem 7.14 remains valid with conditions (~) and (Ira) replaced by (R e) and (Rm),
respectively. 3 ~ Theorem 7.15 remains valid with conditions (I), (1t) and (Ira) replaced by (R), (Rt),
and (Rm), respectively.
7.3
Dirichlet 2 to Neumann 2 (DD-NN)
Iteration
For the same target problem and decomposition of the domain as in the previous section, we take a simultaneous algorithm, Dirichlet 2 to N e u m a n n 2 (DD-NN) iteration. Namely, letting #(0) E V be an initial guess, we solve succecssively the following, where p and q are the relaxation parameters subject to p >_ O, q > O, pq # O, and 0 < p + q <_ 1: -Au~k) = f
in ~"~1
with
u{k)= { 0
it(k)
k u~k) = 0
-Au~k) = f
in f~2
in ~-'~1
on F2
(Ou~k) On =
Ou~k) Ov
v?)={ ~
onr
with
with -Av{ k)- f
{
on FI on ~,
with
k #(k) on 7, v{k) = 0 on F1 OV~ k) OV~k)
{
.(k+l) __ (1 - p - q)p(k) + pu~k)l~ +
on 7,
Ov =
qv{k)l~"
On
on 7,
Introducing the error function ((k) = #(k)_ ~[~ defined on O/, we can derive formally that ~(k+l) __ (1 -- p -- q -- F S 2 1 S l
-- q S 1 1 ~ 2 ) ~(k).
Rigorously we have ~(o) E IS,
~(k+l) = Bp,q~(k) (k = 0, 1 , 2 , - . . ),
256
7. Domain Decomposition M e t h o d
where Bp,q = (1 - p - q)I - p H - qH -1. The operator Bp,q may be called the amplification operator for the error of the DD-NN iteration. Expressed as a rational function of the operator H, it is bounded and self-adjoint in V with respect to the inner product (7.12). We have the follwing. P r o p o s i t i o n 7.20. If both conditions (I e) and (Ira) are satisfied, then we have sup
ra(Bp,q) <
11--p--q--ps--qs-l[,
~-l<s<m
where ra(]Bp,q) denotes the spectral radius of t~p,q. Proof:
guarantees
We have ~-1 _< H _< m by Lemma 7.13. The spectral mapping theorem now
ra(Bp,q)-
sup
sup I i - p - q - p s - q s - l l <
]l - p -
q - ps - q s - l [ .
e- 1<s<m
sea(H)
The proof is complete. For the moment, we pick up the case that p = q and 0 = 2p = 2q. The following theorems are proven similarly to the DN iteration. T h e o r e m 7.21. Suppose both conditions (I e) and (In), and let g <_ m without loss of generality. Suppose, furthermore, that there is a 0 satisfying 0 = 2p = 2q and 0 < 0 < 4 m / ( m 2 + 2m + 1). Then, we have
IIv - coe(~
v
for k = 1, 2, 3 , . . . , where Co > 0 is a domain cortstant and 1
~(0)=
-
[
O
1 ( 1 )
1 is in
(
20
+~ ~ +
0-I
(
4m
1
)
<0<1)
mP + 6 m + 1 -
(o, I).
T h e o r e m 7.22. Under the assumption of Theorem 7.21, the general optimal choice of O is attained by 0 = 4 m / ( m 2 + 6m + 1) with the optimal value m 2 - 2m + 1 7"opt ~- m2 --I-6m + 2"
T h e o r e m 7.23. Theorems 7.21 and 7.22 remain valid with conditions (I e) and (Ira) replaced by (R e) and (Rm), respectively, for g, m >_ 1. We only state the optimal choice of p and q for tile general case, because the convergence result for the DD-NN iteration is complicated then.
257
7.4. Robin to Robin Iteration
T h e o r e m 7.24. Suppose that both conditions (I e) and (Ira) are satisfied with g < m.
Then the general optimal choice of p and q for the DD-NN iteration are attained by p = 2g/am,l and q = 2m/c~m,l with the optimal value
me- 2x/-~+ 1 ropt =O~md
where a,,~,e = me + 2(m + g + ~ )
+ 1.
T h e o r e m 7.25. Theorem 7.24 remains valid with conditions (Ira) and ([) replaced by
(Rm) and (Rl), respectively, for g, m >_ 1.
7.4
R o b i n to R o b i n ( R R ) Iteration
Again we are concerned with problem (7.9), and keep the assumption and the notation of w on the domain f~ and its decomposition. The iterative scheme in this section, which is called the Robin to Robin (RR) iteration, uses the Robin boundary conditions. Given u ~ C H2(f~), we take u[ ~ = u~ and u~~ = u~ as initial guesses. We successively solve {u~ k) = 0
--Au~ k) - f
cgu~k) +
in f~l,
o n 1-' 1
ou~k )
---gV-.
0,tt~k-x) =
t- OU~k - l )
on 7
On
and
{
u~k) = 0
--Au~k) = f
Ou~k)
in f~2,
on ['2
Ou~k) _
(:gu~k-l) ~- OU~k - l )
+
-
on 7
O~
with the relaxation parameter 0 > 0. Convergence of scheme is reduced to the case of f = 0. The following argument is due to Q. Deng. First observation is the identities
vui
I dx--
~r]i -tyui
.(q
i
for i = 1, 2, where
~?~k)_ cgu~k) -
0-7-
Ou~k) +
and
~ k ) _ cgu~k) -
Ou~k)
o---7- +
_(k+l) = 20u~k) _ @k) for i -r j. This implies bk+ 1 = b k - 4Oak for Here, we have qi
1
2
7. Domain Decomposition Method
258 and hence
1 0 _< ak = ~-~(bk - bk+l) follows. The non-negative sequence {bk}k is non-increasing sequence in k and the limit lim bk exists. Consequently, ak ---, 0 as k --, oo. We obtain the following. T h e o r e m 7.26. Sequences {u~ k)} and {'u~k)} generated by the R R iteration converge to the exact solution ~t of (7. 9) in the sense that
I1~~)- ~la, II~l(a,)
+
Ilu~~)- ~ln~ll~l(a~) ~ 0
as k---, c<).
7.5
Exterior
Problem
DDM is effective to the equation on u n b o u n d e d domains. We take the exterior domain, so t h a t f~ = R 2 \ O with O C R 2 being bounded. Consider -Au=f
inf~
with
u{=O -~
on O 0
o
as Ixl-~ +oo,
where f e L2(f]). We employ the polar coordinates (r, ~), and assume that O contains the origin. Taking R > 0 sufficiently large, we put ER = { (r, g)) I r > R} and divide f~ into f~l = En and f~2 = f~\En. Thus, 7 = OER is regarded as an artificial boundary. Letting p(0) be an initial guess, we take --Au]k)= f
in f~l
with
t" UI /
(k) ]--~ 0
{
= p
~k)
~L~~) = 0
in f~2
- A u ~ k) -- f
with
#(~+1) = (1 - o)# (~) + o~)1~,
10'u~k)
R Or
as r --~ + o c on 7,
10u~k) =R
Or
o n OO
ouT,
where 0 E (0, 1] is the relaxation parameter. T h e Green's function for the first problem is explicitly given by 1 R 4 + r2r ' 2 - 2rr'R 2 c o s ( g ) - ~') G(x, .T.') = ~ log R 2 (r 2 + r' 2 _ 2rr' cos(~ - g)')) ' where x = (r, ~) and x ' =
~ / ( ~ , ~) =
(r', g/). We have
r 2 -- /~2 f 2 ~ 27r
Jo
#(k)(~9, ) R 2 + r 2 - 2Rr c o s ( ~ - ~')
dqp' + 9f~ G ( x , x ' ) f ( x ' ) dx' 1
7.5. Exterior Problem
259
and hence
RlOu{k)Or(qo) = ~ 1 ~2~ It(k)(qJ)sin 2 ~-~' &;' +-~ ,~lj~ [~r G(z, z, )] f(z') doe' 2
(7.19)
1
follows in the second problem. We can avoid the unboundedness of the domain, under the cost of the singularity of the integral in this way. Convergence analysis is made similarly. Let V = H1/2(7 ) and ts
= {v E H1(~2) ] rio 0 = 0}.
The DN maps S1 = S(Ftl, 7) and $2 = S(~9, 7) are defined as before. We have J 1 (~, ~)
IlVhl 2
=
2
and =
S1/2~
2
with the harmonic extensions hi c HI(~I) and h2 E _~1(~2) of ~ E g into ~1 and ~2, respectively. The error function ((k) = it(k) _ ul~ satisfies ~(k+l) = (1 - 0)~ (k) - 0 S 2 1 S 1 ~ (k),
and hence the amplification operator is given as
Ao = ( 1 - 0 ) I - 0 H with the bounded extension H of S~1S1 into V. Provided with the inner product (7.12), we have
r,(Ao)=
sup [1 - o - o s [ ,
se~(H)
where ro(Ao) and a ( H ) denote the spectral radius of Ao and the spectrum of H, respectively. For the moment we study the case (.9 is a disc and hence ~2 is a circular ring. L e m m a 7.27. We have
R ~ + R~ 1 <_ H _< R2 - R~)
(7.20)
i f O = {(r, qo) I r < R0} withO< Ro < R. Proof: We have FZ2 = {(r,~)[ R0 < r < R}. Given { c V, we take the harmonic extensions hi and h2 of { into ~'-~1 and t~2, respectively. In terms of the Fourier series, = {(q;) is written as oo
~(~) =
~g~(~) n---1
Z Domain Decomposition Method
260 for gn(~) = an cos n7) + bn sin n~. This implies oo
(r)n
n=l
and
h~(~, ~) = ~ n ~ n - ~ - no~R ~g~(~) n=l Because of
7r
n=l
n=l
we have oo
,.71(~, ~) = E
oo
n(a~ + b~)
and
n--1
n=l
where R~ R -n
-~- R O n R n
<
R 2 -[- R 02
O~n --- D
by R o / R < 1. Hence
R 2 + R~ &(~'~) < n ~ - ng J`(~'~) follows. On the other hand, h~(r, ~) = hl(R2/r, ~) is nothing but the harmonic extension of into (9 from the reflection principle. The variational principle now guarantees al(~,~)
= IlVhlll ~L2(O) -< I I v ~ l l ~ , (M) =
&(~,~),
where It2 denotes the zero extension of h2 into (9. Those inequalities imply (7.20) similarly to the proof of Lemma 7.7. Now we proceed to the general case. L e m m a 7.28. Suppose that (9 C R 2 is a bounded domain containing the origin, and let
R o - i n f { g > 0 I (_gCBe} with Be = {(r, ~ ) [ r < g}. Then, inequality (7.20) follows.
261
7.6. The Stokes System
Proof." Let hi and h2 be the harmonic extensions of ~ c V into ~'-~1and ~2, respectively. The inequality Jx(~, ~) _< &(~, ~) follows similarly to the proof of Lemma 7.27. On the other hand, Lemma 7.27 implies R2
+/702
IlVk~ll ~L2(f~2) -< R2__ R02 Jx({, {) where h2 denotes the harmonic extension of ~ into
(~ = {(<, ~ ) l R o
< < <
~}.
Combining this with the variational principle
we have R 2 + R~ S2(~, ~) ~ /~2 _ ~_~----'Sl ~ (~, ~)-
The proof is complete. Lemma 7.28 implies the following. T h e o r e m 7.29. The error ~(k) decays exponentially if 0 < 0 < 1. More, precisely, we have
I1(~ IIv
(k) IIv for k = 1, 2, 3,-.-, where
,~(0) =
1-R2
+ R~
0<0<
-
2R2 + R~
( R~ + R~ ) 2R 2 + R 2~ <_0< 1 .
20-1
T h e o r e m 7.30. The general optimal choice of 0 for the exterior DN iteration is attained by 0 = (R ~ + R~) / (2R ~ + R~) with the optimal value ro~ = R ~ / ( 2 R ~ + R~).
7.6
The Stokes System
In this section we study DDM applied to the Stokes system -Au+Vp=f
and
V.u=0
in~.
{7.21)
7. Domain Decomposition Method
262
It describes the motion of a viscous incompressible fluid with slow velocity. Here, t2 C R 2 denotes a bounded domain, and u = (u 1,u2), p, and f = (fl, f2) stand for the flow velocity, the pressure, and the external force, respectively. Density and viscosity of the fluid are assumed to be the unity. Homogeneous Dirichlet boundary condition is treated in w u = 0
on 0t2
(7.22)
This assumption indicates that the fluid is non-slipping on the boundary. There exists a unique ( u , p ) E Hi(t2) 2 x {L2(i2)/IR} satisfying
A(u, v) - (p, V . V)L2(~) = ( f , V)L2.n)2 and -(V.
u, q ) L ~ ( ~ ) = 0
for any ( v , q ) E H I ( ~ ) 2 x L2(~), where A ( u , v) =
Vu | Vv dx =
dx.
i,j=l Oxi ~ The second requirement is equivalent to V . u = 0 a.e. In fact, we get IIv 9UllL~(a) = 0 by taking q = V . u . The problem is re-formulated as to find (u, p) C H(~,o(a) 2 x {L2(a)/R} satisfying .A(u, v) - (p, V . V)L2(~) = (f, V)g2(n)2 for v E H(~(f~)2, where H~,o(f~)2 = H~(f~)2 N HJ(f~) ~ with H~(f~) 2 = { v E H l ( f ~ ) 2 I V . v = O } . Its unique solvability is a consequence of the Poincar(~ inequality on H~(Q) and the Helmholtz decomposition Hg(f~) 2 ~ H~,o(l)) 2 x {L2(I))/R}.
Inhomogeneous Boundary Value Problems As in w we take tile case that cOt2 is composed of two smooth curves ~/and F, with and P intersecting transversally at two points. We study the inhomegeneous boundary condition = ~
t
0
As before, n stands for the unit outer normal
on ~/ on F. to F. Let
I(l(t2)= {vEHl(t2)2[ K~(f~)={vcKl(f~)l
for V
=
(7.23)
Vir=0}, V-v=O},
/41/2 "~00 (7) 2. The solenoidal version of Proposition 7.1 is described as follows.
263
7.6. The Stokes S y s t e m
P r o p o s i t i o n 7.31. We have the following. (i) Every v e K I ( ~ ) admits ~ = vl. Y e Vo satisfying II~llv <- C (ii) Every ~ C Vo admits v E KJ(•)
IIvlIH~(~)~.
satisfyin v[~ = ~ and IlvllH,(~)= < C' IIr
Similarly to the Poisson equation, lifting argument now applies to inhomogeneous (7.23) for (7.21). Given (~,f) E Vo x n2(~) 2, we have a unique (u,p) E K J ( ~ ) x L~(~)satisfying ul~ = ~ and (7.24)
A ( u , v) - (p, V . V)L2(a) = ( f , V)L~(~)2
for v C H~ (~t) 2. Here and henceforth, p C L02(~) indicates that p C L2(~) and f ~ p dx = O. We make use of the following lemma due to I. Babugka and A.K. Aziz to prove Proposition 7.31. L e m m a 7.32. Given F C L~o(~t), we have ~ C H~(~) 2 satisfying V. 9 = F
in ~
A(~, ~) <- - ~ IIFll ~L 2 ( g t )
with
,
(7.25)
where 7- is a domain constant. Proof of Proposition 7.31: The first case is an immediate consequence of Proposition 7.1 and the divergence formula of Gauss: O = J~ V . v dx = L ~ v . n ds
The second case is shown as follow. Given ~ C Vo, we take its extension ~p C Kl(~t) according to Proposition 7.1. Here we have F -- ~ . ~ E L~(~) by LF
dx= LV'r
dx= L~'nds=O.
Lemma 7.32 gives ~ e H~(~) 2, provided with (7.25). Then v = ~ - r satisfies the desired properties and the proof is complete. [] We proceed to the mixed boundary condition -~n - pn = g u=0
on 7 on F,
(7.26)
where ~--~ - pn = ( V u ) n - pn =
--nj Ozj
- phi
. i
264
7. D o m a i n D e c o m p o s i t i o n M e t h o d
Smooth u and p solving (7.21) satisfy, by Green's formula, that Vu|
p ( V . v) dx -
~
- pn
. v ds =
f . v dx
for v 9 K~(fl). Problem (7.21) with (7.26) is formulated as to find ( u , p ) 9 K J ( f l ) x L2(fl) satisfying A ( u , v) - (p, V . v) = L g . v ds + ( f , v)
(7.27)
for v C K l ( f l ) . To study the unique solvability of this problem, we introduce the operator G : L2(fl)---+ KI(Q) by (7.28)
(Gq, V)H,(a)2 = (q, V . v)
for v 9 Kl(ft). This is assured by Riesz' representation theorem. L e m m a 7.33. The range of G denoted by n a n (G) f o r m s a closed subspace of K l ( a ) and the orthogonal decomposition K l ( f l ) = R a n (G) @ Kl~(fl) holds. Furthermore, the operator G is an i s o m o r p h i s m between n2(fl) and R a n (G) = Kla(fl) -L. Proof: Taking v --- Gq in (7.28) implies liGqllg,(~)2 <_ C liqilL~(n) and hence the boundedness of G : L2(a) ---, H ' ( a ) 2 follows. Given q 9 L2(f~), we have ~? 9 K l ( f i ) satisfying
V. ~ = q
in f~
II~llm(fl)= ___ c IlqllL=(fl) 9
and
(7.29)
In fact, we have w 9 H2(ft) and r / 9 H1/2(Of~) 2 satisfying Aw = q
in
f~,
Ilwllg~(n) < c Ilqllc=(n),
and
~1~ :
f
0,
/
Jo f~
~ . ~ d~ : ~o # 0,
I1~11,,,~(o~)~ ~ C Ilqll L2(f~),
respectively. The value of r / o n 7 is not specified, and such a choice is possible. Then, rl,r = V w i o a + ecrl E H1/2(Oft) 2 satisfies
/o
7]~ . n d s = 0
for ~ =
n
1L
77o
V w. nds.
n
Therefore, we have v C HI(f~) 2 satisfying V-v - 0
in ~,
vlo~ -
w~,
IlvllHl(~)~_< c IIw~ll.,~=(o~)=
by Proposition 7.31. Relation (7.29) now follows for ~ = Vw - v. Taking v = ~ in (7.28), we have Ilqllt=(~) --< C IIGqlIH,(~)=. This imI)lies the closedness of R ( G ) .
7".6. The Stokes System
265
The relation Ran (G) • = Kl(ft) is easy to prove. It is left to the reader and the proof is complete. [] Problem (7.27) admits a unique solution if (g, f) C L2(F) 2 x L2(f~) 2. In fact, then the linear functional
E KI(~)
~-~
fr g" Fds + (f, ~)
is bounded, and we get a unique u E K I ( ~ ) satisfying
.A(u, ~p) = fr g " ~ ds + (f , ~) for ~ C KJ(f~). For this u, the linear functional V
E
is also bounded, and we have ~ e Ran (G) satisfying
r(r
= (~, r
for r C Ran (G). Theorefore, Lemma 7.33 assures the existence of p E L2(f~) satisfying F ( r = (p, V-r However, F vanishes on K~(f~), and we have F(v) = (p, V - v ) for v 9 K I ( ~ ) , or equivalently (7.27). So far, we have observed that the Dirichlet form A(u, v) on Kl(ft) x Kl(f~) induces the first equation of (7.26) as a natural boundary condition. If we adopt g(u, v) = 2 E
[
1_
ei,y(u)ei,j(v) dx,
(7.30)
d~t.
for A(u, v), then the corresponding boundary condition becomes a(u, p)n = g on 7, where
1 (Ou ~ cOu3~ eid(u) = -2 \Oxj + cOxi,]
and
a(u,p) = (-PSid + 2eid(u)) 9
We note that e(u) = (eid(u)) and a(u,p) indicate the deformation rate and the stress tensors, respectively. In particular, a(u, p)n forms the stress vector.
SDN
map
If (w,p) e K~(fi) • L~(fi) with wl.~ = ~ solves (7.24) for v E H~(f~) 2 with f = 0, then w and p are called the Stokes extension of ~ c Vo into ~ and the accompanying pressure of w, respectively. It is easy to verify that
c' I1~1 ~ _< .A(w, ~) < C I1~11~ 9
(7.31)
7. Domain Decomposition Method
266 A solution h E Kl(ft) of
(
Ah=0
inl2
with
h= ~
[0
on'7 on F
is called the (vector-valued) harmonic extension of ~ C V into f~. Babu~ka-Brezzi-Kikuchi's inf-sup condition is expressed as
(q,V'v)i2(a) fl < inf sup gg(a)vcH~(a)2 IqjJg2(a)IW, II
(7.32)
where fl > 0 is a constant and IlVvll = v/A(v, v). Here, the optimal value of fl is assumed, called the inf-sup constant, and thus it is a constant determined by f~. L e m m a 7.34. Given ~ E Vo, we have
.A(h,h) <_ A ( w , w ) <_ (1 +fl-~)2.A(h,h),
(7.33)
where w and h are the Stokes and the harmonic extensions of ~ into f~, respectively. Proof." We shall write II" II for II 9Iig2(n) or II" Iig2(a)2, in short. The first inequality is a consequence of the variational principle for harmonic functions. If p C L02(f~) denotes the accompanying pressure of w, then we have
Ilpll ~
sup
(p, v . v) =
sup
(Vw, v~) ~ IVwll.
In use of A(w, w) = A(w, ~ - h) + A(w, h) = (p, v
( w - h)) + A(w, h)
-< ]IPII" V h l l + l l V w l l 9 IVhll-< ( 1 + 1 ' ] IIVwll 9IIVhll, /9/ k the inequality I Vwll_<
(1) 1+~
]lVhil
follows and the proof is complete. We put Xr = {~ C X I (~ , n ) x = 0}, where X = L2(7) 2 and (.,.)x denote the usual L2(7) 2 inner product. The bilinear form J : V~ x V~ --, R is defined by J ( ~ , 7]) = A(w, v), where w and v denote the Stokes extensions of ~ and r/, respectively. We call it the (Stokes ) J-form pertaining to (f~, 7). Similarly to the case of the Poisson equation, we get a self-adjoint operator S : D(S) c Xo ~ Xo, satisfying J ( ~ , r/) = (S~, r/) x for (~, 77) r D(S) x Vo. It holds that J ( ~ , r/) =
267
7.6. The Stokes System
(S1/2{,Sll2rl)x for {, 77 E D ( S 112) = Vo and S 1/2 : Va + Xa is an isomorphism. Finally, variational principle holds and the operator S 1/2 is characterized by
for v C K~(f~) satisfying vl~ = ~. The operator S is called the Dirichlet to Neumanu map for the Stokes system, or simply the SDN map pertaining to (f~, 7). We shall often write S = S(f~, ~/) to express this correspondence. Given ~ E Vo, let w be the Stokes extension of ~ into f~. We take P0 C L~(gt) as the accompanying pressure of w. Assume their suitable regularities, say, (w, po) C H2(f~) x H I ( ~ ) . Even in this case, an additive constant c of p = P0 + c is specified for the function Ow/On - pn defined modulo R to belong to Xo. In fact, we have
where I~[ denotes the measure of O'. Therefore,taking 1 L
Ow
.
we ensure that O w / O n - pn E X~. This procedure is automatically done in the abstract setting mentioned above.
Dirichlet to N e u m a n n ( D N ) Iteration The target problem in consideration is (7.21) with (7.22). Keeping the notation introduced in w we study the following scheme, which may be called the DN iteration. We take a vector-valued function #(0) on ~/first, satisfying
L
(v.34)
/~(o). u ds = 0
and then generate successively {u[k), p~k)},
{u~k), p~k)},
{v(k+')},
and
through V - u ~ k) = 0 in ~1, { - A u ~ k ) + V p ~ k ) : f, u~k) = 0 on F1, u~k) = p(k) on 7,
--Aura)+ Vp~k) : f, u~k ) = O
onF2,
V-u~ k)=O
On
inf'2,
Ou
+p~k)u
ony,
7. Domain Decomposition Method
268 and
#(k+l) = (1 - 0)# (k) + Ou2 ~, where 0 E (0, 1] denotes the relaxation parameter. Relation (7.34) and the equation of continuity in t22 imply
f p(k). u ds = 0
(k = 1 , 2 , . . - ) .
Therefore, compatibility condition for the first problem holds at each step of the iteration. The accompanying pressure p~k) of the velocity u~k) is taken uniquely by fal P~k)dx = O. One may take p~k) = 0 at some point in a l . Then, the accompanying pressure p~k) of u~k) is uniquely determined subject to the choice of p]k). Let $1 = S(t2~,3') and $2 = S(t22,3') be the SDN maps pertaining to (t2~,3') and (t22, 3'), respectively. We get the following facts similarly to the Poisson equation. 1~ The operator Ho = S~1S1 admits a bounded extension in Vo, provided with the expression H = S~/2(S~/2S;1/2)*S~/2, where* denotes the adjoint in Xo. 2 ~ The space Vo forms a Hilbert space under the inner product
((%~,'1']))= (S1/2~ ~2 c,1/2 r/jx x
(~, ~ ~ vo)
(7.35)
with the corresponding norm II1" III = ((, .))~/2 equivalent to I1 liE. 3 ~ Under (7.35), the operator H is self-adjoint in Vo. 4 ~ The error on 3' is successively generated in accordance with ~(o) C Vo
and
~(k+l) = Ao~(k)
for k = 0, 1, 2 , . . . , where Ao = (1 - O ) I - O H denotes the amplification operator for the error, self-adjoint and bounded in Vo under (7.35). 5 ~ We have
ro(Ao) =
sup
Aea(H)
11 - 0 - 0~1,
where a(H) and ro(Ao) denote the spectrum of H and the spectral radius of Ao, respectively. For the Stokes system, the spectral radius of the amplification operator is estimated in terms of tile inf-sup constants /-~1 and ~2 corresponding to t21 and t22, respectively. L e m m a 7.35. Under the condition (I), we have 0 < H < (1 +/3-1) 2 for/3 = max(/31,/32).
269
7.6. The Stokes System
Proof: Let f~; = ~Lf~2- The J-forms pertaining to (f~i, "/), (f~2, ~'), and (f~, ")'), are denoted by if1, if2, and if2*, respectively. Given ~ E Vo, we take its harmonic extension hi into f~l and put Ql(~,~) = A n l ( h l , hl),
where Aal ( , ) denotes the Dirichlet form defined on f~l. Lemma 7.34 implies
Qi([,~) ~ ~]'1(~,~) ~ (1 + z~-l) 2 Q l ( [ , [ ) . The bilinear forms Q2 and Q~ are defined similarly, and admit the corresponding inequalities. In particular, we have
j~(~, ~)< (1 + 9;~) ~ QI(~, ~) < (1 + 9 ~ ) ~ &(~, ~)
(7.36)
by Q,(~, ~) < Q;(~, ~) _< &(~, ~). Let w~ be the Stokes extension of [ into f~ and take its zero extension v to f~l \ g/~. Then, the variational principle implies
y~(~, ~) _< A~(~, ~)= Y;(~, ~). Similarly, we have Q1 (~, ~) _< Q~([, ~) by Proposition 7.3. In use of
Q;(~, ~) = Q~(~, ~) < &(~, ~), we have
~]'1(~, ~) ~ J2*(~, ~) ~ (1 + 321) 2 Q;(~,~) ~ (1 Jr-~2i)2~2(~, ~). Combining this with (7.36), we thus obtain
Jl(~, ~) _< (1 + 9-~) ~ &(~, ~). This inequality implies ((H~,~)) _< (1 + ~-1)211~111~ because of ((HI, I)) = IIs~/~11~ = ffl([,[) and Ill,Ill= = IIS~/~ll} = &(~,~). The proof is complete. [] Letting c~ = 1 + (1 + fl-1)~, we get r~(Ao) <_
sup
O
[1-O-AO[
=max{[1-O[,[1-o~O[}=?(O).
This implies the following. Theorem
7.36. Assume condition (I) and take 0 E (0, 1). Then, we have
for k = 1, 2, 3 , - - . , where
~(0)=
I1~
(~)
~1-0
(7.38)
[ c~0-1
(0<0_<2/(oz+1)) (2/(c~+1)_<0<2/c~)
and co > 0 is a constant determined by f~2.
270
7. Domain Decomposition Method
We see that 0 < ?(0) < 1 follows from 0 E (0, 2 / a ) . The following facts are obtained simlarly to the Poisson equation. Theorem
7.37.
Under the assumption of the previous theorem, we have
(k) u~ - ~ ~ l l H ' ( ~ . )
--< c2~(O)kll~ ~
UI~IIH~(~)
for k = 1 , 2 , 3 , . - - . T h e o r e m 7.38. Under the saute assumption, the general optimal choice of the relaxation parameters is attained by 0 = 2 / ( a + 1) with the optimal value ropt = (a - 1 ) / ( a + 1).
Commentary
to Chapter
7
7.1. The space H~/2(3') is familiar among domain decomposition researchers. In other contexts, it appears as a concrete characterization of the real interpolation space [H~(3'), L2(-y)]l/2 (Lions and Magenes [247]), and the domain of L 1/4, where L denotes a minus Laplacian on 3' with the zero Dirichlet boundary condition (Fujiwara [150]). If 3' is a line, rrl/2 --00 (3') has another expression. Let 3' be a line segment on the x-axis. Then --oo rrl/2 (3') is coincident with the set of ~ = }--~-n=lan7),, C L2(3') satisfying E
a n2 /~ n1/2 <
nt-O0 ,
n=l
where )~n and ~an denote the eigenvalue and eigenfunction of de~ dx 2 = ) ~
in 3'
with
~=0
on 03'.
We also have the equivalence
I1,~11ttoo,/~ ('~)
~
an~ ntn= 1
a 2n)~n~/2
,
(7.39)
n-- 1
See Saito and Fujita [332] for the proof. By this form one can prove t h a t H is a compact operator on X. Proposition 7.1 is a particular case of Theorem 1.5.2.3 of Grisvard [164]. If f~ is a polygon, an L p version of Grisvard's result was presented in Arnold, Scott and Vogelius [12]. An elementary proof of Proposition 7.1 is also given by [332] in use of (7.39). The Dirichlet to N e u m a n n map S is sometimes called the Steklov-Poinca% operator. It is regarded as S : H1/2(Of~) --~ H-1/2(Of~) if fi is a Lipschitz domain. An extensive analysis was made by Quarteroni and Valli [319] based on the study of harmonic extensions. There, S is regard as an operator from V into V'. Characterization of D ( S 1/2) by the quadratic
Commentary to Chapter 7
271
form was done by Fujita [141]. The key fact for Proposition 7.2 to hold is the closedness of ,.7({) = J ( ~ , ~ ) ; the conditions ~,~ E V, {,~ --+ {0 in X, and ,.7({n- ~m) --~ 0 imply {0 E V and J ( { n - {0) --+ 0. We refer to Kato [205] for the abstract theory, particularly Chapter VI-w If f~ is a rectangle and "y is one of sides of the rectangle, formula (7.39) implies D(S) = H~(7). Concrete characterization of D(S) for the other case is not known. Algebraic properties of the discretized Dirichlet to Neumann map are discussed in Agoshkov [4]. 7.2. The DN iteration was proposed by Furano, Quarteroni, and Zanolli [152] and Marini-Quateroni [258], [259]. The latter papers adopt FEM, and the DN iteration is described as a Richardson iteration relative to the Schur complement matrix. Actually, the Schur complement matrix is a sort of finite dimensional analogue of the DN map. See [319] for further discussions. We just note that Bj0rstadt and Widlund [37] proposed a conjugate gradient method to solve a system on the Schur complement matrix. A similar idea was given by Bramble, Pasciak and Schatz [50]. Descriptions of this section are based on Fujita [141], Fujita, Fukuhara nd Saito [142], and Fujita and Saito [146]. See Yosida [410] for the spectral mapping theorem. 7.3. The idea of DD-NN iteration was given by Bourgat, Glowinski, LeTallec, and Vidrascu [43]. Mathematical analysis on both discrete and continuous problems was done by Chu and Hu [80]. We followed the arguments of Fujita, Fukuhara, and Saito [142] and Fujita and Saito [146] based on the operator theoretical analysis. 7.4. The RR iteration was proposed by P.L. Lions [251]. The convergence is also discussed there. Theorem 7.26 was due to Deng [110], but the convergence rate is not known. Douglas Jr. and Huang [115] illustrated a simple example illustrating this issue. Also, Douglas, Jr., Paes Leme, Roberts, and Wang [116] derived a rate for a modification of the RR iteration combined with the mixed finite element method. The RR iteration is successfully applied to the Helmholtz equation defined in an exterior domain. See Desp%s [112] and Liu and Kako [253]. Some remarkable works on the Schwarz alternating method were done by Lions [249], [250] and Chan, Hou and Lions [72]. 7.5. The scheme describe in this section was proposed by D.H. Yu [413], [414 I. Theorems 7.29 and 7.30 were also obtained there. Related works were done by Yu [412], [415]. Numerical computation of the singular integral (7.19) was discussed by Yu [411]. 7.6. Standard monographs for mathematical theory of the Stokes and Navier-Stokes equations are Constantin and Foias [93], Galdi [153], [154], Ladyzhenskaya [229], Lions [252], and Temam [380]. Numerical approximations are discussed in Girault and Raviart [159], [160], Glowinski [161], and Quarteroni and Valli [318]. The Helmholtz decomposition is described in those literatures. Lemma 7.32 is shown in Babugka-aziz [19] for the smooth domain and in [159], [160] for the Lipschitz domain.
272
7. Domain Decomposition Method
Our proof of the unique solvability of the mixed boundary value problem (7.21) with (7.26) follows the argument of Solonnikov-Seadilov [355]. Given 9 C H-~/~(~) ~ -
(-0
1/2 2 o (~)
)'
,
we have a similar result; the unique solvability of the problem to find (u, p) E I(~(f~) x L2(f~) satisfying .A(u, v) - (p, 27. v) = (9, v) + (f, v) for any v E KX(Ft) holds, where (., .) denotes the duality between r41/2 -~00 (.y)2 and H-1/2(7)2. Another proof via the mixed weak formulation is described in [319]. For weak formulations using (7.30), see [229], [319]. Marini and Quarteroni [259] studied the finite dimensional version of the DN iteration for the Stokes system. It proved an anologous result of Lemma 7.34. An operator theoretical analysis in the Hilbert space Xo was done by Saito [331]. Lemma 7.35, Theorem 7.36 and Corollary 7.40 were presented there. Condition (7.32) was introduced independently by Babu~ka [18], Brezzi [66], and Kikuchi [218]. The finite dimensional counterpart of this condition guarantees the well-posedness of its mixed finite element approximation as is described in w We have seen that the convergence factor of the error depends explicitly on the infsup constant under condition (I). To evaluate the value of the inf-sup constant is of importance. In this connection, the following well-known lemma is useful, where D C R 2 denotes a simply-connected bounded domain with Lipschitz boundary. See Girault and Raviart [159] or Velte [400] for the proof. L e m m a 7.39. The relation 1//3 = v/~/2 holds for the inf-sup and Korn's constants/3 and ~, respectively. Here, Korn's constant arises in Korn's (second) inequality
A(~, ~) _< ~E(~, ~) for v E H i ( D ) 2 satisfying
/D( OX 0~22 0721) dx = 1 -~X;
0.
(7.40)
Here, g denotes the bilinear form defined by (7.30). The optimal value of ~, which is again denoted by the same symbol, is called Korn's constant. The following facts are known concerning Korn's constant (Horgan [180], Horgan and Payne [182])" 1~ For any bounded domain, ~c > 4 and the equality is attained if D is a disk. 2~ Let D be star-shaped with the boundary represented in plane polar coordinates by r = 9(0). Then we have h: _< 2(1 + maxoC(O)), where
I~'(0)1 G(O) =
1+ \ 9 ( 0 ) j
+ 9(0)
In particular, when D is a regular n-side polygon, we have ~ _< 2/(1 -sin(Tr/n)).
Commentary to Claapter 7
273
3 ~ If fit is a square, we have ec _< 8 + 4x/~. As a corollary of Theorem 7.36, we have C o r o l l a r y 7.40. If condition (I) is satisfied and moreover that both f21 and f22 are squares, then, as long as 0 < 0 < 0.1423, the exponential decay of the error as stated in (7.37) is guaranteed. Furthermore, in this case, we get ropt < 0.8671. The numerical value of the inf-sup constant plays important role also in other areas such as the theoretical analysis for the Navier-Stokes equations (Horgan and Wheeler [183], Ames and Payne [10]) and numerical verifications of finite element solutions for the Stokes equations (Nakao, Yamamoto and W a t a n a b e [282], [283]). Horgan and Payne [182] conjectured that ~ = 7 if f~ is a square. Concerning further discussions about Korn's constant, the survey paper Horgan [181] is useful.
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Index - - regularity, 21, 45 stability, 21, 45
(I)o, 155, 163, 164, 170 (i)o, 146, 147, 148, 151, 153, 170 (II)o, 147, 155, 156, 159, 170 (ii)o, 146, 148, 150, 153, 155, 170 (III)o, 154, 155, 159, 162, 170 (ill)o, 146, 147, 148, 150, 153, 155, 170
adjoint, 248, 268 form, 6, 20, 61, 120, 124, 127, 151 operator(s), 117, 219, 248 atiine space, 183 amplification operator, 247, 259, 268 for the error of the DD-NN iteration, 256 for the error of tile DN iteration, 248 artificial boundary, 246, 258 Ascoli-Arz~la's theorem, 191 asymptotic - - behavior, 35 expansion, 46
(P, S), 154, 155, 156, 159, 160, 162, 163, 164 w-limit set, 209 A acceptability, 146, 169, 170 acceptable, 145, 146, 170 Ao-acceptable, 146, 148 Hol/2 o (7), 244, 249, 262, 270 J-form(s), 245, 249, 251,254, 269 Stokes--, 266 L~ stability, 44, 77, 204 structures, 172
-
Babu~ka and Aziz lemma due t o - - , 263 backward difference, 62, 126, 140, 145, 221,224 barycenters, 172 barycentric - - coordinate, 203 - - d o m a i n , 172, 175, 203, 219 bilinear form(s), 4, 5, 7, 10, 50, 51, 55, 58, 61, 83, 84, 93, 104, 108, 131, 135, 144, 183, 185, 188, 192, 244, 245, 266, 269, 272 blowing-up of the solution, 211 blowup, 211,240 phenomenon, 239 --- solutions, 240 time, 212, 239 one p o i n t - - , 239 three p o i n t s - - , 213, 239 bootstrap argument, 238
L1
-
contraction, 171, 177, 217, 218, 219, 241 estimate, 178, 204 structure(s), 172, 178, 207 theory, 218
-
L2 error analysis, 88 90, 91, 92 inner product, 2, 12, 142, 172, 219 --projection, 16, 174, 178 realization, 50 stability, 19 topology, 118
-
-
-
-
-
-
-
-
e
s
t
i
m
a
t
e
-
,
L; error estimate, 20, 21 stability, 173
WI,p 303
-
-
-
-
304
Index
boundary element(s), 171 - - method, 190, 192, 205 Brezis and Strauss L1 estimate of --, 26 charge points, 195, 205 simulation(s), 171 Chernoff formula nonlinear--, 218, 224, 225, 240, 241 closed range theorem, 6, 42, 185 coercive, 42, 192, 245 strongly--, 7, 9, 10, 15, 58 coerciveness, 93, 144 s t r o n g - - , 8, 22, 51 collocation method, 192 points, 195, 205 method o f - - , 205 compact, 179 operator, 253, 270 compatibility condition, 268 condition (Ie), 252, 253, 255, 256, 257 (Ira), 251, 252, 253, 254, 255, 256, 257 (I), 249, 250, 252, 255, 268, 269 (Re), 254, 255, 256, 257 (Rm), 254, 255, 256, 257 (R), 254, 255 conformal mappings, 206 transformation, 206 conjugate gradient method, 271 constraint(s), 182, 183, 184 continued fraction expansion, 147, 170 contraction, 48, 62, 69, 217, 231,232, 235 property, 51 convection-diffusion, 181 convex, 2, 13, 16, 20, 21, 28, 29, 129 closed set, 238 - - closed subset of a Hilbert space, 42 function, 209 hull, 172 -
-
polygon, 1, 10, 16, 21, 27, 29, 31, 35, 42, 47, 77, 135, 207, 238 smooth domain, 238 Crank-Nicolson approximation, 75 scheme(s), 75, 145, 153 critical point, 207 CSM, 205, 206 curved element(s), 13, 43 cyclic matrix, 196 DD-NN iteration, 255, 256, 257, 271 DDM, 243, 258, 261 deformation rate tensor, 265 degenerate, 208 case, 218 parabolic equation(s), 207, 240 Descloux('s) lemma, 19, 44 lemma of --, 17 Dirichlet boundary condition, 40, 53, 219 condition, 1, 42 form, 2 problem, 182 Dirichlet to Neumann iteration, 246, 267 map, 243, 246, 270, 271 for the Stokes system, 267 Dirichlet 2 to Neumann 2 iteration, 255 dispersive system(s), 2O3 dissipative structure, 171 distributions, 205, 246 DN iteration, 246, 256, 261,267, 271 for the Stokes system, 272 map(s), 243, 246, 248, 259, 271 domain decomposition - - algorithm, 246 method(s), 243 researchers, 270 dual exi)onent , 29 form, 45
305
h~dex --operator, 172, 183, 184, 189 space, 2, 4 duality, 272 argument, 14, 26, 28, 34, 43, 58, 93, 94, 121, 136, 151 -map, 204 method, 139 Dunford integral(s), 51, 59, 63, 67, 75, 93, 109 dynamical systems, 93 Egorov's theorem, 227 electro-magnetic theory, 183 elliptic - - boundary problem(s), 238 boundary value problem(s), 1, 8, 43, 45, 207, 237, 238 nonlinear--, 207 - differential operator, 135 - - e s t i m a t e , 14, 42, 59, 60, 71, 74, 98, 119, 121, 123, 124, 136, 137, 144, 178, 238 --operator(s), 43, 93, 95, 132, 204 - - r e g u l a r i t y , 5, 42, 46, 232, 242 - - Sobolev inequality, 60, 93, 94 uniformly--, 71 energy method, 90, 94, 131, 135, 140 enthalpy formulation, 217 equation of continuity, 268 Euler approximation, 239 - equation, 183 evolution equation(s), 83, 84, 93, 96, 118, 135, 154, 217 operator(s), 94, 96, 98, 99, 101, 107, 108, 118, 121, 122 external force, 262 FDM, 44, 212 FEM, 1, 171 Feyman path integral, 170 finite difference analogues, 211 - - method, 44, 212 --operator, 46, 154
solutions, 207 finite element(s), 205 h p - - , 43 p - - , 43 analogue, 218, 241 approximation(s), 35, 43, 44, 45, 53, 62, 70, 83, 95, 97, 118, 135, 145, 153, 171, 172, 218, 221 discretization, 21,205 method(s), 1, 10, 19, 43, 44, 53, 93, 140, 238 scheme, 207 solution(s), 44 space(s), 70 flow, 181 velocity, 262 fluid, 262 dynamics, 181 mechanics, 205 viscous incompressible, 262 forward difference, 64 Fourier coefficients, 198, 201 series, 259 transformation, 193 inverse --, 193 fractional powers, 43, 51, 58, 93, 101, 144 Frech~t-Kolmogorov's criterion, 228 - - theorem, 242 Fredholm - alternative, 182 - - theory, 191 free boundaries, 217 free boundary problem(s), 42, 206 full-discrete, 93, 140 fundamental solution(s), 171, 190, 191, 195 method, 205 Gauss
divergence formula o f - - , 263 Gel'fand triple, 42, 204 general optimal choice(s), 253, 256, 257, 261, 270
306
hMex
generation theory of Fujie-Tanabe, 99, 144 of Kato-Sobolevskii, 99, 144 of Kato-Tanabe, 104 of Tanabe-Sobolevskii, 98, 144 Kato-Tanabe's--, 123 Green's formula, 2, 192, 264 Green's function, 31, 40, 258 discrete--, 20, 27, 37, 44, 81 regularized--, 39
Babugka-Brezzi-Kikuchi's - - condition, 205, 266 inhomogeneous, 263 boundary condition, 45, 262 boundary value problem, 21, 42, 43 - - Dirichlet boundary value problem, 44 52, 68, 93 --- generators, 119 parabolic equation(s), 95, 144 - - problem, 70 interaction conditions, 243 interpolant, 11 interpolation, 218 inequality, 42 operator(s), 15, 43, 172, 181,219, 231 space, 270 theorem, 19, 44 theory, 60, 93 inverse assumption, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 31, 44, 53, 60, 65, 75, 77, 78, 79, 80, 88, 91, 94, 137, 153, 174, 178, 229 isomorphism, 28, 31, 172, 245, 264, 267 canonical --, 184 - -
- - e q u a t i o n ,
- -
HSlder('s) continuous, 53, 68 inequality, 24, 25, 30 harmonic, 199 equation, 246 extension(s), 201, 245, 249, 250, 251, 253, 254, 259, 260, 261, 266, 269, 270 vector-valued--, 266 function(s), 266 heat equation, 47, 77 Heinz(') inequality, 9, 99, 167 theorem, 61 inequality o f - - , 43 Heinz-Kato's theorem, 59 Helfrich's method, 85, 94 Helmholtz decomposition, 262, 271 higher accuracy, 70, 75, 93, 144, 153 Hille-Yosida('s) theorem, 84 theory o f - - , 48 homotopy invariance, 223 hybrid finite elements, 183 hydrodynamics, 183 hyper-dispersive operator, 240 hyper-dispersivity, 240 hyperbolic equation(s), 47, 53, 83 systems, 203 - -
inf-sup constant(s), 243, 266, 268, 272, 273
Jordan curve, 206 Kato('s) inequality, 221,241 square root problem, 43 theorem b y - - , 65 Korn's constant(s), 272, 273 inequality, 205 second inequality, 272 Lagrangian multiplier, 183, 184 law of conservation, 203 Lax-Milgram's theorem, 4, 7, 42, 105 layer potentials, 205 double--, 191 single --, 191 least energy solution, 207, 238
307
Index
level-set approach, 240 Levi-Tanabe's method, 97 Lie groups, 165 linearized instability, 238 Lipschitz boundary, 190, 246, 272 constant, 224, 225 ---continuous, 181,188, 218, 224, 225, 231,232 domain, 45, 187, 270, 271 lumping, 180, 181,204 of mass, 171,218, 219 operator, 172, 173, 181,219 m a s s - - , 219 scheme of --, 171 m-accretive --operator(s), 231,242 m-sectorial, 5, 8, 51 operator(s), 6, 42, 43, 53, 61, 85, 96, 99, 108, 118, 131, 135, 144 maximum principle, 21, 45, 147, 171,174, 219 discrete--, 20, 21, 22, 44, 46, 181, 182 general--, 44, 45 storong--, 45 mean curvature flows, 240 meromorphic flmction, 151 minimal surface, 44 mixed - - b o u n d a r y condition, 263 - - boundary value problem, 272 elements, 171 - - finite element method, 271 finite elements, 183, 205, 272 weak formulation, 272 Morse index, 208, 238 mountain pass critical value, 208 lemma, 238 multi-step method, 154, 155, 170 Nakao's method, 238 natural boundary condition, 265 Navier-Stokes
--equations, 271,273 system, 205 Nehari('s) iterative sequence, 238 manifold, 208, 238 variational formulation, 208 variational principle, 238 Neumann boundary condition, 12, 44, 240 boundary value problem, 45, 205 condition, 50 problem, 3 Newmark's/3 scheme called--, 94 Newton's method, 238 Nitsche's trick, 43 non-negative type, 20, 45 numerical range, 5, 42, 54, 55, 65, 93 numerical verifications, 238, 273 -
-
-
-
-
-
-
-
one paraeter families of subgroups, 165 orthogonal complements, 189 --decomposition, 185, 186, 264 - - projection, 7, 8, 16, 18, 54, 59, 77, 85, 86, 118, 137, 189, 218 system of eigenfunctions, 48 Pad~ approximation, 146, 147, 170 parabolic equation(s), 47, 51, 54, 62, 93, 95, 118, 145, 153 initial boundary value problem, 101 problem(s), 46, 93 region, 8, 153 parallel computations, 243 peak(s), 211,240 movement o f - - , 240 number o f - - , 240 Phragmdn-LindelSf theorem, 147 Poincard's inequality, 2, 3, 24, 42, 186, 188, 244, 245, 262 Poinca, r~-Wirtinger's inequality, 12, 43,219, 241 Poisson equation, 1, 182, 186 porous media, 217, 240 -
-
308 pressure, 262 accompanying - - , 265, 266, 267, 268 principal value, 191 rapidly decreasing function, 193 Rayleigh's principle, 238 reaction-diffusion system, 240 real method, 93, 94 reflection, 249, 254 principle(s), 243, 254, 260 transformation, 255 regular, 11, 14, 19, 43, 53, 178, 220, 244, 246, 272 triangulations, 13, 21, 171, 181 relaxation parameter(s), 247, 255, 257, 258, 268, 270 resolvent(s), 47, 55, 93, 107, 225, 226, 230 set, 5 Richardson('s) extrapolation, 35 iteration, 271 Riesz(') --representation theorem, 2, 4, 7, 31, 42, 264 theorem, 184 representation theorem of--, 183 Riesz-Thorin's interpolation theorem, 173, 204 Ritz --operator, 7, 16, 19, 20, 21, 45, 119, 120, 135, 179, 182, 218 projection, 85 Ritz-Galerkin approximation, 10 method, 7, 189, 192 Robin to Robin iteration, 257 RR iteration, 257, 258, 271 Runge-Kutta method, 170 saddle point, 183, 184, 185, 186, 187, 188, 189, 204 Schauder's fixed point theorem, 238 Schur complement matrix, 271 Schwarz althernating method, 271 SDN map(s), 265, 267, 268
Index
self-adjoint, 43, 83, 84, 93, 165, 248, 249, 256, 266, 268 --operator(s), 9, 43, 47, 77, 99, 100, 115, 165, 166, 243, 245, 246, 248 semidiscrete, 53, 54, 68, 70, 77, 83, 85, 93, 94, 118, 126, 131, 135, 153, 218 semigroup, 47, 51, 53, 77, 93, 145, 146, 174, 177, 217, 224, 235 (C0) --, 49, 93, 145, 165, 170, 204 of operators, 47, 48 property, 49, 74, 93, 97, 121 theory, 53, 93, 145, 205 analytic --, 165 approximation of--, 53 contraction--, 49, 169, 231 convergence of--, 226, 230 holomorphic--, 5, 42, 51, 93, 97, 98, 148, 154, 170 linear--, 225 nonlinear--, 170, 217, 220, 231,240, 241 rational approximation of--, 170 strongly continuous--, 49 semilinear elliptic equations, 207 parabolic equations, 207, 211,239 discretized--, 240 single-step method, 153 singular integral, 271 skew-adjoint, 84 smoothing property, 47, 50, 51, 54, 62, 74, 94 Sobolev('s) imbedding theorem, 21, 45, 208 inequality, 24, 25, 29, 31, 45, 60, 79, 80, 93, 94 space(s), 1, 42 solenoidal, 262 solid mechanics, 183 spectral --decomposition, 6, 43, 100, 115, 166 mapping theorem, 249, 256, 271 - - radius, 249, 256, 259, 268 spectrum, 5, 8, 76, 153,249, 253, 259, 268 stable sets, 238
309
Index
Stampacchia estimate b y - - , 45 stationary point, 184 Stefan nonlinearities, 240 two p h a s e - problem, 217 Steklov-Poinca% operator, 270 Stokes equations, 271, 273 extension(s), 265, 266, 267, 269 system, 187, 204, 205, 261, 268, 272 continuous, 205 discretized nonstationary--, 205 stationary --, 187, 243 stress tensor, 265 - - vector, 265 strong continuity, 49 convergence, 165 -inhomogeneties, 243 --solution, 2, 5, 53 topological degree, 223, 241 torus, 217 trace, 244 computations, 127 operator, 2, 4, 42 theorem, 40, 244 triple of Hilbert spaces, 4, 7, 42, 83, 108, 131,244 Trotter's product formula, 165 Trotter-Kato's theorem, 226 nonlinear--, 241 type (0, M), 6, 8, 42, 51, 54, 56, 59, 97, 154 type (00, M0), 148, 153 unstable, 237 linearized, 207 -sets, 238 --solutions, 207, 238 upwind, 181 - - difference, 171
finite element method, 181 Ushijima's works, 204 variational inequality, 42 method of mini-max type, 238 principle(s), 243, 246, 249, 260, 261,266, 267, 269 problem, 3, 4, 10, 42, 182, 183, 184, 204 --structure, 1, 171, 188, 208, 221 weak --form(s), 1, 90, 132, 172, 187, 219 - - formulation(s), 184, 272 -solution, 2 Yosida approximation, 230, 241,242
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