Finite Element Methods for Integrodifferential Equations

< C\\Rhu - u^Ww -

RHWW!

+1


^Ch^WRnu-uhMt, and the desire estimate is obtained. Assume that Q is a two-dimensional convex polygon and 5^ is the linear finite element subspace, then the Ritz projection operator R^ is a bounded operator in W1,P(Q) satisfying (see Rannacher and Scott [137]) \\Rhu\\itP < C|M|i l P ,

1 < p < oo.

(3.7)

The above estimate also holds for d-dimensional smooth domains with d <3 (Chen and Huang [22]). For a two-dimensional convex polygonal or smooth domain Q (Schatz and Wahlbin, 1978), the Ritz projection operator R^ is (almost) bounded in L°°, ||/M|o,oo 2. Furthermore, there are the following error estimates of optimal order \Rhu - u\\hp < Chr-1\\u\\r:P, II-R/.W -

M||O,P

< Cphr\\u\\rtP,

2
(3.8)

(3.9) (3.10)

and the maximum norm estimates 2

C/i |ln/i|||w|| \\Rhu-„II0IOO<{—rZ"-°°' Ch ||u|| , p

ri00

2i0O)

« : z v (3-n)

ifr = 2, ifr>2.

3.1. Elliptic Projection

R^u

39

For linear finite elements on a quasiuniform triangulation J^, the logarithm factor \lnh\ on the right-hand side cannot be removed. R e m a r k . If the coefficients of A are dependent on t and differentiate with respect to £, then A(t\ R^u — u, v) = 0 and A{t; Rh,ut — ut,v) = 0 for v G S^. After differentiating the former with respect to t, it becomes A(t] (Rhu)t

-ut,v)

+ At{t\ RHU -u,v) = 0 for v e S^,

and A(t; {Rhu)t - RhUt, v) = -At{t] Rhu - u, v)

for v G Sy ■

Here the coefficients of At are obtained from differentiating the corresponding coefficients of A with respect to t. From this we see that (Rhu)t ^ Rh^t, \\(Rhu)t - flfcUt||i < C\\Rhu - u\\u

(3.12)

and

ll(fl/.«)t||i < CdHU + ||utno. In general, n

||£>?(JM||i < CJ2W&MU

* > 0-

(3.13)

i=o For any ^ G L2(f£), we define the L2-projection P^u G 5^ satisfying (Phu-u,v)=0

for t;€Sf.

(3.14)

Let v — PhU in (3.14), obviously we have

I I ^ I I < II«II. On the quasiuniform triangulation J^, Descloux (1972), and Douglas, Dupont and Wahlbin [42] proved the following most elegant estimates \\Phu\\jiP < C\\u\\jtP,

j = 0,1,

1 < p < oo.

(3.15)

From (3.15) the following error estimates of optimal order can be derived im­ mediately \\Phu - u\\itP < Chr~l \\u\\r,j>, I = 0,1,

1 < p < oo.

(3.16)

Note that the interpolation operator 7^, the Ritz projection operator Rh and the L2 —projection operator Ph are the three most important operators in finite

Chapter

40

3. A Survey on Elliptic Finite

Elements

element analysis. T h e estimates (3.2) and (3.16) hold for all p, with 1 < p < oo. B u t t h e boundedness of Rh is restricted by p , see the estimates from (3.7) to (3.11). In addition, we also introduce the other operators, such as A^.T^ = A^ and T = A - 1 . Here A^ is t h e discrete analogue of A, i.e. for any u £ HQ(Q,), AhU £ 5y satisfies (Ahu, v) = A(u, v)

for

v £ S*.

(3.17)

It is proved t h a t A^u can be uniquely determined even when A is not positive definite. In fact, let \}p)™h be a basis of S^, substituting A^u — X ^ = i aj^Pj{x) into (3.17), it becomes Nh

3=1

Since t h e coefficient matrix of a is a Cramer matrix, which is positive-definite, t h e system of algebraic equations is uniquely solvable. If A is non-negative definite in S £ , t h e n (Ahv,v) = A(v,v) > 0 for v € S£. If u e H2f]H^(n), from t h e following equalities (Ahu, v) = A(u, v) = (Au, v) = (PhAu, v)

for

v <E S*,

we have Ah = PhA

and

\\Ahu\\ < \\Au\\ < C\\u\\2,

(3.18)

and by t h e inverse property of v € 5 ^ ,

We found t h a t t h e operator Ah has a new imbedding property in W 1 , 0 0 ( f i ) which is formulated as L e m m a 10.1 in Chapter 10 and will play an i m p o r t a n t role there. If B is an arbitrary second order partial differential operator, then Bh, the discrete analogue of B, can b e dominated by Ahl i.e. there exists a constant C such t h a t \\Bhv\\
= B(v, Bhv)

= B(w, Bhv) + B(v - w,

= A^v.

Obviously

Bhv)

< C{\\w\\2\\Bhv\\ + ||v - HIill^Hli) < C|M|2||Bhi;||,

3.2. Domains with Curvilinear Boundaries

41

and then \\Bhv\\
= C\\Ahv\\.

Sometimes, when no confusion will be caused, we also use T — A~x to denote the solution operator of the problem Au = f in 1}, u = 0 on dQ,, namely u = Tf. Define the operator Th : L2 -+ S!? such that Thf = Rhu, then Th = RhT. For any v G Sy it is obvious that (/, v) = A{u, v) = A(Rhu, v) = A{Thf, v) = (AhThf, v), thus Th = (Ah)'1. By using the two operators T and Th defined above, the error estimate (3.6) can be rewritten as ||(T/l-r)/||_i<||(JJ/l-/)T/||_z
3.2

-l
(3.20)

Domains with Curvilinear Boundaries

Up to now we have assumed that Q is a convex polygon or a convex domain (only for linear finite element approximation). For a general domain Q with curved boundary dfi, the approximate net domain f^ ^ Q and S^
and

\\u\\Kp,R
1 < p < 00.

Actually, one only needs to extend u from S l t o f 2 * = f£ft|JS7. If £2 is a two-dimensional domain with piecewise smooth boundary, the following better result can be obtained, \\u\\k,P,n* < C\\u\\kiPin

for

l
Assume that u G Wk' p(f2) f] HQ is the solution of the problem Au = / in O, and u and v4 (with coefficients afj extended) are the extensions of u and A to fi* respectively. Define / = Au in Q*, obviously, ||/IU-2,Q* < C||S|UlP,n- < C\\u\\ktPin < C||/|U lPf n.

Chapter 3. A Survey on Elliptic Finite Elements

42

When there is no ambiguity, the extension u is still denoted by u. Here and after it is understood that each function v e S!? is analytically extended from Qh to fi* = Qh U ^> s o ^ *s s t m a piecewise (r — l)th degree polynomial function in Q*. This extension is unique and satisfies \\u\\j, p, n* < C\\u\\j,p, Slh,

j = 0,1,

1 < p < oo

and inf ||u - v\\itPtn* < Chj-l\\u\\jtPfQ,

0 < I < j < r.

(3.21)

Denote respectively the bilinear form and the inner product by A(u, v)h=

I

(aijDjuDjV 4- a^uv)dx

and

(/, v)h = /

fvdx,

and define the Ritz projection RhU E £> such that A(Rhu,v)h

= (f,v)h

for

«GSf.

(3.22)

This algorithm will be convenient for numerical computation. Because all values of a^ and / at nodal points in Q, used in the quadrature formula are given, they are independent of the ways of extensions. On the other hand, integrating the equality Auv = fv in Q,h for v G 5^ and noting i; = 0 o n dti^, we have

/(M)/> = [ = —/

Auvdx a>ij D{uv cos nxj ds + /

(aijDiuDjV -f- a^uv)dx

= A(u,v)^.

(3.23)

Then, from (3.22) and (3.23), we still have the same orthogonal relation ACRfciz-u, !;)& = () for

ve5f.

(3.24)

To estimate the error R^u — u, one needs the following lemmas and the imbed­ ding estimate

IHkan < C||Hli,n.

(3.25)

Let 7 C dfi, be a curved side of the boundary element r and the rth degree curve jh C dQh be the corresponding side of approximate element r^ of r. A

3.2. Domains with Curvilinear Boundaries

43

coordinate transform maps 7 onto {t = 0, 0 < s < h}, and 7^ can be written as t = y(8) = F{s)P{S),

se[0,h]

where P(s) is the rth degree polynomial, y = P = 0 at all points of order r, and F(s) is a smooth function. Then P(s) any polynomials of (r — 3) £/l degree and \P(s)\ < Chr. Lemma 3 . 1 . Assume that d£l is the piecewise smooth and bh is the narrow band of the boundary with width Hh < IHIo,6fc < Chr'2\\w\\ltn

Sj, the Lobatto is orthogonal to boundary of fi, Chr, then

for any w e H1^),

(3.26)

and for any u G H\ [\ H2, \\u\\o,enh < Chr'2\\u\\lthh

For r > I > 3, we

< Chr\\u\\2,n,

(3.27)

\\u\\o,bh < Chr\\u\\hbh

< Cfc 3r / 2 |M| 2| n.

(3.28)

H1'1 and u € Hi fl

H

\

uwds\ < C/i r+z - 2 |M|^|M| z _ ljn .

If

(3.29)

Proof. Construct a piecewise smooth transform which maps bh onto Vh(ZDh = {Q<s< b, \t\ < Hh} and dft onto L = {0 < s < 6, t = 0}. Then / w2d:z
" [

JO

Jbh we

w2dsdt
JL

note that

rv / ut(t,s)dt Jo

= ut(0,s)y+

ry / Jo

utt(y-t)dt,

then [ u2ds < CHh [ Janh Jo

H

[ \Du\2dsdt < CH2h\\u\\2 JL

and [ u2dx< Jbh

[ Jo

h

f JL

u2dsdt
n

Chapter 3. A Survey on Elliptic Finite Elements

44

For r > I > 3, w G Hl~1 and u G H% f] Hl, we have f2/( s )

/ luuds = V / / w(utFP+ Jon,,. J-yh = 0(/ir+i"2)^' /

/ Jo

utt(y{s) - t)dt)ds

\Dl-2(wutF)\dS

+

O(Hh)\\w\\0bh\\u\\2tbn

= 0(fc r + '- 2 )||u|M|HI.-i + 0 ( h 2 ' ) | M | 3 | M l i , thus the proof is complete. Lemma 3.2. For v G S^ and fi* = f^ |J fi, there is N k n - = (1 + 0 ( A ( r - 1 ) / 2 ) ) N k n h )

J = 0,1.

Proof. Consider a boundary element r and its approximation r^, which has at least one side on 9Q and cftl^, respectively. The area of r \ r ^ is 0(fo r + d _ 1 ). Since various norms for u e S j are equivalent, it can be shown that / \Djv\2dx - [ \Djv\2dx = 0{hr+d-l)m^\Djv\2 Th Jr Jrh r 1 = 0(fc - )||IPt;||g fTfc , and by summation over r^ the lemma follows at once. We now estimate the error R^u — u. Theorem 3.1. Assume that dVt is piecewise smooth and locally convex in the neighbourhood of each angular point. Assume also that u G Hr P|ifg(Sl) and RhU G Sy are the solutions of (3.1) and (3.22) respectively. Then WRHU-UWW

< Chr-^uWrv.

(3.30)

l

Suppose that the regularity assumption A 2, 2 < I < r, then \\Rhu - u\\2-i, n* < C / i r + / - 2 | ^ | | r , n .

(3.31)

Proof. By the interpolant I^u G 5^ and (3.12) there is v\Rhu

- Ihv>\\itnh < A(Rhu - Ihu, Rhu = A(u - Ihu, Rhu < C\\u - Ihu\\itnh\\Rhu

Ihu)h Ihu)h -

Ihu\\i,nh.

Eliminating the common factor and using Lemma 3.2, it leads to \\Rhu - Ihu\\ltn*

= (1 4- 0(hW2))\\Rhu

r 1

Ihu\\ltnh


3.2. Domains with Curvilinear Boundaries

45

and
\\RhU - v>\\ltn*

To get the error estimate in L 2 , by the duality argument, we have (Rhu - u, p)h = /

{-DifaijDjiv

= A(Rhu -u,w-

+ a0w)(Rhu

- u))dx

wh)h + e^,

where, from Lemma 3.1, (3.27) and (3.28), £h = /

aijDiWU cos nxjds

< Chr\\w\\2tf\\u\\2in, or

< C/i

r+z 2

- ||n|| z , n\\w\\lt

if

r > 2,

n,

if

r > Z > 3,

and therefore, by (3.30), nD

II

(Rhu-u,

\\Rhu - u|| 2 _ z?

Qh

= sup 1


ip)h

— \m\i-2

^ur+l_2u

< Chr+L

z

\\u\\ri

n.

In addition, denoting by Y2 the summation over all boundary elements r, using the inverse property and (3.26) with b^ replaced by r, and by Lemma 3.1,

T

1

< CbT- £ \\\Rh* - «lll,r + Hll.r) < CVMl

a.

r

Thus by (3.28) we have ||ifcu-u||o,6 f c
or

Cft 3p / 2 ||u|| 2 ,n,

and / (Rhu - u)
Chapter 3. A Survey on Elliptic Finite Elements

46

later in the process of proof, we shall always assume that fth = Q, and admit fi to be smooth enough. Remark. In the case of curved boundary, if we take v = 0 outside Slh for v G S!? and r > 2, then we have only, by Lemma 3.1, \\Rhu - tx||ifn\nfc = N | i , 6 l h <

Chr/2\\u\\2.

It will damage the optimal accuracy \\RHU — ^ H i , ^ =

3.3

0(hr~l).

Ritz—Volterra Projection VhU

In order to analyse the semidiscrete finite element approximation to a parabolic problem, Wheeler (1971) first used the corresponding elliptic projection R^u as a comparison function. The error analysis is greatly simplified, because Rhu(t) is quite close to the parabolic semidiscrete finite element solution in the sense of the maximum norm estimate as well as superconvergence. With regard to the finite element of PIDE, the earlier works (as Thomee and Zhang [156], and Yanik and Fairweather [167]) used also the Ritz projection as a comparison function and obtained a number of nice results. Later Cannon and Lin [12] (1990) introduced a new Ritz—Volterra projection (or quasielliptic projection) Vh,u(t) E S^ which satisfies A(Vhu(t) - u(t), v)=

[ B(Vhu(s) - ti(a), v)ds, Jo

v
(3.32)

It is shown that using Vh u as a comparison function the error analysis of the finite element for PIDE can be simplified. Its applications to the proof of superconvergence of the finite element approximation are more successful (see Chapter 10). Therefore the Ritz—Volterra projection operator Vh is the fourth important operator in this book. In the following its various properties will be discussed. Theorem 3.2. Assume that the regularity assumption A1^2 holds and that Vh(t) € S!? is the Ritz—Volterra projection of u, then \\Vhu(t) - u(t)U

< Chr+l(\\u(t)\\r

+ / \\u{s)\\rd8),

- 1 < I < r - 2.

Proof. Denote by Rhu(t) the Ritz projection of u, and p=

VHU

-u = (Vhu - Rhu) + (Rhu - u) = 6 + £,

3.3. Ritz—Volterra Projection Vh,u

47

where 6 = Vh,u — RhU G Sp satisfies A(0, v) = [ B{9{s) + £{s), v)ds Jo Taking v — 6, then u\\9\\\ < A(9,9) = [ Jo

for

v G S*.

(3.33)

B(6{8)+£{8),9(t))d8

< C f\\\9(*)lli + Jo

IK(»)lli)
or

\W)h
Jo and by Gronwall's inequality and (3.6), ||0||i
Jo

< ChT~x f \\u\\rds. Jo

Thus the estimate in H1 follows. To get other estimates, construct an auxiliary problem Aw — if in ft, w — 0 on d£l. By Wh = R^w and (3.2), there is (p, ip) = A(p, w) = A(p, w-wh)

+ /

B{p{s),wh)ds

Jo

= A(p,w -wh)

+ / B{p,wh-w)ds+ Jo

I Jo

< CdlPlli + /

\\p\\ida)\\wh - wh +C f

Jo


(p,B*w)ds \\pUds\\w\\l+2

Jo

f\\phds)+ Jo

f \\p\UdaWiph. Jo

Using the definition of the norm ||p||_ s and Gronwall's lemma, < Cft I + 1 (||p||i+ /

Jo

| | p | | i ^ ) < C / i r + z ( l l ^ ) l l . + / \\u\\rds). Jo

The theorem is thus proved. Theorem 3.3. For p(t) = Vhu(t) - u(t), there is ||AVWII+fc|lAVWIIl<^fcr(^|PJu(0||r+ t=0

/ Jo

Mrda),

71=1,2,3, ■

48

Chapter 3. A Survey on Elliptic Finite Elements

Proof. Only the case of n — 1 will be proved. Differentiating (3.26) with respect to £, A(pt,v)

= -At(p,v)

+ B(p{t),v)+

I

Bt(p(s),v)ds,

Jo

and writing Pt = (Vhu)t -ut

= {{Vhu)t - Rhut) + {RhUt -ut)

= 0 + £,

then v\\pt\\i
= A(pt,9) +

= -At(p,6)+B(p,9)+

A(pt,0 f Bt(p(s),e)ds Jo

+

A(Pt,0


Noting ||0||i < |M|i-t-||f||i and using Young's inequality to eliminate the term ||p||i on the right-hand side, it becomes

\\Pth

+ \\ut\\r + /

|M| r d*).

Jo

Next we shall estimate the L 2 -norm by the duality argument. As in the proof of Theorem 3.2, there are (pt,ip) = A(puw)

= A(pt,w-wh)

+At(piwh

+ / Bt(p(s),w-wh)ds-(piAtw)-\-{p,B*w)-\' Jo

-w) + / Jo

B(pyw-wh) {p{s),B*w)ds

and

INI < CMIIPtlli + IIPIII + / ||p||iA») + C(\\P\\ + /* \\p\\ds) Jo r


+ \\ut\\r+

Jo

\\u\\rds). Jo

The proof of Theorem 3.3 is complete.

3.3. Ritz—Volterra Projection VhU

49

From Theorems 3.2 and 3.3 one can observe that the following estimate \\D?(Vhum\i
/"'Hlick,

n>0

are valid and will be useful in the analysis of the time discretization. Finally, turn to the Lp error estimates of Vhu(t). Following Rannacher and Scott [137], a new gradient type Green function G(t) G HQ can be defined as follows: A(G, v)+ [ B(G(s), v)ds = Dz8hcre(t) in QTi (3.34) Jo and its discrete analogue Gh{t) G S£ are studied by Lin and Zhang [100], where (T£(t) is a modifier function satisfying the following conditions: 0<(Te(t)ZCS?{J), /
/ f{t)*e{t)dt = f(t0). ->°Jo Use the weighted norm method it can be proved that \\Gh(t)-G(t)\\ltl
I

ae{s)ds).

Jo

A simplified proof in its weaker form will be given in Chapter 9. T h e o r e m 3.4. Let O C R2 be a convex polygon or a domain with piecewise smooth boundary dQ which is locally convex in the neighbourhood of each angular point on dfl. Let Vhu(t) G S^ be the Ritz—Volterra projection of u(t) G Hi, then ||VW*)||i >P < C(||u(t)|| 1 | P + /

\\u(s)\\hpds),

2 < p < oo,

(3.35)

Jo

and \\Vhu(t)\\0tOO < C\lnh\\ (||u(*)||o,oo + /

\\u(s)\\Qt00ds).

(3.36)

Jo

rmore, there exist the following optimal order error estimates \VhU - u\\ltV < Chr-l(\\u(t)\\riP

+ / \\u(8)\\rtPd8), Jo

I = 0, 1,

(3.37)

Chapter

50

3. A Survey on Elliptic Finite

Elements

where 2 < p < oo, with I — 1 or 2 < p < oo, with I = 0, and \\Vhu - ix||0,oo < Cfc r |ln/i|(||u(t)|| r ,oo + /

\\u(s)\\rj00ds).

(3.38)

Jo

For s m o o t h domains of any dimension and r > 2, Lin, Thomee and Wahlbin [97] (1991) presented a simpler duality argument (without using Green function) and got the following result. T h e o r e m 3 . 5 . Let ft be a smooth domain of any dimension, p = Vhu(t) — u(t), 2 < p < oo, then \\p(t)\\o,p + h\\p{t)\\liP

< Cphr(\\u(t)\\riP

+ f

\\u(s)\\r,pds).

(3.39)

Jo

P r o o f . For any

p ' < C*p IM|o,p' = Cp , 1 < p < oo, which is valid at least for the smooth domain ft. Taking Wh = RhW, then (Dtp, (f) = (p, -Dup)

= A(p, w)

— A(p,w-wh)

+

B(p(s),wh)ds Jo

= A(Rhu-u,w)

+

B(p(s),wh)ds Jo

= (Di(Rhu-u),(p)+

B(p(s),wh)ds Jo

< \\RhU-u\\liP\\ip\\oiP'+C

/

H/o^lli^ll^fclli^/ds.

Jo

By t h e boundedness of Rh in W1,p

, | | ^ ^ | | i , P ' < Cp||w||i j P / < CP, we have

\\p(t)\\i,P
+ C [

\\p\\itPds.

Jo

By Gronwall lemma the desired estimate in W1,p follows. T h e derivation of t h e error estimate in Lp is similar to t h e case when p = 2. In addition, t h e following L°°-estimates ||pWI|0,oo + ^ | b ( t ) | | l , o o < C / l r - e ( | | ^ ) l | r , o o +

/ ||li(5)||r>00ds), Jo

€>0

(3.40)

3.3. Ritz—Volterra

Projection

51

VhU

and t h e L p -estimates of t h e derivatives with respect to t can also be derived. Finally a super convergence estimate between V^u and Rh,u will be useful. T h e o r e m 3 . 6 . Assume t h a t the triangulation is quasiuniform. Let a(t, x) be a n arbitrary function and B\ be an arbitrary first-order operator. Assume t h a t B can be expressed as B — a A + B\, then the super convergence estimate \\Vhu(t)

- Rhu(t)\\x

< Chr

< C [\hU(s)h Jo f

\\u(s)\\rds,

+

te

U(*)\\o)ds

J.

(3.41)

= 0 + £, 6 = Vhu-Rhu

G S^ satisfies (3.33).

JO

holds, where £ = R^u — u. P r o o f . From (3.32), Vhu-u Rewriting t h e bilinear form as

B(u, v) = A(u, av) + (u, Bxv), Denoting w — av and w^ — Ih{&v) G S^ B(H, v) = A(£, w-wh)

v G S£.

and using

+ ( £ , B l V ) < CU\U\\w

-wh\\1+

C\\t\\o\Mi

and t h e super approximation estimate for v G 5 ^ ,

Ik - «*ll?.n = E I™ - «"ill?,r < Ch2r~2 £ II^Hlt r

r


11% < C f \\6hds + C / W i l l + U\\o)ds. Jo

Jo

By Gronwall inequality t h e desired estimate follows.

Chapter 4

Semidiscrete and Fully Discrete Schemes In this chapter t h e following parabolic integrodifferential equation (

ft (uuv)

+ A(u,v)=

B(u(s),v)ds

+ (f,v),

v G #0\

/ Jo j u = 0 on dQ x J, ^ ^(0) = UQ in ft,

(4.1)

where J = (0, T ] , the coefficients of A depend on t G J and the coefficients of 5 depend on £ and s, satisfying 0 < 5 < t < T , will be discussed.

4.1

Semidiscretization

Consider the semidiscrete approximation Uh(t) G Si? satisfying

{

(u>ht,v) +A(uh,v)

= / B(uh(s),v)ds

+ (/»,

veS?,

^

^

where the initial value UQ^ G S^ is an approximation of i^o, which can take IhUo, RhUo or PhUo, b u t in each case it should have the following accuracy \\uoh - u0\\i < Chr-l\\u0\\r,

1 = 0,1.

(4.3)

Assume t h a t {(fj}ih is t h e basis of the finite element space 5 ^ . T h e semidis­ crete finite element approximation Uh(t) G S^ c a n be represented in the fol53

54

Chapter

4. Semidiscrete

and Fully Discrete

Schemes

lowing form Nh

h

where {ctj(t)}1

satisfies

Y^i^i^jWjit)

+ Aiifu^ajit)

at(0)=7i,

2= 1,2,.-.,^.

-

/ B{ipi, ipj)aj{s)ds] =

{f{t),(pi),

This is a first-order system of ordinary differential equations and here ji is the coefficient of (fj(x) in the representation of UQ^, i.e. UQ^ = X^?=i ljfj(x)' Due to t h e mass m a t r i x [(j)\ is a Cramer matrix which is positive-definite and invertible, this system is uniquely solvable in Cl(J). T h e o r e m 4 . 1 . Let u(t) and Uh(t) be the solutions of (4.1) and (4.2) respectively, then we have the following error estimate K ( t ) ~ " t o l l < C/lr(||wo||r +

/

\\ut{8)\\rds),

t G J.

JO

P r o o f . According to Lin, Thomee and Wahlbin [97], choose t h e Ritz— Volterra projection Vhu(i) € 5 ^ as a comparison function and express the error as e = uh - u = (uh - Vhu) 4- (VHU - u) = 0 + p. Noting t h a t u(t) = u(0) -f J0 ut(s)ds

and Theorem 3.2, it is known t h a t

l | p ( * ) l l i < C f c r■- I ( H | | u*()*H)r| + | r + /[* t°
\\u(s)\\rd8) ds) \\u(8)\\

\\ut\\rds),

(4.4) Z = 0, 1.

/o Jo

Since 6 G Si? satisfies {0uv)+A(0,v)=

[ B(9(s),v)ds-(Puv), Jo 6(0) = UQ — RhUQ in £2,

veS?,

where ||0(O)|| < ||«0h - " 0 | | + I N - RhM

< C7l r ||tlo|| P .

(4.5)

4.1.

Semidiscretization

55

Setting v = 6 in (4.5), it becomes 1

SDtWef + upwiKcJ^Phdswmh

+ Wpt

2J

<^\\8\\l + cJ*\\e(s)\\lds + \\Pt\\\\ei /o and after eliminating the first term on the right-hand side and integrating in t, / V l l ? < f a < C F ( 0 ) | | a + C7 / ' f\\e{T)\\\drds Jo Jo Jo Applying the Gronwall inequality, it leads to P(t)f+

H0(*)ii2 + f \\n\ds < c\\m\\2+c

+ C f \\pt\\\\0\\ds. Jo

f \\Pt\\\\e\\ds

Jo

Jo


(4.6)

f\\pt\\ds) Jo

s
Letting ||0(t)||=sup||0(a)||,

0
s
and using (4.6), we have

l|0(t)|| < \\e(i)\\

[ \\Ut\\rd8), Jo

and Theorem 4.1 follows from (4.4). Note that if the Ritz-projection Rhu(t) G S% is used as a comparison function and let e = (uh — RHU) + (RHU — u) = 6 + p, then the error equation is of another form (0t, v) + A(0, v)=

[ B(6(s) + p(s), v)ds - (pu v), v € S?. Jo In this case the argument is more complicated and it was given by Thomee and Zhang [156]. If the initial value was chosen to be uoh = Rh^o 6 S^, i.e. 6(0) = 0, then by (4.6) one can derive ( / ||0||?d*)*
\\ut\\rds.

56

Chapter 4. Semidiscrete and Fully Discrete Schemes

This is a super convergence estimate for the gradient y # in L 2 . In the following we shall further improve the estimate in L°°( J; L2) and get a L°°-error estimate of 9 in two-dimensional case. Lemma 4.1. Assume that UQ^ — RHUQ, then

( Ch^ifwutWl^ds)1'2, \\uh{t)-Vhu(t)\\Hi{n)<{ r

Ch (

^

Jo

J° ||«t||?dfl) 1/2 .

Proof. Let v = 6t in (4.5), then

||0tf + ±DtA(9,9) = ±At(9,8) + DtJ B(9(s),9(t))ds -B(6(t),6{t))-

f Jo

Bt(9(s),9(t))ds-(pu9t)

< Dt j * B(9(s), 8(t))ds + \\\9tf +C\\9\\\+C

Jo

+ l\\Pt\\2

f\\9\\\ds.

Using the argument used in the proof of (4.6), it leads to \\9{t)\\\
\\Pt\\2ds,

and, by Gronwall lemma,

\\9(t)\\l
Thus the lemma follows from (4.4). Theorem 4.2. There is \\v(uh(t)-u(t))\\
+ / \\ut\\rds + ([ Jo

Wutf^ds)1/2).

Jo

Proof. From the estimate of \jp in Chapter 3 and Lemma 4.1, the theo­ rem follows immediately. Lemma 4.2. If 0(0) = 0, then ||flt||
Jo

||u„||r
4.1. Semidiscretization

57

Proof. Differentiating (4.5) with respect to £,

(OtUv)+A(euv)+At(e,v)=B(0(t),v)+

f Bt(6(s),v)ds-(ptuv),

v e S*,

Jo

taking v = 0U noting 6{t) = 0(0) + / 6t(s)ds and 0(0) = 0, we have Jo

\Dt\\6tf

+ Hl*t||? < c||«t||? + C(e)||0||? + C j f ||0||?da + llpttllll^H

By the same argument used in the proof of Theorem 4.1, it leads to l|et||
(4.7)

JO

Letting t=0 in (4.5), (0t(O),t;) = -(MO),tf),

«€Srh,

and using the estimate of pt in Chapter 3, we have ||0t(O)|| < C\\Pt(0)\\ < Chr(\\u0\\r + |K(0)|| r ). From the estimate of ptt and (4.7), the lemma follows. Making use of Lemmas 4.1 and 4.2, the L p -error estimate can be derived as follows. T h e o r e m 4.3 (Lin, Thomee and Wahlbin, 1991) Assume that u(t) and uh(t) are the solutions of (4.1) and (4.2) respectively, and the initial value UQh = Rhuo, then for cases where d = 1, 2 < p < o o and d = 2, 2 < p < oo, there is \ut(t)-u(t)\\Lp{n)


\\ut\\r,Pds + ( / Jo

Jo

||ut||J!ds)2 )•

In addition, for dimension d — 3 or 4, under the assumption A^, p G (1, oo), we have \\Uh(t)

- u{t)\\Lp{n)

< Cphr{\\u0\\r,p

+ |k(0)||r,p +

/ (||tit||r,p + Jo

\\Utt\\r)ds).

58

Chapter

4. Semidiscrete

and Fully Discrete

Schemes

P r o o f . Denote t h e error e = 6 + p , here p = VhU — u has been estimated in C h a p t e r 3. If d = 1, 2 < p < oo or d = 2, 2 < p < oo, by t h e following Sobolev imbedding inequality l|0(t)llLp(n)
^ CP^\\

+ IHI + / ' H*Ho,P
(4-8)

For this purpose construct w G # o ( ^ ) satisfying Au> — (^ in Q, and thus by (4.5), (6,ip) = A(9,w)

=

A(6,Rhw)

= -(6t+pt,Rhw)+

/ B(0(s),.R ft t<;)ds. Jo

By t h e assumption j4p, 1 < p < oo, a n d t h e inverse property, one can estimate B(9(a),

Rhw) = B(0(a), flhu; - «;) + (0(a), B*w)

, where t h e index p' = p/(p — 1) maybe very close t o 1. In order t o study t h e upper bound of H i ^ H N using t h e following Sobolev imbedding theorem for 1 < p' < q < 2, there are

\\Rhw\\ < C\\Rhwh,q < CHIi,,,

- <\ +K q

2

d

(the latter inequality is t h e boundedness of Ritz projection operator R^ in W1'*, 1 < q < oo) and 1

IMk
1

1

for _ < - + - , p

q

t h e n for \ + ^ > p- (i.e. 1 < d < 4), we have

HiihHI < c\\w\\2iP, < cy\\0,P., and

d

4.2. Full Discretization

59

Therefore (4.8) follows and then the proof is complete by applying Gronwall inequality in (4.8) and the estimate ||#t|| in Lemma 4.2. Remark. By using the imbedding estimate ||0|U-(n)
for 2

and the superconvergence estimate ||0||i = 0(h ), l|0||L~(n) < Ch2\ ]nn\i

for

d = 2, one can obtain d = 2.

This is a simpler and more efficient way to get the maximum norm estimate. For more refined estimates in L°°(fi), see Chapters 8 and 9.

4.2

Full Discretization

We now turn to fully discrete schemes. Let k be the step-size in time, tn = nk, n = 0,1, • • •, N = T/k. Let fn = / ( t n ) , fn~^2 = / ( t n _ 1 / 2 ) and An{u, w) = A(tn;u,w). Denote by Un G 5^ the approximate solution and by dtUn = n n l (U — U ~ )/k the backward difference quotient of Un. Two classes of fully discrete schemes are studied below. Backward Euler Scheme (BES): find Un G S? such that f (BtUn,v) + An{Un, v) = QMB{u, v)) + ( f \ v), \ U° = u0h in fi, ra= 1,2,3, ■■-,

v G# ,

u [

Q, }

*

where the left rectangular rule n-l

fny(s)ds

Qi(v) = Z > t e ) « Jo

3=0

Unn is not included in Q™. is used to discretize thee Volterra intergral term and U n C r a n k - N i c o l s o n Scheme (CNS): find U € S? such that I (dtUn,v) + An-7(Un-v,v)=Qt *(B(U,v)) + (fn-?,v), \ U° = u0h in fi, n = l , 2 , 3 , - ••, where Un

x 2

l — (Un + Un

x

(4.10) )/2 and the following trapezoidal rule is used:

n 2

~

QTHV)

h

= ky£(y(ti) + v(tJ+i))/2 + 7;y(tn-i) z

3=0 t

= o2/(*o) + k Y^y(tj) 1

v e S?,

7=1

~ / ^o

2

2/(«)rf5.

60

Chapter 4. Semidiscrete and Fully Discrete Schemes

Un is included in both An~2 and Q£~*. In addition, the second order difference scheme of second order accuracy will be discussed in Chapters 8 and 13. In order to study the error estimates of the fully discrete approximate solution, one needs some auxiliary tools. We begin with the error estimates of quadrature rules. The (Left) Rectangular Rule: n-l 3=0

Noting that the error in (tj-i,tj) f3

y{s)ds - kyfa-!)

= fJ

Jtj—i

is expressed by j

y'(T)dTds=

Jtj _ i Jtj—\

f'

(tj - T)y'(r)dT,

Jtj—i

hence the quadrature error is of first order accuracy, Qi(y) = \Qi(y) - f " y(s)ds\ < k f Jo Jo

\y'(r)\ds.

(4.11)

The Trapezoidal Rule: n-l

Ql *(y) = *X](y(M + y(*i+i))/2 + | y ( * n - i ) « / " J

Z

j=0

\{s)ds.

°

In an interval (—/,/), there is j

y{s)ds = (y(s)s - y'(s)(s2

- Z2)/2)|<_, + \ j \ s

2

-

l2)y"(s)ds

= W) + y(-0) + \ J J** - i2)y"(s)ds. Hence ftj+1 k 1 / y(s)ds - - ( y f o + i ) + y(tj)) = -

ftj+l (s - tj)(s -

and the error of trapezoidal rule is of second order accuracy,

tj+1)yn{s)ds,

4.2. Full Discretization

61

i y(s)ds\ fnf'tnh~v{s)ds\

n 92 "*(y) = Q2^(y)<£""*(») = I!<£"*(»)-

< \ j

Jo h. Jo \y"{s)\ds + ^j

< \ j

\y"{s)\ds + ^j

1.2

/"tn-i


rtri_\

^\y'{s)\ds

~*\V\8)\d8

f {\y'(s)\ + \y"(s)\)ds. \y"(s)\)ds.

(4.12)

Jo Jo

'

T h e Simpson's Rule: — 7 n 11

pt-i /"£2

B Q3 (l/) = -»»5E (>»f( *o2)i ) + 44^(%+i) l / ( ^ + l ) ++» (w(*«+2))« *2i+2))« / QT(y)

""»(*)<**• y(fi)^-

Denote the error in a typical interval (-&, fc) by £(3/) and let P2(t) be the quadratic interpolant of y(t) such that e = y(t) - i^(t) = 0 at t = -fc,0,fc, then e(y) = f / c(y)

V(s)ds - |(y(fc) + 4y(0) + y(-fc)) V(s)da

e(t)dt= -k

/ e(i)dt+ / JO

e(i)dt e(t)dt.

J—k

Using integration by parts and choosing the appropriate constants, we have €{y) €{y)

= =

[" » £ _ 2 + 1 fc2)dt + / ° e " (t)( f! + lkt f" ee / /((tt )) (( ^ _ 2 fct f° e„{t){tl + lH Jo 2 3f c t + 16k2)dt + J_ 2 3 k 1

t

,

4

3

2 2

+

+

lk2)dt 6lk2)dt

4x (4), N

1 /[ \ ^3*4 ~ 3 + 6 f c2i 2 4^-k)y f K) l (t)dt ^ = 72 3 i K) = 72 / ^ * ~ +6fc -k)y (t)dt + _L 8fe3 + 6 6fc 2t2 __ fc4k)2/^4) (4){t)(t)dt^ fc2t2 + JL // °° (3 (3tt44 ++ gA;t 3+ Hence the quadrature error is of fourth order accuracy, n 4 ql 932n{y) (y) = = \Qf(y) I0§"(») -- f/ "" y{s)ds\ »(*)d«l < < ^^ // ' "" |y< |y ( 4 )>(*)|^. (^)l^-

(4.13)

L e m m a 4.3 (discrete Gronwall inequality). Assume that w n > 0, / „ > 0 and that for n = 0,1,2, • • •, yn > 0 satisfies Vn < fn +

^UjVj, 7=0 7=0

62

Chapter 4. Semidiscrete and Fully Discrete Schemes

then for any TV > 0, N-l

N-l

VN < IN + ^

exp( ^2

n=0

Vj)ujnfn

j=n+l

and N-l VN

< exp( V Ui) max fn. *■—' i=0

0
Proof. Let n n

CTn

z — e~

n

UJ

^2 jyj^

where

z~X = 0,

<jn = ^ ^ j ,

i=0

.7=0

then n-l zn

_ zn-l

= e-°nu)nyn

+ ( 1 _ g^n) £

^

j=0 n-l

< e~aTiu;nyn - Un^Ljjyj

< e~Gllujnfn,

j=o

and N

N

j-0

n=0

By assumptions of the lemma, N-l VN < fN

N-l U

+ J2 j=0

M

N+

= f

N 1

e^-'z ~

< f

N +

Y , n=0

e^-'-^LJnfn,

thus the first inequality of the lemma follows. In order to get the second conclusion, one needs only to apply a simple inequality successively, N-l

N-2

1 + ] P U)n e x p ( > i V - l - <7n) = 1 + &N-1 + J ^ Un e^P(aN-l n=0 n=0 N-2 1

< e ^ " (1 + ] T un exp(°N-2 ~ (Tn)) < n=0

< eaN~l.

~ &n)

63

4.2. Full Discretization

We now return to analyze the error of fully discrete schemes and take W(t) = Vhu(t) G S^, the Ritz—Volterra projection of u(t), as a comparison function. Rewrite

un - un = (un - wn) + {wn - un) = en + Pn, where pn — Wn — un has the following estimates IIP^II* < Chr-i(\\un\\r+

fn\\ur\\ds), Jo

1 = 0,1,

and

^ E ii^nn= £ 11 / tow ^ Chr / Kiir^n=l

n=l

Jtn-i

JO

What remains is to estimate 6n = Un — Wn. First consider the backward Euler scheme (BES). Theorem 4.4. For BES, there is \\Un - U(tn)\\

< Chr(\\uo\\r

+ Ck f\\\u\\2 Jo

+

/ Jo

Tl

\\ut\\rds)

+ \\ut\\2 + \\utt\\)ds.

Proof. At t = tn, (4.1) is of the form f « , v) + An(un,

v) = j " Bn(u(s), v)ds + (/», v),

[ u(0) =

ii.

UQ

in

v € S?,

Subtracting it from (4.9) and using the projection W(t), 9n = u^ — un satisfies f (dt6n,v) + An(9n,v) = QUB(e,v)) + (rn,v), \ 6° = u0h-Rhu0 in fi, n = 1,2,3,-••,

v e SP\

, K

where rn = r™ + r j 4- r j and r" = —dtpn was estimated as before, while r j = u? - a t u n , Obviously we have

and

r j = V fcSJW - / " BJJW(a)da =
M\\=\\*?-dtunw< r \\uu\\ds. Jtn-l

. '

64

Chapter 4. Semidiscrete and Fully Discrete Schemes

Recalling \\BhW\\ < C\\AhW\\ < C\\W\\2, by the left rectangular rule, there is K l l < ck f\\\W\\2 Jo

+ \\Wth)ds

< Ck f'\\\u\\2 Jo

+ \\ut\\2)ds.

So we have N

= kJ2 \K\\
r

\\ut\\rds + Ck [ T\\\u\\2 + K | | 2 + \\utt\\)ds

= 0(hr + k). By the energy argument, it can be shown that N

ll*"ll<W°ll+*£ini)-

(4.15)

ra=l

For this purpose, taking v = 0n in (4.14) and noting that (5t9n,9n)

= (\\9n\\2 -

(9n-\9n))/k

= 77r[2||0"ll2 - 2{en-\en) + w^-H* - ll*?"-1!!2] 2k n 2

we haveldt\\e

\\ + hdte\\2 + An{Bn,en) = Y^kBn(e^en) + (rn,en), n 2

2

n

n

n

nj=o -l

and then3*H0 H 4- -lld + 0ll 4- A (9 . Q ) =

n-l 3=0

Eliminating the first term on the right-hand side and summing over n, it becomes

ii^ii2+A:^iini2
n=l j=0

n=l

4.2. Full Discretization

65

The second term on the right-hand side can be cancelled by Gronwall inequal­ ity, and then (4.15) can be derived by N

N

n=l

n=l

\\eN\\2 < c\\e°\\2 + ckJ2 \\rn\\\\0n\\ < c(\\e°\\ + \\r c*n\\)£ KID suP \\n\. n
Finally, the desired result follows from (4.15) and the estimate of pn. Remark. The error Qn consists of: (i) ||0°|| + \\dtpn\\ = 0(hr), and (ii) \\dtun-u?\\ + the quadrature error ||
n

\\ut\\rds)

Jo

+Ck2 [ T\\\u\\2 + \\uth + 1 M b + \\uttt\\)ds. Jo Proof. Let t = tn_1/2 in (4.1), then (ur1/2,v) = / ^ ^ Bn^2{u{s\ Jo

An-1/2(un-1/2,v)

+

v)ds + ( r " 1 / 2 , v),

v€

S?.

IT - u(tn) = (Un - Wn) + {Wn - u(tn)) = 0n±

pn ,

Write here pn can be estimated as before. Following the standard argument (see also the monograph for purely parabolic case by V. Thomee [150]), 9n satisfies ( (5t0n,v) + An-*(0n-i,v) = Qn(Bn~H^v)) I { 6° = u0h - Rhu0 in ft,

+ (rn,v),

where rn = J2j=i r^ •> r? =

-dtpn,

rj = - ( 5 t t i n - u r 1 / 2 ) ,

v £ S*, (4.16)

66

Chapter 4. Semidiscrete and Fully Discrete Schemes

and r j = Qr1/2(Br1/2W)

- ftTl~l/2 Bl~l/2W{s)ds

=

q^1/2{BhW).

Jo

By the midpoint formula and the trapezoidal rule, we have *IKII = II /

W

< Ck2 [ " \\utuWd8,

Ms) - ut(tn_1/2))ds\\

Iks II
\\uuhds,

Jtn-i

IK I = \\qr1/2(BhW)\\ < Ck2 AlMla + ||«t||2 + ||«„||2)da Jo

and then r/l

=

fcV||rn||


7L

\\ut\\rds + Ck2

Jo

n=l

[ (||Ti||2 + ||tit||2 + ||Titt||2 + | h t « | | ) d s .

Jo

Now setting v = Q71'1!2 = (6n + <9n~1)/2 in (4.16) and noting that (Stfl n ,fl n - 1 / 2 ) = ^ ( | | f l n | | 2 - | | » | | 2 ) , there is ||^||2_|| (9 n-l||2

+

2 / c z y || ( 9 n-l/2||2

n-1

2

< cife ^ii^+^iiiii^-^iii + CAii^iiii^-1/2!! n-l

< fci/H^-1/2!!? + c&2 ] T ||0*+1/2||? + Cifcii^ini^-1/2!! Eliminating, summing and using the Gronwall inequality, we get

ll^ll2 < c\\eY + ckJ2 IkKini + ll^1/2ll), n=l

and then

jv

nfl ii
n=l

The theorem follows from the estimates of 9°, rn and pn.

4.3. The Lumped

4.3

Mass

67

Method

The Lumped Mass Method

We recall t h a t in Section 4.1 t h e semidiscrete finite element approximation Uh{t) € 5£ c a n ^ e expressed in t h e form Nh

where {(f}^h is t h e basis of S% and the coefficient a(t) = {aii(£), • • •, a.Nh(t)} satisfies t h e system of first-order ordinary integrodifferential equations Ma'(t)

4- Aa(t)

- / Ba(s)ds

= F,

a(0) = 7.

Jo In general, the mass matrix M = [rriij] is not diagonal and hence the solution of this system is a cumbersome task. T h e lumped mass method is to replace the mass m a t r i x M by a diagonal matrix M. T h e diagonal elements of M are Nh

ma = ^jTmij,

i = 1,2, ■■■,iV/l,

j=i

i.e. group all t h e mass on each row into its diagonal element. Hence it leads to a new system of integro differential equations, ~Ma'(t) + Aa(t)

= [ Ba(s)ds

+ F,

a(0) = 7.

(4.17)

Jo Now it is easy to find the inverse of M. T h e algorithm can be mathematically explained as follows: A simple quadrature rule is applied to the mass matrix, namely, on each triangular element r with vertices P r j , j — 1,2,3. Consider the following q u a d r a t u r e formula with second order accuracy, 3 1 f Or(/) = d r | X ; / ( P T | i ) « f{x)dx, 6

3=1

^

and t h e corresponding approximate inner product in S%,

(¥>,V0ft= Z ) Q T ( ^ ) . T€Jh

L e m m a 4 . 4 . If

,V0fc - (y», V) < CA 2 || v VIIII'VV'II.

(4.18)

Chapter 4. Semidiscrete and Fully Discrete Schemes

68

The lumped mass method denned by (4.17) is equivalent to the following version:

{

(uht,v)h+A(uh,v)=

B(uh{s),v)ds

+ (f1v),

ueSj

^ ^

UH(0) = uoh in ft. Then 0 = uh — Vhu G S% satisfies

I (Ouv)h + A(0,v)= \

0(0) =

f B(0(a),v)d8-(pt,v)

+ eh(Vhu,v),

v € S?

uh(0)-Vhu(0).

(4.20) The proof is similar to that in Section 4.1. The lumped mass method for purely parabolic case was proposed by P. Raviart (1973). Chen and Thomee [25] (1985) proved the following error esti­ mate of optimal order

IK - «|| + h\\ v K - «)|| < ch2(\\uoh + lk|| 2 + {j ikll!
RHUQ)

|| v M O - Rhu(t))\\ < Ch2(\\ut(t)h + ( jf*(IKIIl + IMI?)^) 1/2 ). Further, when / = 0 and all angles of the triangles r € Jh are acute, then for the semidiscrete scheme there exists the following maximum norm principle \Wh{t)\\Loo^

<

\\uoh\\Loo(ny

The fully discrete schemes are also discussed. These facts show that the lumped mass method has better numerical stability and some monotonity. The weakly nonlinear parabolic case was also discussed by Nie and Thomee [123], and the maximum norm estimates were studied by Thomee, Xu and Zhang [158]. For the lumped mass method of PIDE, Y.P. Lin and T. Zhang [98] studied the semidiscrete and fully discrete schemes for weakly nonlinear case and ob­ tained similar error estimates. Recently, Pani and Peterson [127] exhaustively discussed linear cases with smooth and nonsmooth data by the energy method. For example, when / = 0 and the initial value UQ € H2 f] JF7Q(Q), they proved that, for the semidiscrete solution Uh(t) £ S^-, there are \\uh{t) - «(«)!!,• < C 7 i 2 - ^ - ( 1 + ^ 2 | k | | 2 ,

3 = 0,1,

4.3. The Lumped Mass Method

69

and \\uh{t) - u(t)\\LOO{Q) < Cetf-H-^uoh,

e > 0,

n

and, for the fully discrete solution U G Slf, \\Un - U^Wi

< C(h2-H-1'2

+ k)t-^2\\u0\\2,

j = 0, 1,

and \\Un - u(t„)IU~ <

C£(h2-H^+C\lnk\kt^2)\\u0h.

Chapter 5

Saving of Storage In this chapter the full discretization of PIDE will be further discussed, but we shall focus on the time discretization of the Volterra integral term by various quadrature rules. Consider an abstract integrodifferential equation (IDE) in a Hilbert space H, while PIDE in both space and time will be discussed in Chapter 6 later. The operator method will be a principal one, though the energy method will also be used. Consider the time discretization of the following problem, iH + Au = I B(t, s)u(s)ds + / = Bu{t) + /,

t e J = (0, T],

,g ^

u(0) = i£0, where A is a self-adjoint positive-definite but not necessarily bounded operator with dense domain D(A), and B is a time-dependent operator on H with domain D(B) C D(A). Assume always that B and all D\B (derivatives with respect to s) are dominated by A, i.e. \\A-\DiB)g\\

< C\\g\\.

Here and later all the constants C are uniformly bounded with respect to h in the case of the space-discretized equation.

5.1

Economic Backward Euler Scheme (EBES)

Consider the following economic backward Euler scheme (EBES), UU

~^n

+ AUn = Qn(BU) + / n , 71

n>0,

U° = ti 0 ,

(5.2)

Chapter 5. Saving of Storage

72

where Qn(y) is a quadrature rule with weights cjnj as follows: 71-1

n

Q (y) = YJUJnjy{tj) « /

rtn

y(s)ds.

J

3=0

°

Following Sloan and Thomee [147] (1986), we assume that the quadrature weights Ljnj are dominated in the sense that there exists ujj such that n

\vnj\ < Vj

f°

r

0 < j < n,

with ^2^j

< C

for

tn £ J.

^ < 3 < ™> ~ ^,

tn G J,

(5.3)

If the dominated weights u>n also satisfy n

] P l^rn+lj ~ Wm,j\ < ^h

(5.4)

771=1

then one says that the quadrature rule has persistent dominated quadrature weights. The simplest quadrature rule is the (left) rectangular rule, with unj — A:, for 0 < j < n — 1, which is obviously persistent dominated, and is consistent with the accuracy 0{k) of the EBES. A drawback of this rule is that, unless B(t, s) has a special structure, to compute Un one must use and thus store all previous values of C/-7. Hence a vast amount of storages and computations will be needed. More precisely, during the evaluation of UN all the solutions computed must be stored. This causes a major obstacle in practical calcu­ lations. In order to reduce the storage significantly, a new idea proposed by Sloan and Thomee [147] is to employ the quadrature formulas with higher or­ der truncation error, so that a larger stepsize, or fewer quadrature points, may be used without losing the order of accuracy of the scheme. Their results are generalized by N.Y. Zhang [168, 169] later. The Modified Trapezoidal Rule. For the basic time stepsize k, let tn = nk and m\ = [A;-2], where [x] denotes the largest integer less than or equal to #, and set k\ = m\k to be the larger stepsize and £' = jk\. Apply the trapezoidal rule with stepsize k\ in [0, t[] and the rectangular rule with stepsize k to the remaining part [£{,£n]j i.e. introduce the following modified trapezoidal rule:

« n (y) = y l > ( * ; + i ) + * E 3=0

j=lm\

y(ti) = Q2(y) + Qi(y)~

/'"»(«)&»• (5.5) °

5.1. Economic Backward Euler Scheme (EBES)

73

An upper bound of the storage for this rule is given by Smax = T/k\ -\-mi = 0 ( / c - 1 / 2 ) . Let ujj = u'j + u'!, where J- = k and u'j = fci if j = 0(mod mi) or ujj = 0 otherwise, then it can be shown that \unj \
n—1

W

T

Y1 J^

I

and

Y^ui-Ylki^tf3^CT-

3=0

3=0

j=l

Thus the Uj are the dominating weights. Lemma 5.1. Let Qn(y) be the modified trapezoidal rule (5.5) and qn{y) be the corresponding quadrature error, then \\qn(y)\\
■ Jo JO

(\\y'ss(s)\\ ( (\\y' +

\\y':s(s)\\)ds.

Proof. By the definition and error expressions in Chapter 4, Qn(y) = (Q5(y) - / ' y{s)da) + (Q?(y) - / " y(s)ds)

= [lM*)y"(s)ds+

[nMs)y'(s)ds

Jo

Jt'

and thus t'

\\qn{y)\\
t

\\y"{s)\\ds + Ck / " \\y'{s)\\ds Jt\

< Ck■,f\\\y"{s)\\ Jo

+ \\y'{s)\\)ds.

The Modified Simpson's Quadrature Rule can be similarly considered so that the number of quadrature points can be further reduced to 0(k~1^4). As an example one can first use Simpson's rule on subintervals of stepsize /cx = 0(/c 1 / 4 ), the number of such subintervals in [0,£n] is 0(Ar _1 / 4 ), the length of the remaining subintervals are less than 0(A: 1 / 4 ). On the remainder, the trapezoidal rule with stepsizes first fc2 = 0(k1/2) and then k% = 0(fc3//4) are used. Now the remaining subintervals are of length 0(/c 3 / 4 ) < A:3 and here we use the rectangular rule with stepsize k. The error of the combined rule is then of O(k) and the storage requirement is reduced to 0 ( / c - 1 / 4 ) , but higher regularity is required, i.e.

\\qn(y)\\
i=i

Chapter 5. Saving of Storage

74

We now turn to the stability of EBES. Theorem 5.1 (Sloan and Thomee, 1986). Assume that B is dominated by A and the modified trapezoidal rule (5.5) is used to EBES, then n

\\u^
for

tn
3=0

Proof. (5.2) can be written in the form Un = EkU71-1 + kEk{Qn(BU)

+ /n),

U° =

Wo,

(5.6)

where Ek = (I + kA)~ exists as a bounded operator on H because of the positive definiteness of A. In fact, since the eigenvalues of A are bounded away from 0, there is \\Ek\\ < 1. It is convenient to write Ek = r{kA), where r(A) = (1 + A) - is a rational function. Note that Ek is an approximation of the semigroup E(k). Using repeatedly the iterative scheme (5.6) with initial value U°, there are n

un = E%U° + kYJK~j~\Qj(Bu)

+ fj)

and

fcllE^"1^

lit/1 < II^H +

{BU)\\ + kJT\\f3\l

3=1

3=1

Exchange the order of summation in the second term, it becomes

rh = jri%-i-iQi(BU) = ^ r

i _ 1

E^BJUi

^iEnk-^A{A-'BW%

=E E and hence

n—1

n

k\\rh\\
WK^'^MWW'I

j=i+l

By a spectral argument it can be shown that

E ii^rj'_1^n < SUP E ^r-^x < s u P - ^ - = i.

(5-7)

5.2. Economic Crank—Nicolson Scheme (ECNS)

75

Thus from (5.17),

3=1

j=l

The theorem thus follows by the discrete Gronwall inequality. From the above discussion on stability one can immediately derive the following theorem on error estimate. Theorem 5.2 (Sloan and Theomee, 1986). Assume that B,Bt and Bu are dominated by A, and \\utt\\> \\Bu\\, ||Bi6t||, and ||-Bn£t|| are integrable on J. If in the EBES (5.2) the modified trapezoidal rule (5.5) with kx = Oik1/2) is used, then \\Un - u(tn)\\ < C(T)k for tn < T. Proof. Let en = Un — u(tn). following error equation:

k

From (5.2) and (5.1) it is easy to derive the

+ Aen = Qn(Be) + rn,

n > 1,

e° = U° - u0,

where rn is the truncation error,

and it can be shown as before that N

t*

kJ2\\rn\\
n=l

°

Now the desired estimate follows from Theorem 5.1. If the solution u is more smooth, one can directly adopt the following ECNS.

5.2

Economic Crank—Nicolson Scheme (ECNS)

We now turn to the economic Crank—Nicolson scheme (ECNS):

(

jjn _ jjn-1

k U°=u0,

jjn + A

,

jjn-1

2 n=l,2,3,...

=Q"{BU)

+ r-*,

(5.8)

Chapter 5. Saving of Storage

76 where

QU(y) = yZ^njVitj)

« /

y(s)ds,

which is of second order accuracy. The Modified Simpson's Rule proposed by Sloan and Thomee is stated in the following. Taking rrii = [A;-1/2] and the time stepsize k\ = m\k = Oft1/2), t'j = jki. Let I = l(n) be the largest integer such that t'2l = 2lk\ < tn — nk, and split the interval of integration into [0, £2Z] U [£2Z,£n_i] U [£n-i»*n-i/2]- We shall use Simpson's rule with stepsize ki in [0,£2Z], the trapezoidal rule with stepsize k in [£2Z,£n_i] and a single rectangu­ lar rule with stepsize k/2 in [£ n _i, £71-1/2] > respectively. The amount of storages is N = T/ki + 2fci/k + 1 = 0(/c~ 1 / 2 ). The modified Simpson's rule is now of the form

Qn(y) = | b t t ) + M%) + M%) + • • + vtei)] + 2 \yfai) + 2^(*2ifci + *) + - + s/(*»-i)] + 2 ^ ( ^ - 1 ) /**2*

« / Jo

/**n-l

y(s)ds -f / Jt'2l

/•«„_£

2/(s)d5 4- / Jtn-i

/■«„_£

j/(s)ds = / Jo

y{s)ds.

(5.9) If I = 0 or n = 2lmi + 1 there would be no contribution from Simpson's rule or the trapezoidal rule respectively, and the first or second square bracket will thus disappear. According to the error expressions of Simpson's, trapezoidal and rectan­ gular rules in Chapter 4, we have

qn(y) = (Qsiv) - f"y(s)ds) + (gj(») - / " " y(a)da) Jo

Jf„ n U

HQiiv)-

f ~ \{s)ds) Jtn-i

= f 2'Ms)y{4\s)ds JO

+ fn Jt'2l

1

f"~i/2\pi(s)y'(s)ds,

^2(s)y"(s)ds+ Jt„_!

and thus, by the imbedding estimate \y'(t)\ < C / 0 (\y'(s)\ + \y"(s)\)ds, \\qn(y)\\ < Ck\ [ " \yW(s)\ds + Ck2 [ " " \y"(s)\ds + Ck / ' " " * \y'{s)\ds JO

Jt',

■/*„-!

5.2. Economic

Crank— Nicolson Scheme ft'

< C{k\ + k\kx + fc2) /

77

(ECNS)

4

V

= 0(fc 2 ).

\yU)(8)\ds

Hence t h e consistency of E C N S is ensured. Rewrite equation (5.8) in the following form Un = EkU71'1

+ kGk{Qn(BU)

+ /"-*),

(5.10)

where Ek = (I + kA)-1^

- ^kA)

Gk = {I + kA-1)

=

=

r(kA),

g{kA),

and Gk = - i ( £ f c - J)i4" with notations r(A) = ( l - A / 2 ) / ( l + A/2),

and

5 (A)

= 1/(1 + - A ) .

It is easy to see t h a t \\Ek\\ < 1, ||Gfc|| < 1, b u t t h e current E% no longer has the smooth property as the one in E B E S . There is an interesting difference between t h e current one and t h a t in E B E S : because r(A) takes negative values for A < 2, the sum of r°(A) becomes an alternative series for large A. In addition, r(A) —> — 1 as A —► + o o , and therefore t h e estimates obtained from replacing LUJS by their upper bounds are no longer useful. For example,

is unbounded for A > 2. But the essence of the m a t t e r being t h a t the summa­ tion of r J (A) involves the substantial cancellation when A is larger. For this purpose Sloan and Themee have proposed the "persistence" property (5.4) of q u a d r a t u r e rule (which says, essentially, t h a t within each of the two ranges of n, t h e coefficients cunj for fixed j are independent of n). N.Y. Zhang [168] has further improved t h e arguments so t h a t they are also valid when B is an arbitrary time-dependent operator. T h e o r e m 5 . 3 (Sloan and Thomee). Assume t h a t B and Bt are dominated by A, then t h e E C N S (5.8) with the modified Simpson's rule (5.9) yields t h e following stable estimate n

II^II^CCTKII^II+fc^H^-ill) for tn
78

Chapter 5. Saving of Storage Proof. Iterating (5.9) with the initial value U° we have n

jjn

=

E nn o

+

j j k^2E^ Gk{Q {BU)

fj~i)

+

and hence ||C/n|| < ||tf°|| + k\\ f^ErJGkQi(BU)\\+kJ2

ll/ j -*ll-

(5-11)

What remains is to estimate the second term. Applying the equality kGk = — {Ek — I)A~X and summing by parts,

F(U) = kJ2E%-jGkQl(BU) = -J2(K+1~J ~ K'nA-'Q^BU) 3=1

3=1

n

= ~^2Ek~3^_1(Qj+1

- Qj)(BU) - E^A~lQ1{BU) + A~1Qn{BU)

3=1

= F\ + F% + -F3. By the stability of £"£, since B is dominated by A and the quadrature rule has the dominated weights, we obtain immediately IIjy < \\u10A-lB(t1/2M)U»\\

< Ca,0||£/0||

and 11*311 = II £ A- 1 a; n i B(t n _ i l t i )^'ll < c £ W j - | | ^ | | . j=0

,7=0

To estimate i*i, split it into three terms as follows:

j= l

2=0

z=0

n-1

3= 1

- ] T E^A-1 3=1 n—1

(f>,- + 1 ,< - ^ ) 5 ( t i + i , ti)U*) i=0 j—1

-7YJEnk-iA-\Y,UJ^B^+hM) 3=1 i=0

~ ^(*i-i.*i))^') = A + h + h-

5.2. Economic Crank—Nicolson Scheme (ECNS)

79

Similar to the estimation of F2, one can derive the following bounds

||/i||
and

l|/3|| = l l E ^ - M - 1 ' ^ ^ /ti+* Btfatjdrirw j=l

1=0

j=l

7

*i-i

i=0

Changing the order of summation and using the assumption that the quadra­ ture rule has the persistent dominated weights, we also have n—l n—2

11*211 = II £ i=0

£

EZ-lfaw-w^A-iB^um

j=i+l

n—l

n—2

n—l

< c £ ( £ k-+M-^l)ll^ll
i=0

These estimates lead to n-l

||F 1 ||<||/ 1 +J 2 + / 3 || i ||t/ i ||. i=0

Finally the desired stability estimate follows from (5.11) by the discrete Gronwall inequality. The error estimate for ECNS with the modified Simpson's rule can be derived from Theorem 5.3 as follows. Theorem 5.4 (Sloan and Thomee). Let u and Un be the solutions of (5.1) and (5.8) with the modified Simpson's rule (5.9) respectively. Assume that B, Bt and D\B, 2 = 1,2,3,4, are dominated by A. Then for 0 < tn < T,

||t^-ti(t n )ll
[tnJ2\\Di(Bu)\\ds)-

J

°

j=2

n

Proof. Let u = u(tn). From (5.8) and (5.1) at t — £ n _i/2, the error en = Un — un satisfies the following equation + AeU

+

f

= Qn(Be) + r n ,

n > 0,

e° = 0,

Chapter 5. Saving of Storage

80 where rn is the truncation error, ,n

n

r = (Mtn-1/2)

~

U

oin—^ >

' ^ ^ ' )

~

qn{Bu)

-

Since kfZ\\rn\\
5.3

Additive Schemes

Another way of saving storage was suggested by Y.Q. Huang [84]. Let KBu = / Jo

K(t,s)Bu(s)ds,

where the operator B is independent of t and s, and the kernel K(t,s) smooth enough. If

is

m

tf(M)

= $>;(*)^(s),

(5.12)

then K is called a degenerate kernel. In this case the Additive Backward Euler Scheme (ABES) is of the form dtUn + AUn = £ [ U° = U0,

4>j(tn)Qi(^BU)

+ fn,

(5.13)

n=l,2,3,---,

where the quadrature rule satisfies n-l

Qi(y) = kJ2y(tj) = Qr\y) + M*n-i) ~ / "y(s)ds J

i=0

<>

I Q? = o. The Additive Crank—Nicolson Scheme (ACNS) is of the form m n

n

WtU + AU ~i [ U° = U0,

=^

Mtn-^QWjBU)

n=l,2,3,---,

+ /»"*,

(5.14)

5.3. Additive

Schemes

81

where the trapezoidal rule is of the following form

<

Q2 (y) = -^y(o) + k J2 y(u) = Qr\v) + *y(*n-i)« / " _ i y(s)ds, l i=o

Jo

[ QUy) = ~y{0). Note that in the above two schemes the quadrature rule has an additive property. This property determines the advantages of the algorithms as fol­ lows. To compute the solution Un one only needs m + 2 levels of storage, namely two levels to store Un~~x and Un , and m levels to store the sum Q^i^jBU), j = 1,2, • • •,m. In order to solve t / n + 1 , one only needs to add a new term kifjj(tn)BUn to the cell where Qn(ipjBU) is stored. If K(t, s) is a smooth function in t and s, 0 < s < t < T, it can be approximated by the piecewise interpolant of degree m— 1, 771— 1

Km(t,s)

= ] T K^t'^P^s),

0<s
3=0

where t'- — jL/(m — 1). When K(t, s) is sufficiently smooth (in particular, if it is analytic) and I Z ^ i f (£, s)\ < M for all m > 2, then the remainder satisfies jm

\rm\ = \Km-i(t,s)

—M. ml One can choose the integer m large enough such that jm

—M ml

= 0(ka),

- K(t, s)\ <

a = 1 (for ABES) or 2 (for ACNS).

As an example, one can choose m = 0(|lnA:|), because m! « v 2 m ( m / e ) m . But, in general, K(t, s) would not be so smooth. For example, if \D™K(t, s)\ < MinO<s
a - l o r 2,

Chapter 5. Saving of Storage

82 is added to the remainder, and then *El|rJ||
7

||£u||
°

The following error estimates are obtained by Y.Q. Huang [84](1994). Theorem 5.5. Assume that A~1B is bounded in H and that ||w«||, ||£^|| and \\But\\ are integrable in [0,T], then for ABES (5.13), \\Un-u(tn)\\
tn
Theorem 5.6. Assume that A~1B is bounded in H and that ||wm||> \\Bu\\ and \\Butt\\ are integrable in [0,T*], then for ACNS (5.14), \\Un - u(tn)|| < C ( 7 » A ; 2 ,

£n
From these results one can see that the regularity requirements on the solution are much weaker in comparison with the ecomomic schemes where ||2?Mtt|| (f° r EBES) and ||£i&tttt|| (f° r ECNS) are also required to be integrable in[0,T]. The differences between the economic schemes and the additive schemes are stated as follows: Firstly, the integral / KBuds instead of the individual U3 is stored, Jo hence the required storage can be reduced in several ways provided that K(t,s) is smooth. Secondly, instead of quadrature rules with higher order, lower order quadra­ ture rules matching the accuracy of the scheme are employed, hence extra regularities of the solution are not required. The Huang's idea was used by Meclean, Thomee and Wahlbin [119] to discuss the following evolution equation with positive memory ut+

Jo

K{t-s)Au(s)ds

= f(t),

u(Q) = u0,

where A is a self-adjoint positive-definite linear operator in a Hilbert space H, and the kernel K is positive-definite. In particular, the weak singularity of K(t) at t = 0 can be admitted, for example, K(t) — t~a, 0 < a < 1, will be studied in Chapter 13. In addition to the two algorithms mentioned above, when the kernel is of convolution type, there are also other two remarkable algorithms for saving

5.3. Additive

Schemes

83

storage and computation, i.e. the convolution quadrature techniques (see Lubich [107,108]) and the fast inverse Laplace transform (see Y. Yan, Fast inverse Laplace transform for some time-dependent problems with memory, preprint, Dept. of Math., University of Kentucky, Lexington, 1995).

Chapter 6

Cases with Nonsmooth Initial Values Consider the following homogeneous PIDE, (l

( ut+Au=

B(t,s)u(s)ds

= Bu(t)

in

fixJ,

J=(0,**]

(6>1)

u = o on an x j , [ u(0) =

2

u0£L {ty,

where A is a symmetric positive-definite elliptic operator of second order with coefficients independent of £, and B is an arbitrary partial differential operator of order (3 < 2, and the initial value u0 € L2(Q) is nonsmooth. Results about the semidiscrete and fully discrete approximations exhibited here were presented by Thomee and Zhang [156,157] .

6.1

Regularities of the Solution

First recall that in Chapter 2 it was proved that the solution for the homoge­ neous parabolic differential equation (2.10) with the nonsmooth initial value UQ G L2(Q) has the following estimates of regularity \\DtAjE(t)u0\\

< Ct-^WuoW

for

ij

> 0, t > 0 ,

(6.2)

where u = E(t)uo is the solution of (2.10) and E(t) is the analytical semigroup generated by A. These results can be generalized to the PIDE (6.1), but the 85

86

Chapter 6. Cases with Nonsmooth Initial Values

index j , which measures the differentiability with respect to x, will be greatly limited. By using £"(£), (6.1) can be written as an equivalent form u(t) = E(t)u0 + / E(ts)Bu(s)ds Jo /o = E(t)uo + E * Bu(t) = E(t)u0 + Gu(t).

(6.3)

Now w = Gu(t) = u — E(t)uo satisfies wt + Aw = Bu = Bw + BE(t)u0,

w(0) = 0,

(6.4)

and by Duhamel principle, (6.4) is equivalent to the following Volterra integral equation w{t) = E* (Bw + BE(t)u0)

= Gw + F(t),

F = E* BEu0,

(6.5)

where G — E * B is a Volterra integral operator on C(H2) (see Lemma 2.4) and \\AGg(t)\\
[ \\Ag(s)\\ds for Jo Lemma 6.1. For UQ G L2(Q) there are

geL\j;H2).

(6.6)

\\BEu0\\
(6.7)

and \\AE*BEuo\\


(6.8)

1

Proof. Using E = —A~ DtE and integration by parts we have BE(t)u0=

/ Jo

B(t,s)(-A~1DsE(s))u0ds

= -B(t1s)A~1E(s)u0\t0+

5 s (^5)A~ 1 £ , (5)w 0 cZ5

/

= -B(t,t)A-1E(t)uo-B(t,0)A~1uo-\-

/

J B s (t,s)A"

1

E(5)w 0 ^.

Since B and i? s are dominated by A, (6.7) holds. To show (6.8), similarly one can write AE*BEu0

= / AE(t-s) Jo

/ Jo

B{s)r)E{r)uQdTds

6.1. Regularities of the Solution

87

= BE(t)u0

l

- / E(tJo

s)B(s,s)E(s)u0ds

_

t

E(t - s)BtE(s)u0ds

o

= n + r 2 -f- r 3 .

By (6.7) ri is bounded as desired and similarly so is BtE(s)uo As for r 2 , using the smoothness property (6.2) of E(t), that rt/2

ft

{t - s)-1\\A-1B(s,s)E{s)u0\\ds

IN| < C /

^ C /

\\AE(s)u0\\ds Jt/2

JO

rt/2


and hence r%.

rt

(t-s)-1\\u0\\ds

+ C / s^WuoWds^ C\\u0\\. t/2 Jo Jt/2 The proof of Lemma 6.1 is complete. Theorem 6 . 1 . Assume that B(t,s) and all its derivatives with respect to t and s are dominated by A, then (6.1) has a unique solution u G C( J; L2) P| C°°(J]H2) satisfying \\D\Aju(t)\\

< Ct^-i\\uQ||

for

i > 0, j = 0,1, t G J,

and for u>(£) = Gn, there is \\D\Ajw(t)\\

< C^-^Hwoll

for

i > 0, j = 0,1, z + j > 1.

(6.9)

Proof. By (6.2) and u = E(t)u0 + w, the first conclusion follows. It remains to show (6.9), which means that w(t) has a weaker singularity at t — 0 than E(t)uo has. Start from the case i+j = 1. From w = E*B(w + E(t)uo) — Gw -f- F(t) and (6.6), we have \\Aw(t)\\ < \\AGw\\ + P F | | < C /

| | ^ ( S ) | | ^ + CHiioll,

Jo

and by Gronwall Lemma, \\Aw{t)\\ < C\\u0l

(6.10)

i.e. (6.9) holds for i = 0, j = 1. Note, in particular, that ||5u(t)|| = \\B(E(t)uo+ w(t))\\ < C\\u0\\. From (6.4) we have ||iyt|| < \\Aw\\ + \\Bu\\ < C\\u0\\, and therefore (6.9) for i + j = 1 is proved. We shall now carry out the proof of (6.9) by induction for i + j = n > 1 and assume that the results are valid for m < n. Begin with the case i — ra,

88

Chapter

6. Cases with Nonsmooth

Initial

Values

,7 = 1. In order to estimate D™Aw(t) = D™AE * Bu we shall first prove the following L e m m a 6 . 2 . For g appropriately smooth and n > 0, with C = C n , we have tn+1\\D?AE*g(t)\\
+ C

fs^g^+'Hs^ds. J°

P r o o f . Since rt/2 /»t/z

E * g(t) =

rt

E(t-

s)g(s)ds

+ / E(t Jt/2

E(t-

s)g(s)ds

+ / Jo

J0 rt/2

s)g(s)ds

rt/2

= Jo

E(s)g(t

by straightforward manipulation, using AE(t) coefficients C n j ,

- s)ds = Pg(t) +

= —Et(t),

n

D?APg(t)

Rg(t),

we have, with some

ft/2

= Y^CnjE^(t/2)g^-^(t/2)

Here, using t h e smoothness of E(t), tn+1\\EW(t/2)gln-»(t/2)\\

-

/

E^^(t

-

s)g(s)ds.

there are, for the first term,

< Ctn+1^\\g^'^(t/2)\\

< Ctsup(S^'||p(^)(5)||) s
and for t h e second t e r m , rt/2 tn+l||

/

^ + l ) ( t _ s ) p ( s ) ( i s | | < C t SUP ||^(5) ||. s
JO

In addition, there are rt/2

ARg{t) and

=

/ Jo

rt/2

-Es{s)g{t-s)ds

= -E(t/2)g(t/2)+g(t)-

/ Jo

E(s)g'(t-s)ds

6.1. Regularities of the Solution

89

The first two terms on the right-hand side are similarly estimated, while the last term satisfies t n + l | | ^ ( n + l ) ( t ) | | = t „+l||

j

E{s)g(n+l){t

_

^

JO


(t-8)n+1\\gln+V(t-8)\\d8

(s)||cfe. Jo

Thus the proof of the lemma is complete. Now return to the proof of (6.9) for i = m, j = 1. As a result of Lemma 6.2, recalling w = E * Bu, we have m

~ t \\AD?w(t)\\
1 r* + C- / ^ p ^ W * ) ) ! ! * t Jv (6.11) f ~ . r and B[j)g(i) = / B^j\t,s)g(s)ds, it can be Jo

m

Setting Blj\t,s)

= D3tB(t,s)

shown that i-i

\\D{(Bu(t))\\ < CJ2 \\AuW(t)\\ + \\B^u(t)\\

for 0 < j < m + 1 , t G J. (6.12)

i=0

In fact, since DtBu(t)

= B(t,t)u(t)

+ B^u(t), hence

£>^ U (t) = i p r 1 ^ , *)«(«))+i?r 1 (s t (1) U (t)) = £r>j(B t ( i - 1 - i ) (t ) tMt)) + Bt0)«(*) 2=0

= E E ( j ) ^-,^-1-*)(*>t)A-1(A«<'>(t)) +B^u(t). i=0 1=0 ^

/

From our assumption, the estimate (6.12) is obtained. Using the induction assumption, (6.12) and an analogue of (6.7) imply that 3-1

tj\\D3tBu(t)\\

< Ctj ^ i=0

\\AuW{t)\\+tj\\B{tj)u{t)\\

< C\\u0\\ for j < m, (6.13)

Chapter 6. Cases with Nonsmooth Initial Values

90 and similarly

tm+1\\D™+1Bu(t)\\

+Ctm+1\\AD™u{t)\\

< C||uo||

tm™WII < C\\uo\\ + C f

sm\\ADrw(8)\\ds.

Jo

The desired estimate for AD™w follows by Gronwall lemma. The correspond­ ing estimate for D^^w is now derived from wt = —Aw -f Bu and (6.13), and then the proof of the theorem is complete. The second aim in this section is to discuss the optimal regularity of the solution u(i) with respect to x. For the purely parabolic case (i.e. B = f = 0, see (2.14) in Chapter 2), there is \E(t)uo\p+q
for

u0eHq,

p>0,q>0.

(6.14)

In the case of homogeneous PIDE (6.1), we know, by Theorem 6.1, that this estimate holds for p < 2 (i.e. for j = 0 , l ) . Thomee and Zhang [156] (1989) proved the following sharp regularity results for j3 < 2. T h e o r e m 6.2. Assume that the initial value UQ € L2(Q), then (6.1) has a unique solution u{t) e C{J; H4~P f| H2) and \Ht)\\x

< C£- A / 2 ||uo||

for

A < 4 - /?, t G J,

here the index A = 4 — (3 is optimal. To prove the theorem one needs a series of auxiliary lemmas. To begin with, recall the expression u = E(t)u0 + w, where w = Gu satisfies a Volterra integral equation w = Gw + F, (6.15) with G = E*B

and

F = GEu0 = E * BEu0. 2

It will be shown that for UQ 6 L (Q) this Volterra integral equation has a unique solution w(t) G C(J; H4~p f| H2) and that |K0I|4-/3
for

tel.

(6.16)

In view of the well-known estimate (6.14) for E(t)uo, Theorem 6.2 will follow. Note, in particular, that the term w(t) does not have the singular behavior of E(t)uQ at t = 0.

6.1. Regularities of the Solution

91

First note the following equalities

Dt(E*f)=

[ E'(t-s)f(s)ds

+ f(t)

Jo

and f E'(t-s)f(s)ds = Jo Using the inverse operator T = and integration by parts, there

[ E(t-s)f'(s)-f(t)+E(t)f(0). Jo A'1, E(t - s) = -TDtE(t - s) = TDsE(t - s) is an important Thomee—Zhang formula:

/ E(t - s)f(s)ds Jo

s)E(t - s)f(s) - TE(t - s)(f(s)s)'}ds

= \t f[(tJo

+

Tf(t).

(6.17) Lemma 6.3. For I = p + q € [0,4 - /?], p > 0, q > 0, and f(t),tf'(t) e Hi(fl), then [\t-s)-^2\f(s)\qds Jo

WEtfm^C \\E * f(t)\\t

if

0
(6.18)

< Ct-1

[\t - sf-^dfis)], + \sf(s)\q)ds Jo +C||/(t)||i- 2 if Z > 2 , 0 < p < 4 .

(6.19)

Proof. (6.18) directly follows from (6.14). (6.19) can be obtained from (6.17), (6.14) and the regularity of A " 1 / (cf. Chapter 2). Lemma 6.4. The operator G = E * B is bounded in C(J; H2), with \Gg(t)\2
[ \g(s)\2ds Jo

for

geC(J-H

2

Furthermore, for g G C(1\H2) there are Gg{t) G C(J;H4) Gg(t) G C(J\ H3 0 H2) for /3 = 1, and \\Gg{t)\\i-p
for

). for (3 = 0 and

t G 7, £ < 2.

Proof. Note that ||#|| 2 < C\\Ag\\ < C ^ f o r g G H2. The first conclusion is a result of (6.19) where / is replaced by Bg and taking I = p = 2, # = 0, namely \Gg(t)\2 < Ct-1

[ (\Bg\0 + \sB'sg\0)ds + C\Bg\o Jo

< Csup(|B 5 |o + \sB'sg\o) < C [ \\g\\0 < C [ \g\2ds. s
Chapter 6. Cases with Nonsmooth Initial Values

92

This also shows the second conclusion for (5 = 2. For the case /3 = 0, the operator B is simply a multiplication by a scalar function, and Bg € i? 2 , we have, by (6.19) with p = q = 2, that \Gg(t)\4 < Ct-1

[ (\Bg\2 + \sB's\2)ds + C\\Bg\\2 Jo

< Csup(\Bg\2

+ \sB'sg\2) + C\\Bg\\2 < C /

\g\2da.

JO

s
For the case (5 = 1 (note that Bg ^ 0 on 0ft), by (6.19) with p = 3, q = 0, there is IIGfllls < C-i- 1 A t Jo < Ct'1

S)-

1/2

( | S 5 | o + | S S ^ | o ) ^ + CIIBflH,

f (t- s)-1'2 JO

[' \\ghdrds + C\g\2 < Csup \g(a)\2, Jo

s
and the lemma is proved. We now consider the term F = E * BEu0 in (6.15), which cannot be estimated directly from Lemma 6.4. L e m m a 6.5. If u0 € L2{Q), then / = BEu0 £ C(J; H2"^). In particular, / G C( J; H2) for (5 = 0. Furthermore, there are H/Wlln-zs + l l t / ' W ^ - z J ^ q i i i o l l

(6.20)

ll/(*)ll + l | « / ' ( t ) l l < ^ 1 ^ / 2 | l « o | | .

(6.21)

and

Proof. Using E(t) = —TDtE(t) fit) = -B(t,t)TE(t)u0

and integration by parts we have + B{t,0)Tuo

+

BsTEuo.

Hence, by the boundedness of the operator T : 1? —» H2, it leads to II/WH2-/3 < C7sup||TS(a)tio|| 2 < C\\u0\\. S
When (3 = 0 and (3 = 1, directly, there is ||/(i)|| < \\BEuoW < C f Jo

\\E(s)uo\\0ds

< C [ s-P^WuoWds < C ^ - ^ H u o l l . Jo

6.1. Regularities of the Solution

93

For the case 0 = 0, B(t, s) is a multiplication by a scalar function and hence BE(t)u0 G H2 for t e l . Since Dtf = B(t,t)E(t)uo -f BtEuo, we also have t||A/H2-/3 < Ct(\\E(t)uo\\2 + | | A ^ 0 | | 2 - / 3 ) < C | M I and t | | A / | | < Ct(p(*)«o||/J + \\BtEu0\\) <

Ct'-^Wuol

By these estimates, the theorem is complete. Lemma 6.6. For u0 G L2(Q) there are F(t) = £* * JB£U 0 G C(J; # 4 _ / 3 ) for /? = 0 and 0 = 2, and F(t) G C(J; if 3 f| # 2 ) for /? = 1. Furthermore ||F(t)|| 4 -/3
for

tel,

0<2.

Proof. For 0 = 0 and 0 = 2, / - £ £ u 0 G C(J;H2~P), Lemmas 6.3 and 6.5, that \\F(t)\U-f> < Ct-1

f\\f(a)\2-p Jo

+ \8f'{8)\2-P)da

+ C\\f(t)h-P

we have by

< C||«o||.

In the case /5 = 1 it can be obtained similarly, by Lemmas 6.3 (with p = 3 and q = 0) and 6.4, that ||F(t)|| 3 < Ct'1

f\t Jo

- s ) - 1 / 2 ( | | / ( s ) | | + \\sf'(s)\\)ds + C\\f(t)\\i < C\\u0\\.

These complete the proof of the lemma. Proof of Theorem 6.2. Consider the Volterra integral equation w = F -h Gw. Since by Lemma 6.4, the operator G is bounded in C(J; H2) and by Lemma 6.6, F G C(J;H2). By a standard argument for Volterra integral equation, it can be concluded that this equation has a unique solution w G C(J]H2) and \w(t)\2
By Lemmas 6.5 and 6.6, we can get the regularity estimate (6.16) for w, i.e. IH0IU-/3 < \\F\\4-p + HGHI4-/3 < K | | + Csup \w(s)\2 < C\\uo\\. s
Now Theorem 6.2 follows from u = E(t)v + w(t) and the estimate of E(t)uo. The following results will be useful in studying the error estimate for semidiscrete schemes.

Chapter 6. Cases with Nonsmooth Initial Values

94

Theorem 6.3. Let u(t) be the solution of (6.1) with UQ G L2(Vt), then ||BM(t)||2_/3 + ||5ttt(*)|| 2 -/3
for

tel,

||5ti(t)|| + \\Btu{t)\\
for

teJ.

and Proof. Since Bu = BEu0 + Bw, the bound for Bu follows easily from Lemma 6.5 and (6.16). Btu can be estimated similarly. This completes the proof. Thomee and Zhang gave an example to show that the result of Theorem 6.2 is sharp in the sense that, for the general operator B of order (3 < 2, regularity with order higher than H4~@ cannot be attained for UQ G L2(Tt). The problem that whether D^~^u(t) can be further differentiated with respect to t is still open. For example, as a special case, there is the unproved estimate: ||DJTx(t)||4-/3 < C t - ' - ^ - ^ H t i o l l

6.2

for some

i > 0.

Error Estimates for Semidiscretization

We begin with the purely parabolic case (i.e. B — 0 in (6.1)) and its semidiscrete approximation u^it) G S^ satisfying {uh,t,v)+A{uh,v)

= 0,

v€S£,

uh(o) = u0heL2(n).

,R99^ [0 ZZ)

'

First recall the following important result (see Thomee [150]). Theorem 6.4. Let u be the solution of (6.1) with B = 0 and uo G L2(Q). Let uh{t) G S^ be the solution of (6.22) with u0h = Phu0 G S£. Then \\uh(t) - u(t)\\ < Chrt-^2\\u0\\

for

t G J.

Proof. Using the solution operators T = A"1 and Th — (Aft) -1 , (6.1) with B — 0 and (6.22) can be written in the forms Tut + u = 0 and

Thuhyt +uh = 0.

Obviously the error e = u^ — u satisfies Thet + e = -p,

6.2. Error Estimates for Semidiscretization

95

where p — (Th — T)ut = (Th — T)Au = Rhu — u and Th is positive semidefinite in L2(Q), then e) + \\e\\2 = -(p, e) < ±\\p\\* + i||e|| 2 .

\Dt(The,

After integration in t it becomes [ \\e\\2ds<(The(0),e(0)) Jo

(The(t),e(t))+

+ / Jo

\\pfds.

Note that The(0) = 0 when uoh = PhU0. In fact, for w G L 2 (fJ), Thw € 5^, there is {The(0),w) = (e(p),THw) = 0. Then it gives the following estimate:

f \\efds < f \\p\\2ds. Jo

(6.23)

Jo

Next, from the equality (Theuet)

+ ^Dt\\e\\2 =

-(P)et),

we have Dt\\e\\2 < -2(p,et)

= -2Dt(p,e)

+ 2(pt,e),

or by multiplying t, A(i||e|| 2 ) < -2Dt(t(p,e))

+ 2t(pt, e) + \\ef + 2(p, e).

After integration with respect to £, we have t||e|| 2 < 2t|H|||e|| + A | | e | | 2 + 2||p||||e||+2«||p t ||||e||)da. Jo By (6.23) we can further get

IMI2 < C(\\pf + ± j T ||p||2 + s2\\pt\\2)ds,

(6.24)

or for any e > 0, ||e«||2 < j f

s2\\pt\\2ds + C£(\\p(t)\\2 + \ £

\\p\\2ds)

96

Chapter 6. Cases with Nonsmooth Initial Values

and therefore \\e{t)\\<esup(s\\pt(s)\\)+C£Sup\\p(s)\\. S
(6.25)

S
By the approximation property of Ritz projection, \\p(t)\\ = \\Rhu - u\\ < Cfc||u(i)||i <

Cht-^Wuol

and by definition of norm in if 1 , we have

f\\p{t)\\2ds
Jo

Jo


f\\u\\\ Y,XMs),fj)2ds

/ Jo

j=1

ft

°o

< Ch2 / Jo

VA^e-^wo,^)2^ i=i

oo

OO

< CT^A^o,^)2 /

-2A7s,

3= oo

Similarly, there is / s2||/>t||2d* < Ch2 f s2\\ut\\\ds < Ch2 I s2\\u\\2ds Jo Jo Jo OO



2 s2-2A e~27^sds

(uo^A'M■ J

3=1

°

< C / i 2 ^ ( u o , ^ ) 2 = C^||n0||2. Thus by (6.24) a preliminary estimate is obtained, Wem^Cht-^Wuol

(6.26)

The theorem will be proved below by an iterative argument. Denoting by Eh(t) the solution operator in (6.22), i.e. Uh(t) = Eh(t)uoh — E ^ ^ P ^ o , then e(t) = uh(t)-u{i)

= Fh{t)u0

and

Fh(t) = Eh{t)Ph -

E(t).

6.2. Error Estimates for Semidiscretization

97

Our aim is to prove \\Fh(t)uo\\
(6.27)

2

In view of the boundedness of Fh(t) in L (Q,), without loss of generality, one can suppose ht~xl2 < 1. We are going to prove the following equality Fh(t) = Fh(t/2)E(t/2)

+ E(t/2)Fh(t/2)

Fh(t/2)2.

+

In fact, by definition and property of semigroup, the right-hand side is (Eh(t/2)Ph

- E(t/2))E(t/2)

+(Eh(t/2)Ph 2

= Eh{t/2) Ph

-

+ E(t/2)(Eh(t/2)Ph

-

E(t/2))

2

E{t/2))

- E{t/2f

= Eh{t)Ph - E(t) =

Fh(t).

Note that both Fh(t/2) and E(t/2) are self-adjoint, then E{t/2)Fh{t/2) is adjoint to Fh,{t/2)E(t/2), and they have the same norm in L2(Cl). This gives \\E(t/2)Fh(t/2)u0\\

= ||Fh(t/2)JE7(t/2)tio||

r

< C/i^- r / 2 ||wo||-

< Ch \\E(t/2)u0\\r By (6.26), we have ||F fc (t/2) 2 «o|| <

Cht-V2\\Fh{t/2)u0\\,

and then \\Fh{t)uQ\\ < Chrt-^2\\u0\\

+Cht-1/2\\Fh(t/2)u0\\.

Apply repeatedly this formula, and note that ht~xl2 < 1, we get \\Fh(t)u0\\ < Cft r t- p / 2 ||uo|| +C(ht-1'2y\\Fh(t/2r)u0\\

< C7^-r/2||Uo||.

The theorem thus follows. In general (cf. Bramble, Schatz, Thomee and Wahlbin [9]), H^hW^oll < Chn-(q-pV2\u0\p,

0
(6.28)

We now turn to the PIDE (6.1) and its semidiscrete approximation Uh(t) € 5^ satisfying

{

uh,t + Ahuh = / Bhuh(s)ds uh{0) = uQh.

= Bhuh(t),

, g 2Qs

98

Chapter 6. Cases with Nonsmooth Initial Values

The main result in this section is the following Theorem 6.5. Let u and Uh G S> be the solutions of (6.1) and (6.29) respectively, and B be a partial differential operator of order /3, (3 < 2, UQ G L2(Q) and u0h = ^i^o G 5^, then \\uh(t) - u(t)\\ < Chxt~X/2\\u0\\

for

A<min(4-/?,r).

Obviously, in view of Theorem 6.2, the power A in this theorem is the best possible. For exhibiting the main idea of the proof, B is assumed to be independent of t and s. By DuhameFs principle, from (6.1) and (6.29) we have u(t) = E(t)u0 + / E(tJo uh(t) = Eh(t)Phu0

s)Bu(s)ds,

+ / Eh(t Jo

s)Bhuh(s)ds,

and then e(t) - uh(t) - u(t) = (Eh{t)Ph + / Eh(t - s)Bhuh(s)ds Jo = Fh(t)u0 + / F h (* Jo

E(t))uo - / Jo

E(t-s)Bu(s)ds

s)Bu{s)ds

+ / Eh(t - s)(Bhuh{s) Jo

-

PhBu(s))ds

= e0-hei-he2,

(6.30)

where eo = F/j,(£)uo is estimated as before, i.e. ||eo|| <

Chrt-r'2\\u0\\.

Set e = ei 4- e 2 , below we shall prove, by estimating ei and e2 separately, that ||e(t)|| < Ch4-e\\u0\\.

(6.31)

It means that the contribution of e to the error does not exhibit any singularity at t = 0. Together with the estimate for eo, this will complete the proof of Theorem 6.5. The proof of (6.31) will be based on a sequence of lemmas.

6.2. Error Estimates for Semidiscretization

99

To avoid the singular behaviour of Fh(t) = Eh{t)Ph — E(t) at t = 0, the following self-adjoint operator Hh(t) : L2 —> HQ is introduced: Hh(t) = ThEh{t)Ph

- TE(t) = Eh(t)Th -

= ThFh(t) +

TE(t)

(Th-T)E(t),

which can be understood as an integral of — Fh(t) since - / Fh(s)ds = - / (Eh(s)Ph Jo Jo

-

E(s))ds

= [ (ThE'h(s)Ph - TE'(s))ds Jo = ThEh{t)Ph - TE(t) - (ThPh - T) = Hh(t) -

Hh(0),

and Hh(0) =

Th-T,

i.e. H'h{t) = -Fh{t). Lemma 6.7. There are the following two estimates:

\Hh{t)uo\-q < C / i ^ - ^ - ^ I K H ,

l<
and \\Hh(t)u0\\ < Ch*\\u0\\2

for

tel,

r> 4.

Proof. From the approximation property oiTh — T and the boundedness of E(t) in L 2 (Q), we have \(Th-T)E(t)u0\-q

< Ch*>\E(t)u0\p_q_2 < ChHl-^-q^2\\u^\l

q>-l,p-q>2,p
For purely parabolic case what remains is to bound the term ThFh(t)uo = T^eo, where CQ = Fh(t)uo is the error of semidiscrete projection. Introduce the following discrete negative norm (see Thomee [150], Chapter 6) \\v\\-a,h = {Tshv,v)^2,

(for v £ 5r^ also for s = - 1 )

and the corresponding inner product (viw)-8th

= {Tshv,w).

100

Chapter 6. Cases with Nonsmooth Initial Values

It is proved, for 0 < s < r, that \\v\\-s,h < C(\\v\\-a + hs\\v\\),

||«||_, < C d M U f c + h'\\v\\).

Which shows that, with the exception of a very small error, HuH-,^ is equiva­ lent to ||f ll-s- Thus we have, for 0 < q < r — 2, \The0\_q < C(|T h eo|_, >h + Cft«||Tfceo||), < C(|co|_,_ 2 ,fc+C/i fl ||eo||_2,fc) < C|eo|_,_ 2 + Cft«|eo|_2 + C7i«+2||eo||, and for q = —1, |T h e 0 |i = [A{The0,The0)}1/'2

(AhTheo,Theo)^2

=

= (e 0 ,T h e 0 ) 1 / 2 = |e 0 |_i, h < C(|e 0 |_i + h\\e0\\). For any
0
whence |eo(*)|-i < Chh-V-WWuoW,

for

0 < i < j < r.

Together with the above three estimates, this shows iTHFhWuol-g^Ch^-b-M^uoll

for

1 < q + 2 < p < r,

which completes the proof of the first conclusion. To prove the second conclusion note that Theott + e0 = p0 = ~{Rh - I)E(t)u0,

T%e0,t + The0 = -ThPo,

and hence, by (6.25) for TheQ instead of eo, (6.26) and the negative norm estimate of (R^ — I)E(t)uQ, we have \\The0\\ = |e0|-2,/k < Csup(s|^(s)|_ 2 j / l +

\p0(s)\.2yh)

s
< C s u p ( S | ^ | _ 2 + |po( S )|_ 2 ) +C/i 2 sup( S ||p( ) ( s )|| + \\Po(s)\\) s
s
< Ch4sup(s\E'(s)u0\2 S
The proof is complete.

+ \E(s)u0\2)

<

Ch4\u0\2.

6.2. Error Estimates for Semidiscretization

101

L e m m a 6.8. For the integral Fh*g{t)=

I Fh(t - s)g(s)ds Jo

for

t e 7,

there are \\Fh*g(t)\\
1 = 0,2, r> 1 + 2,

S
\\Fh * g(t)\\ < Ch3(sup Wgtfh +1~^2 sup(||
r > 3,

S
and \\Fh*g(t)\\i

< ChSup(\\g(s)\\+s\\g'(s)\\),

r > 2.

S
Proof. By the analogue of (6.17) and Fh — —H'h, we write tFh * g(t) = [ (t - s)Fh(t - s)g(s)ds + / sFh(t Jo Jo = tHh(0)g(t)

s)g(s)ds

+ / [ ( * - s)Fh(t - s)g(s) - Hh(t Jo

s)(sg(s))'}ds.

The first conclusion now follows immediately from Lemma 6.7 and (6.28) (note tha.tHh(0) = Th-T). Similarly, for r > 3 it becomes \\Fh * g(t)\\ < C/i3{||s(*)lli + 1 - 1 f\t

- sy^iWgisn

+ s\\g'(s)\\)ds}

Jo

< Ch3(\\g(t)\U +t-V2sup(\\g(s)\\

+ s\\g'(s)\\)),

S
(the last conclusion follows in the same way) where we have also used the fact that

WFnWgWi < Ckt-iyW,

r>2,

geL2.

To show this, note that eo = Fh{t)u0 = (Eh(t)Ph

- RhE(t))u0

+ (Rh - I)E(t)u0

= 60 + Po

and by the inverse property, ||0o||i < Ch-'PoW

< Ch-^Heoll + Hftll) < Cht-^luoll-

The desired result follows from \\po\\ < Ch\\E(t)u0\\2 < C/ii _1 ||wo||-

102

Chapter 6. Cases with Nonsmooth Initial Values We are now in a position to estimate e\ in (6.30). Lemma 6.9. For e\ — Fh * (Bu)(t) there are ||ei(*)|| < Ch*-?||u01|,

p<2,

and

lleiWIIi^C-ftlluoll,

/3 = 2.

Proof. When (3 = 0 and /? = 2 , by Theorems 6.2 and 6.3 we have \Bu(a)\2-p

+ s\{Bu)'(s)\2_p

< C\\u0\\ + s|3(s, sMa)| 2 _/j

+ s | 5 t u ( 5 ) | 2 _ / 3 < C||uo|| + Cs\u(s)\2 + 5 | S t w ( 5 ) | 2 ^ < C||u 0 ||. The first conclusion then follows from the first estimate in Lemma 6.8 with I = 2 — p. For P — 1 the result follows similarly by the second estimate in Lemma 6.8 and Theorem 6.3. The second conclusion is a consequence of the third inequality in Lemma 6.8. We now turn to the third term e2 G S% in (6.30), and we need only to bound (e2,i>) for v £ S^. By definition, (Eh{t-s)(Bhuh(8)-PhBu{s)),

v)

= {Bhuh{s) - PhBu{s), Eh(t - s)v) = /

J5(e(r), Eh{t -

s)v)dr,

Jo

and hence, since e(t) — Fh(t)uQ + e(t) and from (6.30), (e2(t),v)= / / Jo Jo + / I Jo Jo = e2i{v)+e22(v).

B{Fh(r)u0,Eh(t-s)v)drds B(e(r),Eh(t-s)v)dTds (6.32)

For the purpose of estimating e2\, define the following functional

Bi(f,g)=

f f B(f(T),g(t-a))dTd8,

Jo Jo

and denote simply f(t) = J0f(s)ds. Lemma 6.10. There are | £ i ( / , 0 ) | < Csup(|/( 5 )|,3_ fc + s\f(8)\p-k) s
sup(|3f(a)|fc + 8\g(s)\k), s
0 < k < 2,

P = 0 and 2,

6.2. Error Estimates for Semidiscretization and

103

|£i(/,s)l < Csup^^n/^iDsup^1/2!^)!!), 0 = I. s
s
Proof. By integration by parts, we have

tB1{f,g) = t [

Jo

B(f(s),g(t-s))ds

= [ (t-s)B(f(s),g{t-s))d3+ Jo

f Jo

sB(f(s),g{t-s))ds

= [ (t-s)B(f(s),g(t-s))ds+ Jo

[ Jo

B(f(s),g(t-s))ds

+ / Jo

sB(f(s),g{t-s))ds.

For (3 = 0 and 2, 0 < k < 2, note that \B(fi9)\
^ = 0,2,

0
(6.33)

and the first conclusion follows. For j3 = 1 we get.directly

\B!(f,g)\
[ 11/(5)111^-5)1!^

Jo

< Csnp(s^2\\f(s)\\) s
sup((t - sf'2\g(t

- s ) | i ) f (t -

s
s)-l'2s~^ds,

Jo

then the second conclusion follows. Remark. The reason why we are treating (3 = 1 separately is that for (3 = 1, k = 2, the factor | / | - i in (6.33) would have to be replaced by | | / | | _ i , since in general Vg does not vanish on dQ and this last norm is undesirable in our application below. We are now ready to estimate the term e2i in (6.32). Lemma 6.11. There are \e21(v)\
veS^

(3<2

\e2i{v)\
veS?,

13 = 2.

and Proof. First discuss the cases that (3 = 0 and 2. By Fh =

EHPH

— E we

have e21(v) = Bi(Fhu0,

Ehv) = B^FHUQ, Fhv) + Bi(Fhu0, Ev) = e211(v) + e212(v).

Chapter 6. Cases with Nonsmooth Initial Values

104

Since F^v — —Hh(t)v-{-Hh(ti)v, we find, by the first estimates in Lemmas 6.10 and 6.7, (6.27) and (6.33) with k = min(/?, 1), that |e2ii0>)| < Csup(\Hh(s)u0\(3-k 2 (/5 fc)

+ s\Fh(s)uo\p-k)

2 fc

sup(\Hh(s)v\k

+

s\Fh(s)v\k)

4

< Cft " - /i - ||t4o||||t;|| < Ch -P\\uo\\\\v\\. Since E(t)u0 = —TE(t)u0 -f Tu0, we obtain similarly |e 2 i 2 (v)l < Csup(\Hh(s)u0\p-2

+ s\Fh{s)uoif3-2)sup(\TE(s)v\2

+ s\E{s)v\2)

< CTr^KIIIMI. From these estimates it implies the first conclusion. When (3 = 2 since Eh{t)v = —ThEh(t)v + ThV, we also have |c2iWI < Csup(\Hh(s)u0\i + s\Fh(s)u0\i)siip(\ThEh(8)v\1 < Ch\\u0\\ \v\_hh,

+ s|JEk(s)v|i)

where in the last step we have used the following inequality (see Thomee [150], Lemma 3 in Chapter 6) for the case p = — 1 and q = 1 \Eh{t)v\qth
p
(6.34)

When P = 1, by the second inequality in Lemmas 6.10, 6.7 and (6.34), \e21(v)\ < C s u p ^ ^ n ^ ^ ^ i i ) s u p ^ i ^ ) ^ ) < Ch3\\u0\\\\v\\. Therefore the proof of the lemma is complete. Proof of Theorem 6.5. We shall now finish the proof of Theorem 6.5. First consider the case (5 < 1. Recalling the notation of (6.32) and using (6.34), \e22{v)\
Jo

s)v\\pdrds

f\\e{T)\\dr\\v\\.

Hence, using also Lemmas 6.9 and 6.11, we find \\e{t)\\ < || e i (t)|| + ||e 2 (t)|| < Ch^WuoW + C f \\e\\ds, Jo

and the desired result follows from Gronwall lemma.

j3 < 1,

6.2. Error Estimates for Semidiscretization

105

When P = 2, for v G S£, e22{v)=

[ f Jo Jo + /

B(e(r),Fh(t-s)v)dTds / (e(r), £*£(£ - r)v)drds = e 221 + e 222 .

Exchanging the order of integration and integrating by parts, we have e 22 i = /

I

B(e(r),Fh(t-s)v)dsdr

= / B(e{T),Hh(0)v)dTJo Using Lemma 6.7 with p=l, |e 2 2 i| < C f

/ B(e(T),ff fc (t-T)t;)dT. Jo

q = —1, it becomes

||e||i
Jo

s
||e||ida||v||.

Jo

Similarly, |e 2 2 2 |
a
Jo

Thus, ||e22(t>)||
[ ||e||ids + C / \\e\\ds. Jo

Jo

We shall now prove that, when f3 = 2, there is P(s)||i < Cfc||uo||

for

s £ J.

(6.35)

If this is true, then using Lemma 6.9 we can obtain ||e(i)|| < ||e 1 (t)|| + ||e 2 (i)||
f Jo

which concludes the proof of (6.31).

\\e\\ds,

106

Chapter 6. Cases with Nonsmooth Initial Values What remains now is to prove (6.35). Rewrite 622(f) as pt

e22(v) = /

ft

/

B(e(r),Eh(t-s)v)dsdT

= f B{e{r),ThEh{0)v)drJo

f B(e{r),ThEh(t Jo

-

r)v)dr,

whence by (6.34), \e22(v)\
HellidasupllTfc^^ll! < C f

Jo

s
||e||i
\\v\\-hh.

Jo

Applying the second estimate of Lemma 6.11, we have \(e2(t),v)\
f llelUdaJH-Lh, Jo

and hence, by duality, \\e2(t)\\i
fllehds).

Jo

Together with the second estimate of Lemma 6.9, this shows \\e(t)h < Ik!Will + IMOIU < Ch\\u0\\+C

/ \\e\Uds.

Jo Jo

By Gronwall lemma, the proof of (6.35) and hence that of the theorem is now complete. The proof mentioned above appears quite complicated! Can it be simpli­ fied by the Ritz—Volterra projection Vhu(t)l This is an open question.

6.3

Backward Euler Scheme

In this section, the time discretization will be discussed. First consider an abstract initial value problem in a Hilbert space H, ut + Au= Let Un « u(tn),

/ Bu(s)ds = Bu(s), Jo

t e J,

u(0) = u0.

n = 1,2, • • ■, N,

U° = UQ,

(6.36)

(6.36) is approximated by

dtUn + AUn = Qn(BU),

(6.37)

6.3. Backward Euler Scheme

107

where A is a self-adjoint positive-definite, not necessary bounded, operator with dense domain D(A), B is a time-dependent operator with domain D(B) C D(A), the quadrature rule is Qn(y) = V LOnjy(tj) » / " y(s)ds, and Qn(BU)

= Qn{B(tn)U).

(6.38)

Denote the quadrature error in (6.38) by

qn{v) = Qn(y) - f" y(s)ds. JO

Assume that B and all D\B (derivatives with respect to s) are always domi­ nated by A. To analyse the stability and convergence of the numerical scheme, Thomee and Zhang [157] proposed the following assumptions: the quadrature coefficients unj in (6.38) are dominated in the sense that there exists u>j such that n

(i). \ujnj\ < ujj

for

0 < j < n,

with

2^WJ

—^

^ or

^n — **»

j=o

and satisfies n

(ii). ^ujit'1

< C\1nk\ for

tn < t*.

In addition, in order to get the optimal error 0(k\nk) for the discrete solu­ tion t/ n , we further require that the quadrature rule Qn be appropriate for nonsmooth data in the sense that there exists m > 0 such that (hi). \qn{y)\ < Ck\\nk\ if ||£>^(£)|| < Ct~J for tn < t*, 0 < j < m. Later we shall verify that the rectangular rule satisfies the above three con­ ditions. In addition to this, Thomee and Zhang also proposed a new graded grid to treat the singularity of the solution at t — 0 and discussed the corre­ sponding economic scheme. Let Ek = (I + /cA) - 1 and rewrite (6.37) in the following form Un = EkU71-1 +

kEkQn{BU),

and by iteration, Un = E%U° + Sn(U), where

n-l

Sn(U) =

k^EZ-jQj+\BU). 3=0

(6.39)

Chapter 6. Cases with Nonsmooth Initial Values

108

We now recall some important properties of the operator E%. Denote by SP(A) the spectrum of A and r(A) = (1 + A) - 1 . Lemma 6.12 (smoothing property). For n > 1,

ll^ll^n1Proof. It is well-known that (1 + A)n > nX for A > 0, and sup A(l + A)~ n < A/(nA) =

n'1,

A>0

thus \\AE%\\ =

|Ar(A:A)n| < AT1 sup A(l + A)~ n < k^n'1

sup xesp(A)

-

l/tn.

A>0

Recall that E(t) = exp(—tA) is the solution operator for the purely parabolic problem (6.36) with B = 0, and the error of the approximate so­ lution Un of (6.37) with B — 0 can be expressed by Un - u(tn) = E%u0 - E(tn)u0

= F>0,

where F£=EZ-En(k),

E(tn) = E(nk) =

En(k).

Lemma 6.13 (approximation property). For n > 1 and tn < £*, Ili^uoll^CArt^lluoll. n

Proof. Since F (X) = r(X) — e~nX, what we need is to prove that \Fn(\)|


j = l,2,.-.,JV.

Let Co > 0 be a positive number such that |r(A)| < e- C o A

for

0 < A < 1,

then for 0 < A < 1 we have \r(X)-ex\


and

l^n(A)| = lE r ( A r H ( r W-e- A )e- jA | ra-l

x

< C\r{\) - e~ \ ] T | e - C b < n - 1 - r t A e - ' A | < CX2ne-ConX

< C(An) 2 e- C o ( n A V« < C/n.

6.3. Backward Euler Scheme

109

On the other hand, for A > 1, e-

nX

< e~n < C/n,

and there exists a positive number C\ > 0 such that

Thus r n (A) < e~Cin < C/n

for

A > 1.

n

The above estimates conclude that |F (A)| < C/n and the lemma follows by the spectrum analysis as in Lemma 6.12. Now return to PIDE and (6.39). Lemma 6.14. Assume that the quadrature rule Qn satisfies the condition (i), then there is the following stability estimate n-1 J

n

\\S (y)\\
if y° = y(0) = 0.

3= 1

Proof. Using the definition of Qn and exchanging the order of summation, n—1n—1

n

S (y) =

fc££^w?r^(*;+i^y i=0

j=i

n—1n—1

= ^^W;+1,iA£r^"1B(*i+i.W)2=1

j=l

Since B is dominated by A, and by the smoothing property of E%, we have

\\Sn(y)\\ <

fcj;£||a;j+lii^^|||^-1B(ti+1)ti)i/i|| i=l

j=i

<^££^M£riii2/ii 2=1

j=i

n—1

n—1

i

i\\v \\T,

SU

P k\r(k\)n-i,

where n—1

V sup kXr(kX)n~j J ^ AeSp(A)

n—1

< Y] sup Xr(X)n~j < sup A r ^ = 1. ~ ^ A>O A>O 1 - r(X)

Chapter 6. Cases with Nonsmooth Initial Values

110

This proves the lemma. We now show a preliminary error estimate where the precise quadrature rule has not yet been chosen. Theorem 6.6. Assume that By Bt and Bs are dominated by A, and that Qn satisfies the conditions (i) and (ii). Then for the solutions u of (6.36) and Un of (6.37), we have \\Un-u(tn)\\


tn £ J,

l<j
where kYjErJqj+\Bu)

e"{u) = 3=0

is the global quadrature error, and qn(Bu) = qn(B(tn)u). Proof. Let F£ = E% - E(tn) = E% - E(k)n denote the error operator for the purely parabolic problem, and en = Un-u(tn)

= F£u0 + en,

where F£ can be estimated by Lemma 6.13. It remains to show that ||e n ||
max | | e ^ ) | | ,

tn < t\

(6.40)

l<j
With notation (Fkv)n (6.3), en = Un-

(Fku0)n

= F£v and I3 = (tj,tj+i), - u(tn) = Sn{U) + E{tn)u0 -

= Sn(U) - [n E(tn Jo = S (Fku0)

n

u{tn)

s)Bu(s)ds n_x

n

we have by (6.39) and

n

+ S (e) + S (u) - V 3=1

r / E(tn -

Jl

s)Bu(s)ds

*

n-l

= Sn(Fku0)

+ Sn(e) + A; ^

E^~j{Q^\Bu)

-

Bu(tj+1))

3=0 n—1 lb—X

n—1 lb—J.

rn

+ E /

K~\Bu(t3+i)-Bu(s))ds

+ J2

3=0 JlJ n—1 „

+ T2 = Sn(Fku0)

j=0

(E^n-j)

~ E{tn -

p

s))Bu{s)ds

+ Sn(e) + en(u) +r^+r^+

rj.

K^BuWds Jl

J

I111 ll

6.3. Backward Euler Scheme Then by Lemma 6.14 and discrete Gronwall inequality, P I < C max (||S'"(Ffcu0)|| + ||e*{u)\\ + V ||r?'||). \<j
f-'

i =f -l '

The Sn(Fku0) and r? shall be estimated below separately. First, using Lemmas 6.14, 6.13 and condition (ii), n-l

n-l

\\Sn(Fkfc u0u)\\ 0 | | ^^Ck^^tj'WuoW Cfc^^r'KH 0 )|| <


Note that r? are independent of the quadrature rule, and w{t) = ix(t) E{t)uo = Bu(s) has better regularity (see Theorem 6.3), lllDiA'wWll^Ct^-'lluoll lZJJA^tJH^C^-'-^lluoll,

,l, ij ==00,l,

zt >> 0 ,

2-hj t + j > l1..

It can be seen easily that ||r?|| ||B U (Jfc)-Bti( S )||d a +V IKH < /[fkc\\Bu(k)-Bu(a)\\d8 + V1 Jo

j=1

/I

Jlj

| | B \\Bu{h+i)-Bu{s)\\ds u(ti+1)-B«(a)||d5

n—1 n—1 2

< Cfc||«o|| +CA: ^i-1||Uo|| || 1 < CA||«o|| +C/c2^t.7 = ||1Wo

< CAIlnAIHuoll, C*|lnfc|HH,

and similarly IKII < C f c ] T / t-^WBuisWs ||rj||
KCe^j'W^oW j=l < Ck^tJ^uoW


Since for 5 € Ij, \\E(tn_ \\Eitn-j)3) - E(tn - 5 )|| = || f ""~'3 Er{r)dr\\ Jtn-S


Jl^^j-x

r"1 <

Ckt'l^,

thus finally we have HrJH \VnA < < /[

\\(E(k)-E(tn-s))Bu(s)\\ds

+ +

Vt"1,. Ck2Yt-\


J M

Chapter 6. Cases with Nonsmooth Initial Values

112

Theorem 6.7. Assume that B = B(t, s) and its derivatives with respect to t and s are dominated by A, and that the quadrature rule Qn satisfies conditions (i), (ii) and (iii). Then for the solutions u of (6.36) and Un of (6.37) we have \\Un - u(tn)\\ < Ck(t-X + | lnfc| 2 )|K||,

tn G J.

Proof. It remains only to estimate en(u) in Theorem 6.6. By our as­ sumptions and the regularity of solution we know WDKA^Bit,

s)u(s))\\ < COItioll

for

s
Hence, by the condition (iii) and the linearity of e n , ||yrV(2?«)|| = l l g " ^ - 1 ^ ) ! ! < Cfc|lnfc|||«o||, using also Lemma 6.12 and self-adjointness of A, we find ||€»(u)|| = | | f c £ i 4 J ^ - V " + 1 ( i 4 - 1 B u ) | | 3=0 n-1


^CkJ^tn-jkl^klholl j=o

The theorem follows from Theorem 6.6. As an application of Theorem 6.7 we now consider the following fully discrete backward Euler scheme for the PIDE (6.1), dtUn + AhUn = Qn{BhU)

for

n > 1,

U° = Phu0,

(6.41)

where B/^t, s) and its derivatives with respect to t and s are dominated by A^ (see Chapter 3). Theorem 6.7 yields the following Theorem 6.8. Under the conditions of Theorem 6.7, and let Sy be a subspace of piecewise polynomials of degree r — 1, then for the solutions of (6.1) and (6.41) we have with A = min(r, 4 — /?), \\Un - u(tn)\\ < C{hH-x'2

+ k^1

+ \lnk\2))\\u0\\,

tn e J.

The remaining part of this section is to give some examples of the quadra­ ture rules which are appropriate for nonsmooth data and have the advanta­ geous storage properties. A numerical test is cited at the end of this section.

6.3. Backward Euler Scheme

113

a). The Rectangular Rule with u>nj = u>j = k was described in Chapter 4. In addition to (i), the condition (ii) obviously holds, n—1

^cjjtj1

n—1

= ^ T j - 1 < Clnn < C|lnfc|,

j=i

tn £ J.

j=i

In this case we have, using the assumption of (iii) with m = 1, that

\\qn(y)\\<\\knJ2y^3=0

J

[tny(s)ds\\

°

<2k max \\y(s)\\+k 0<s
/

\\yt\\ds < Ck\ In k\,

Jk

and hence (iii) also holds. b). The Modified Trapezoidal Rule mentioned in Chapter 5 is sparser than the above rectangular rule, but nevertheless retains the order of accuracy of the backward Euler discretization when the solution has better regularity. Take the larger stepsize k\ = mk, where m — [A;-1/2] is the integral part of A; -1 / 2 , and set t'3; = jh\, j = 0,1, • • •, / with I the largest integer such that t\ < tn. The rule consists of two parts such that the trapezoidal rule with stepsize &i is applied to [0, tj] and the left rectangular rule with stepsize k on the remaining [£{,£n]. Then, for y smooth enough, \\qn(y)\\ =Ck\\l Jo

\\ytt(s)\\ds + Ck f Jt[

\\yt(a)\\da

= 0(kf) + 0(kk1) = 0(k). The number of levels of Un needed to be stored thus reduces to 0(k^1) + 0{m) = 0{k~ll2), with the same order of accuracy. This rule satisfies the conditions (i) and (ii). However, the main contribution to the bound for the quadrature error is Ck\ I

||y tt ||ds, and since we may now assume

||T/«(5)||

=

Jk±

0(s~2) for small 5, this gives an error bound of order 0(k\) = 0(k1^2), hence with a loss of accuracy. To overcome this drawback we shall therefore present an alternative quadra­ ture rule as follows. c). The Graded Trapezoidal Rule. This rule has locally refined steps near t = 0 and behaves better for nonsmooth data, but still performs well for smooth data. We introduce the sequence of time steps defined by £' =

114

Chapter

6. Cases' with Nonsmooth

Initial

Values

j2k, j = 1, 2, • • •, a n d let I be t h e largest integer for which t\ < £ n - i - Now propose to use t h e basic trapezoidal rule on each interval ( ^ , ^ + 1 ) and then the rectangular rule with time stepsize k on ( t j , t n _ i ) , i.e.

Qn(y) = \ E(^+i " *J)(y(^+i) +y(*J))+ * E v&) « / ' " *(*)*■ 1

Jo

j=l-

3=0

It is easy to see t h a t t h e number of nonzero weights in Qn(y) is n — I2 + /, and since n < (I - f 1 ) 2 this is bounded by 31 + 1 < C / c - 1 / 2 , so t h a t , like t h e earlier modified trapezoidal rule, t h e number of levels of t h e approximate solution U3'(which needs to be stored) is 0 ( / c - 1 / 2 ) . It is easy to prove t h a t this rule satisfies (i) and (ii) with LJJ = t ^ + 1 — tj_1 for j = i2,i > 2, and LJJ = k for other j . In order to show t h a t (hi) holds, we note t h a t

qn(y) = l(y(0) + y(k))- j

y(s)ds

+\ X>;+i - *J-)(y(*S+i) + y(*J)) + k £ v(h) - F y(s)dai j=i2

i=i

whence || g n fo)|| < Ck max||j,( a )|| + ( 7 ^ + 1 - ^ )

3

, max

||y tt ( S )||

ptrt

+k /

||2/ t ||d5.

This yields, using t h e assumption of (hi) with m = 2,

ikn(y)ii < cfc + ^c(t;. + 1 -«j.)3(«i-r2 + cfc f s - 1 ^ , j=i

Jt

i

where £ ( * i + i - *J03(*J-)-2 = * ! > ' + l ) 2 - i 2 ) 3 J - 4 < C f c ^ r i=i

i=i

1

< C7fc|lnfc|.

j=i

These estimates complete t h e proof of (iii). T h o m e e a n d Zhang also considered using Simpson's rule on t h e larger basic intervals Ij = (t^, t ' + 1 ) , where t'j = j 4 / c , j = 0 , 1 , 2, • • •. It requires only

6.3. Backward

Euler Scheme

115

0(/c~ 1 /4) l e v e l s o f jjn t o b e s t o r e d 7 w i t h t h e a c c u r a c y \\qn(y)\\ < CA;|lnA:|. We finally cite a numerical example [157] considering the simple special case when A = -Dl on [0,1], with B = A. Here the integro-differential equation may be differentiated with respect to t to yield the following second order differential equation utt + Aut - An = 0

for

t > 0,

with u(0) = uo,

and

ut(0) =

-Au0.

T h e exact solution may easily be obtained by separation of variables. In this case t h e rectangular q u a d r a t u r e rule may be stored separately, see the addi­ tive scheme in C h a p t e r 5. Choose the initial value u0(x) = 1 in (0,1), for which t h e solution is discontinuous at x — 0 , 1 . As comparison, we have also treated t h e smooth initial value UQ(X) =' — sin7rx which is the first term in the Fourier expansion of above discontinuous initial value. T h e errors e 2 and e ^ are measured in discrete l
Method a b c a b

1 1 1

N 160

320

c a b

c

640

^max

160 17 28 320 32 48 640 40 40

10 4 e 2 119 156 100 60 80 50 30 42 25

Smooth data 4 P2 10 e o o 137 180 115 69 1.0 92 1.0 1.0 58 1.0 35 0.9 49 1.0 29

Poo

1.0 1.0 1.0 1.0 0.9 1.0

Nonsmooth data io 4 e o o 10 4 e 2 92 120 168 914 357 114 96 84 1.0 60 0.6 662 237 1.0 49 60 42 1.0 30 494 163 0.5 31 25 1.0

| Poo 1

1.0 0.5 0.9 1.0 0.4 ! 0.9

It can be seen t h a t the modified trapezoidal rule (b) seriously damages the accuracy reached by the backward Euler scheme, but the graded trapezoidal rule (c) greatly decreases the storage requirement and reserves good accuracy.

Chapter 6. Cases with Nonsmooth Initial Values

116

Remark. Le Roux and Thomee [93] also studied the following semilinear equation ut -f Au = / /(£, 5, x,u(s, Jo

x))ds,

with u=0

on

<9Q,

and

^(0) =

UQ

in

Q,

where A is a self-adjoint elliptic second order operator with time-independent coefficients. If UQ 6 L2 with ||wo|| < R they show that, for any v < 1, \\uh(t) - u(t)\\ < C{v,R)h2vt~\

0
and a substantially better estimate is not possible. For backward Euler scheme they used the modified trapezoidal rule with quadrature points j2k, j = 0,1, 2,... (thus the total number of points used is of the order 0(k~1^2)) and obtained the following error estimate \\Un-u{tn)\\


+ kl')t-l/

for

\\uo\\
In comparison with purely parabolic case, see Johnson, Larsson, Thomee and Wahlbin in Math. Comp. 49 (1987), pp. 331-357 and Crouzeix and Thomee in Math. Comp. 49 (1987), pp. 35-93.

Chapter 7

Cases with Weakly Singular Kernels Consider PIDE of the form ut + Au= < y u(0) = uo

K(t - s)Bu(s)ds + /

in

fJ x J,

J° in

(71) fi,

where A and B are smooth operators independent of t (and 5), and K(t) is a weakly singular kernel such that \K(t)\ < Ct~a

with

0 < a < 1,

teJ=

(0,**].

(7.2)

Always denote by K * / the convolution K*f(t)=

[ K(t-s)f(s)ds= [ K(s)f(t-s)ds. Jo Jo The existence and regularity of the solution of (7.1), and its finite element approximations are considered by Chen, Thomee and Wahlbin [26] (1992). Note that in the case of weakly singular kernel, the regularity of the solution with respect to time is limited, which makes higher order quadrature formulas as well as economic schemes mentioned before less attractive.

7.1

Continuous Problems

We shall need the following two lemmas. 117

Chapter 7. Cases with Weakly Singular Kernels

118

Lemma 7.1. (generalized Gronwall inequality). Assume that y is a nonnegative function in LX{J) satisfying y(t) < F{t) + v j (t- s)-*y(s)ds, Jo where F(t) > 0, v > 0, then there is a constant C = C(t*) such that y(t) < F{t) + C j {t- s)~aF(s)ds, Jo

(7.3)

t G J.

Proof. Let Ki(t) = vt~a for t e J. Recall that the convolution oper­ ator i^i* is bounded in L 1 (J). Denote by K{ the kernel of i times iterated convolution, we have Ki(t) = if i * Kx * • • * Ki = C(z, a ) ^ 1 " ^ " 1 . i times

It is easy to see that Ki * F(t) < CK\ * F(t) for z > 1, t € J. Hence, applying ifi* to (7.3) n times in succession, we obtain n-l

3/W < F(*) + J2 Ki * F ^) + K" * ^(*) < F(t) + CKX * F(t) + Kn * j/(t). For n ( l — a) > 1, we have i f n * 3 / W < C / y{s)ds. Jo Applying the Gronwall lemma, the proof is complete. Lemma 7.2. Let K e LX(J). Then for each e > 0 there is a constant Ce = Ce{\\K\\Lr{J)) such that t*T

nt

T

I/ / K(t-s)f{s)f{t)dsdt\ ' Jo Jo

T

< e [ f(t)dt+Ce ' Jo

[ \K(T-t)\ Jo

t

f Jo

f(s)dsdt.

(7.4) Proof. In this proof, (•, •) and || • || denote respectively the inner product and the norm in L2(J). Using the Cauchy—Schwarz inequality, we have

\(K*f)(t)\2 < (jfV(*)l 1/2 l^(a)| 1/2 |/(*-*)|d«) 2 <\\K\\LHJ)

Jo

ft\K(s)\f(t-s)ds.

7.1. Continuous Problems

119

Integrating with respect to t and changing the order of integration and the variables, \\K * ff

< \\K\\LHJ) j

\K(a)\J

= \\K\\LHJ)

\K(T-T)\

I Jo

f2(t-s)dtds I' Jo

f2(
hence, for the left-hand side of (7.4), we get |(^*/,/)|<||^*/||||/||<6||/||2 + i||ir*/||2

< ell/ll2 +

^WKWLHJ)

£

\K(T - r ) | j f

f(s)dsdr,

which is the desired inequality. In the case of a weakly singular kernel, the regularity of solution of (7.1) is limited. In general, utt blows up as t —> 0. Consider an example = / (t - s)~aAu(s)ds in ft x J Jo u = 0 on dft x J, u(0) = (/) in ft,

ut+Au

where (/) is an eigenfunction of A corresponding to the eigenvalue A. Setting u(t, x) — y(t)<j)(x), for the scalar function y, we have yi + \y = \ [ (t-s)-ay{s)ds, Jo

t > 0, y(0) = 1

and hence y"

= \t~a - Xyf(t) + \ [ (t- s)-ayf(s)ds Jo

«

0(t~a).

Since yi G C ( J ) , we conclude that for this particular function, WuuW^Xt-*

as *-><)

and / \\utt\\pdt
Chapter 7. Cases with Weakly Singular Kernels

120

Theorem 7.1. Assume that u0 £ HT\H%(Sl), f £ C(J;HV~2) and tfft £ L°°(J; Hu~2) with u > 2, 0 < 7 < 1. Then (7.1) has a unique solution u £ C(J) L 2 (0)). Furthermore, u £ C(J; H2 f| H]), ut £ C(J, L 2 ) f| L 1 (J; # 2 n^o^and^GLHJ;^2)The regularity of the solution is proved in an improved way which is slightly different from that used in the paper by Chen, Thomee and Wahlbin. Differentiating (7.1) with respect to t, then z = ut satisfies zt + Az = K*Bz + F(t), z = 0 on fflxJ, z(0) = /(0) - Au0 in HU-2(Q),

(7.5)

where F(t) = fl + K(t)Bu. Let Mun H K I L + 11/(0)1^-2 + s u p | | ^ / / ( s ) | I / _ 2 j with the semigroup E(t) generated by A, the solution of (7.5) can be expressed in the form z{i) = E(t)z(0) + E*K*Bz + E*F, (7.6) for 2 < 6 < v
\\E(t)z(0)\\6 < C&-2-6^2{\\uQ\\v

+ 11/(0)11,-2) <

C&-2-dV*M„n,

and from Lemma 6.2, \\E*F\\S < C

f\t-s)^-2-6V2\\F{s)\\u_2ds

Jo

< C f\t Jo

<

- s)^-2-6V2S-\s1\\fX-2

+ \\*L0\\v-2)d8

C&-6^2-^Mvn.

In the following we shall analyze the operator G = E * K * B. By Dirichlet formula and the Thomee—Zhang formula (6.17), we have Gz=

[ [ E(tJo Jo

s)K(s -

t)Bz(r)drds

E(t - s)K{s -

r)Bz{t)dsdr

t

JO Jr

7.1. Continuous Problems pt

121

pt-T

Jo Jo nt

nt — T

- A«-r)-jf

[\(t-T-Z)E(t-T-Z)K{Z) (t-T-f)B(t-T-OTO

- TE(t Z)(K(t)t)']Bz(t)dtdT + [f/ ' TKlt TK(t - s)Bz(s)ds s)Bz(s)ds i)(K{£)Z)']Bz(t)dtdT s)Bz{s)ds T£(« -T- rT - 0(^K)0']S«(t)dedT Jo and, using Lemma 6.3, pt

\\Gz\\s
Pt-T pt-T

a CaC\\Bz(T)\\ \\Bz(T)\\SS-2dtdT -2dtdT

/ Jo JO

\\Bz{s)\\62-ds + C ff (t- s)-aa\\Bz(s)\\s2ds Jo


aa s)\\z(s)\\ s)\\z(s)\\66ds. ds.

Hence, from (7.6) we can derive a new inequality 1+( 0/2 ||*(t)ll«
+C / Jo

a (t-s)(t-s)-a\\z(a)\\eds \\z(a)\\eds

and, by the generalized Gronwall inequality, it leads to \\z(t)\\ ||*(t)IL S <

Ct-^-^M^. ct-^-^M^.

In particular, (( II \\z(s)||fds) \\z(s)Hfds)1/PV P < < CpM CpM„, for some some pp = = p{v-8)> p{v-8)> r for v>r

1. 1.

Finally, from (7.5) we also get a Ilkt|U-2 N U - 2 < C||z(*)||« + ||F(*)||«-2 ||F(t)|U_2 C\\z{t)\\s + C [I (t - s)a)-a\\z(s)\\sd \\z(a)\\sda S Jo

< CM < CM^(t-^-W vn(t-»W*

+ + |j\t *(t -

8)-8-W-Wda) s)-»s-^-Wds)

a + \\ft\\s-2+Ct\\u0\\S +U/t'IU_ 2 + c*-°iK||s

< cit-i+W2 cit-W-W <

+ t-"+^-*)/2)M £-"+<"-«>/2)MJ/>7 Ct-a||Uo||, J/,7 + ||/ t '|| 7 _2 + Ct-^Wuoll

Ct~xMva,

where A = max(l max(l - (y (i/ - S)/2,7, 6)12,7, a) < 1, 1,

Chapter

122

7. Cases with Weakly Singular

Kernels

and hence

d

)

II // P P

< CvMvn

for

1 < p < 1/A.

These estimates complete the proof of Theorem 7.1. R e m a r k 1. Assume t h a t v > 4 and 7 < a, then by taking 8 = 2, A = a , we have

|M| = |NI < ^""M^ and

tt

\\utt\\pds)

/o

i/p y


for

p
R e m a r k 2. If the initial value -uo, t h e boundary dQ and the free t e r m / are appropriately smooth and further if some compatibility conditions are satisfied, then it is believed t h a t the solution u(t) of (7.1) has the following regularities

ueL°°(j',w2>q(n)), uteL%J',W2«{tt)), ixttGLP(J;L9(n)), where q »

7.2

1 and 1 < p < 1/a.

But so far no proof seems to have been given.

Discretization in Space

We now t u r n to t h e semidiscrete problem of finding Uh (t) G S% such t h a t I

{uh,t,v)

+ A{uh,v)=

K{t-8)B(uh(s),v)ds

+ (f,v),

v G S£,

^

?.

[ uh(G) = UOH, where UQ^ G Sh = S^ is an approximation to ^0 satisfying ||^o/k-^o||i
j = 0,1.

T h e main result is the following T h e o r e m 7 . 2 . Assume t h a t u(t) and uh(t) (7.1) and (7.7) respectively, then \\uh(t) - u(t)\\ < Ch2 (\\uoh

+ /

G Sh are the solutions of

||ut||2ds),

tel.

7.2. Discretization in Space

123

The proof of the theorem is shown at the end of this section. For the analysis we introduce the Ritz—Volterra projection V^u G Sh satisfying A(Vhu(t) - u(t),v) = f K(tJo

s)B(Vhu(s)

v G Sh.

- u(s),v)ds,

(7. 8)

Lemma 7.3. For the Ritz—Volterra projection, we have Il(vft« - «)(*)!! +fclKVhti- «)(*)||i < C ^ s u p | | « ( s ) | | 2 < C^(||«o||a + /

||ut||2ds).

JO

S
Proof. Let W = V^u and p = W — u. We begin with an H1 -estimate, and introduce also the standard Ritz-projection Rh defined by A(R^u — u, v) = 0 for v G Sh. Recalling that \\Rhu - u\\ + h\\Rhu - u\\i < Ch2\\u\\2, we have, using the definition of W and notation 0 — W — Rh,u G Sh, that v\\0(t)\\2
= A(p,9)=

f

K(t-s)B(p(s),e(t))ds

Jo


- s^Ms^ds

+ \\(Rhu - u)(t)\U.

Lemma 7.1 now implies ||p(*)||i < CTsup \\(Rhu - u)(*)||i < CThsup s
\\u(s)\\2.

s
Consider now the L 2 -estimate, which will be derived from a duality argu­ ment. Using \\p(t)\\= sup (p(t),y) llwll=i and the solution of Aw = y in

HQ(0,),

||W||2

< C, we have

(p(t),y) = A(p{t),w) = A{p,w - v) + A(p,v),

v £ Sh,

124

Chapter 7. Cases with Weakly Singular Kernels

where A{p,v)=

/ K(t-s)B(p(s)iv-w)ds+

/ K(t - s)(p(s)1 B*w)ds. JO

JO

Hence, with v = RhW,

|| P (OII
- ayiWMWxWRkW

< Ch2(\\u\\2 + J* \\u\\2ds)+C

-

W ||!

+ \\p(s)\\\\w\\2)ds

j \ t-

s)-a\\p(s)\\ds,

by using Lemma 7.1, the proof of Lemma 7.3 is complete. For the time derivatives of p in the Ritz—Volterra projection there is the following estimate. Lemma 7.4. We have for p = V^u — u, y*CII/^ll -I" /*||A^|U)rf« < C7/i2(||TXo||2 H- y * II^^Harf^).

Proof. Writing (7.8) in the form A(p(t),v)

= j K(s)B(p(t Jo

- s),v)ds,

v e S£,

and by differentiating with respect to t, A(pt(t), v) = K(t)B(p(0),

v)+ [ K(s)B(pt(t

- *), v)dx.

Jo

Starting with the H1 -estimate and let W = Vh,u, then u\\Wt - Rhut\\\

< A{Wt - Rhuu Wt -

Rhut)

= A(pu Wt - Rhut) = K(t)B(p(0), + / Jo

Wt -

Rhut)

K(t-s)B{pt{s),Wt-Rhut)ds,

hence \\{Wt - ii h u t (*))lli < Ct-a\\p{0)h

+ C [ (t Jo

s )-«|| f t («)||ida,

(7.9)

7.2. Discretization in Space

125

or \\pt(t)\\i < C*- a ||p(0)||i -f \\Rhut - utHi +C [ (t-

syoWptWhda

Jo

< Ch(t-a\\uoh + IMOIh) + C [ (t- ^ - " i i ^ i i i d * . Jo

Thus by Lemma 7.1, \\pt(t)\\i < Ch(t-"\\uo\\2

+ \\ut(t)h + J\t

-

s)-a\\ut(s)\\2ds),

and finally j

\\pt(s)\\ids
+J

\\ut\\2ds + J

+J

Wuthds).


j\s-T)-<*\\ut(T)\\2dTds)

Consider now the L 2 -estimate. Following the notation used in Lemma 7.3 and using (7.9), we have (pt(t),y)

=

A{pt(t),w)

= A(puw-v)+

/ K(t-s)[B(pt(s),w-v) Jo

+K(t)(B(p(0),w-

+

(pt(s),B*w)]ds

v) + (p(0),S*n;)).

With an appropriate choice of v it leads to

Ch(\\(H(t)\\l+j\t-8)-a\\Pt(8)\\1d8)

\\Pt(t)\\ <

+Ch2t-a\\u0\\2

s)-a\\Pt(s)\\ds.

+ C f (t Jo

Together with Lemma 7.1, this gives ||p,(t)|| < Ch(\\pt(t)\\i

+ j \ * ~ s)-a\\Pt{s)hda)

+ C7fc2*-tt||tio||a.

After integration and using the H1 -estimate already derived, we get

/ \\pt\\ds
Jo

Jo


+J

\\ut(s)\\2ds),

126

Chapter 7. Cases with Weakly Singular Kernels

and thus the proof is complete. Proof of Theorem 7.2. In a standard fashion we write uh - u = (uh - Vhu) + (Vhu - u) = 6 + p. Lemma 7.3 immediately gives the desired estimate for p, so it remains to get a bound for 6. We have the following directly from our definition, (9U v) + A(0, v)=

[ K(t - s)B(6(s), v)ds - (Pu v), Jo

v e S£,

setting v = 6, it leads to

\Dt\\e\? + A(9,6)
l|fl(r)||2+/T||fl||?* JO

2


j\t-8)-a\\e{8)\\i\\0(t)\\id8dt

+ cJ

||pt||||»||dt).

Using Lemma 7.2 with a suitable choice of e, we thus have

\\0{t)\\2 + £ <

Mldt

C(\\0(0)\\2 + jf ||ft|P||
- t)-« jf* WOtfWldadt).

and obtain by Lemma 7.1 the bound

\\8(T)\\2+Jo

rT

\\9\\ldt
+

jo

\\pt\\\\e\\dt).

Further, using Lemma 7.4 and noting that 1^(0) = Rh, it becomes ||0(T)|| < C T (||0(O)|| + /

\\Pt\\dt) < Ch2(\\u0\\2

+ f

\\uthdt).

This completes the proof of the desired estimate for 0, and thus of the theorem. Under some assumptions on UQ and / , the maximum norm error estimate for (7.2) is also derived by Chen and Wahlbin [27].

7.3. The Completely

7.3

Discrete Scheme

127

The Completely Discrete Scheme

In this section we shall consider the completely discrete scheme. Since the kernel as well as t h e integrand are now singular, even when the solution is smooth, t h e product integration should be used: approximate y in Jn(y) = fQn K(tn — s)y(s)ds by a piecewise constant function, which take the value y(tj) in ( t j j t j + i ) , and t h u s Jn{y) is approximated by

K Qn(y) = E / 3+l K^ ~ s)v(h)ds = J2n-Mh)

(7.io)

where

Kn_j = f

3+l

K(tn - s)ds = f " ' K{e><^.

Jtj

(7.11)

Jtrt — j — \

Denote its error by n

"1

rtn

n

V (y) = Z ) ^-Mh)

~ /

K(tn - s)y(s)ds.

T h e completely discrete scheme is therefore n-l

(dtUn,v)

+ A{IT,v)

U° = u0h,

= ^2Kn^B{U\v)

+ (r,V),

3=0

v G S£,

^ '

'

n = l,2,.--,]V.

T h e following error estimate is the main result of this section. T h e o r e m 7 . 3 . For each t* > 0, there is a constant C = C(t*) such t h a t for t h e solutions of (7.1) and (7.12), \\Un - u(tn)\\

< C(h2

+ fc)(||u0||2 4- j

( I k t H + \\uth)ds),

t
In t h e case of a weakly singular kernel, the regularity of t h e solution with respect to time is limited, which makes higher order q u a d r a t u r e formulas as well as q u a d r a t u r e formulas based on sparser sets of time levels less attractive, see t h e economic scheme in Chapter 5. To prove Theorem 7.3 we need some lemmas. L e m m a 7 . 5 . If yt € Ll(J;L2), then kf"

\\qn(y)\\


I"

\\yt(s)\\ds

for

Nk < t*.

128

Chapter 7. Cases with Weakly Singular Kernels Proof. By the definition of K3 we have


j=0

3

and for each i G l ) , since |if (£)| < C£_c*, \qn(y)\ < £

l^(*n - a)\ f3+l

/

< C2^,^-3

I Jt

j=o

\yt{*)\dads

\yt(cr)\da,

J

where v3 = r

8-*ds = (t)-a - t]-?)/(l

- a) > 0

(7.13)

Jtj-l

and JT v3 = fN 3=1

J

(t*)1-"/^

s-«ds
- a).

°

Using integration in x and Minkowski inequality, this yields n-l J

j=0

tj

and, by interchanging the order of summation,

n-l

Jt

j=0 n = j + l

i

^°

This completes the proof. The following two Lemmas are discrete analogues of Lemmas 7.1 and 7.2 respectively, and can be proved similarly. Lemma 7.6. Let v3 be defined by (7.13) and assume that yn > 0 satisfies n-l

Un < bn + /3 ^2 un-jyj

for

n > 0,

j=o

where bn > 0, (5 > 0. Then for each t* > 0 there is a constant C = C(t*) such that n-l

yn
for

nA: < £-*.

7.3. The Completely Discrete Scheme

129

Lemma 7.7. Let K e £*( 0 there is a constant Ce(||Zf ||i,i(j)) s u c n that N

N-l

N

n=l

j=l

n=l

N-l

n-l

n=0

.7=0

Proof of Theorem 7.3. With the Ritz—Volterra projection W = VhU introduced in (7.8), we write

un - u{tn) = {un - wn) + (wn - u{tn)) = en + Pn. The term pn has been estimated in Lemma 7.3. For the term 0 n , by our definition, we have n-l

(dtff", v) + A(6n, v) = J2 Kn-jB(6i,v) i=o

+ (r B , t/),

(7.14)

where rn = u?-dtWn

qn(BhW).

+

We show by an energy argument that N

N

rn

for

\\Q \\
Nk
(7.15)

n=l

In fact, choosing v = 9n in (7.14), which yields V | | 0 | | 2 + \k\\dt6n\\2 2

V ,lMVM

,

n 2

2

k\\7)+f) \\

+ A(On,n n

n

4- A(0 .Q )

n = nY,Kn-jB(6i,e ) -l

=

+ (rn,n,

3=0

whence

dt\\e\? + i r ii? < c^2 ^riiiii^iii+cur* and, after summation,

iKii+fcgini? 2

n=l

< ||^|| + Cfc E " E ^ r i l i i r i l i + c* E lkn||im n = l jf=0

n=l

130

Chapter 7. Cases with Weakly Singular Kernels

Using Lemma 7.7 with K(t) = Ct~a, we conclude that ll^ll 2 + k £

||0"||? < C||0°||2 + Ck J2 \\rn\\\\en\\

n=\

n=l

N

Combining with Lemma 7.6, in which yn is replaced by y^ — k ]T] II@n111> ^ n i s n=l

shows N

\\8N\\2 \\2 + k1£\\rn\\\\en\\)

for

Nk
and (7.15) follows. Going back to estimate r n , we write rn = r7? + r% + r J + rJ where r? = u? - a t tt n , r j = 5 t (« n - W n ) = dtpn, r% = qn(Bhu) and r%=qn{Bhp). Obviously, kf^\W\\
fN \\utt\\ds ^

and by Lemma 4, *Z>2ii<£/

11*11* = /

n=lJt—1

n=l


J

Hftll^

°

\\uthda).

To estimate r j , we note that when u is smooth, B^u = PhBu and hence, by Lemma 7.5, N

t

t

*y>311
Jo

7.4. Cases with Nonsmooth Data

131

Using the inverse assumption, we have (BhP,v)

= B(p,v) < CWpUvh

<

Ch-'WpUvl

so that \\BhP\\ <

Ch-'M,.

Hence, for r j , by Lemmas 7.5 and 7.6r it follows kV

||rj||
fN

\\BhPt\\ds < Ckh-1

f

Jo

n=l

N

\\pt\Uds

7o


\\ut\\2ds).

Substituting these estimates into (7.15), it leads to ll^ll < C\\u0 - Rhu0\\ + C(h2 + k)(\\uoh

+ /

N

(\\utt\\ + I M W d s ) .

This completes the proof.

7.4

Cases with Nonsmooth Data

Consider the homogeneous PIDE with a weakly singular kernel and nonsmooth data

(

f*

ut + Au = / K(t — s)Bu(s)ds | iA = 0 on 0ft x J, [ u(0) = uQeL2(n),

in

Q x J, ^ 7 ' 16 ^

where A is a symmetric positive-definite elliptic operator of second order, B is an arbitrary partial differential operator of order /3, (/? < 2) and their coefficients are independent of t (and 5). Assume that the weakly singular kernel satisfies the following condition \K(t)\ + \K'(t)t\ + \K"(t)t2\

< Ct~a,

0 < a < 1.

(7.17)

(7.16) is a more difficult problem and requires more complicated techniques to deal with. The following regularity result is obtained by Chen and Wahlbin (1991) at Cornell University.

Chapter 7. Cases with Weakly Singular Kernels

132

Theorem 7.4. Assume that the kernel K(t) satisfies (7.17) and the initial value u0 € L2(ft), then (7.16) has a solution u(t) € C(J] H4'? f)H2) and \\u(t)\\x < C f ^ M ,

0 < A < 4 - p.

(7.18)

This result shows that the optimal index A = 4 — /? is the same as that obtained in Chapter 6 for the case of smooth kernel and which is independent of a. The proof of this theorem will be shown at the end of this section. We need some lemmas. Similar to what have been done in Chapter 6, by means of the semigroup E(t) generated by A, the solution of (7.16) can be expressed as pt

u(t) = E(t)u0 + / E{t-s) Jo

pS

Jo

K(s - r)Bu{r)drds

= Eu0 + w(t).

(7.19)

Introduce now the following Volterra integral equation w

= V(t) + Gw,

(7.20)

where G = E*K*B,

V(t) = GEu0 = E*ip,

Here a particular difficulty in the integral term K*Bu appears, which is caused by the singular function K(t). For example, the method used in the proof of Lemma 6.3 is no longer valid and more careful analysis should be carried out. Lemma 7.8. The operator G = E * K * B is bounded in C(J; H2) and for/GC(J;i^), | G / | 7 < Ct?-a

* U/IU p' = (2 - 7 ) / 2 > 0.

(7.21)

In particular, |G/|20.

(7.22)

If p = 1, then Gf € C(J; H3 fl # 2 ) and ||G/|| 3 < Ct-1 f\t - s)i-"(\\f(s)h

+ *||/'(«)lli)d» + Ct~a * ||/|| 2 .

(7.23)

7.4. Cases with Nonsmooth Data

133

Proof. Using Dirichlet formula and Thomee—Zhang formula (6.17), we have Gf = j

j E(t- s)K(s - r)Bf{r)dsdr

= [ [ Jo Jo

T

E{t-T-S)K(g)Bf{T)dt-dT

= /V-r)"1 f Jo -TE(t

Jo - T-

(7.24)

T

[(t-T-S)E(t-T-OK(0

0(K(00'}Bf(T)d£dT

+ f TK(t - s)Bf{s)ds. Jo Hence, by Lemma 6.2 and (7.25),

(7.25)

|G/|2 < C A t - r ) - 1 f \\K{£)\ + \ZK'{Z)\)\\Bf{T)\\dZdr Jo

Jo

+C f \K(t - s)\\\Bf(8)\\d8
s)-a\\f(s)yds

\t-r-i)-^2ra\\Bf{T)\\didr

|G/|7 < C f f Jo Jo

< C f\t - r)^-||/(r)||^r, Jo

p' = (2 - 7)/2

and (7.21) follows. If 0 = 0 or 2 and / G C{J\ i f 2 ) , then Bf e H2~P. We have, by (7.24) for 2 - / 3 < 7 < 4 - / ? , that |G/|7

T

(t-s)-^-2^/2(s-r)-a\Bf(r)\2.f3dsdr

f{t-sY~a\f{r)\2dr

Jo and, by (7.25) for 7 = 4 - /?, that

|G/|7 < C A i - r ) - 1 /■ t " T (t-r-0 1 - 2/a (|ii:(OI 7o Jo +\ZK\t)\)\Bf{T)\2-pdtdT

+C [ \K(t-T)\\Bf(T)\2-pda
[ Jo

(t-T)-a\f(T)\2dr.

Chapter 7. Cases with Weakly Singular Kernels

134

If (3 = 1 and / e C(J; H2), this time BfeH\ since Bf / 0 on 0ft. If (7.25) is used to estimate \Gf\s as in Chapter 6, then we can only get \Gf\z < C f\t Jo

- r)"1 f \ t - T Jo

< C f\t - rr^^Wf^hdr

Ol-Z/2ra\\Bf{r)\\d^dr

+ Ct~a * \\Bfh

+ Ct"° * ||/||2.

Jo In this case, a should satisfy 0 < a < 1/2. /3 = 1 is a more difficult case and has to be tackled in another way. For example, by estimating ||G/||3 via ||/t||i, Thomee—Zhang's technique can be used as follows, tGf = [ [ [(t - s) + (5 - r) + r]E(t - s)K{t Jo Jo = E*K*Bf + E*K*Bf + E*K*Bf, where ip(t) = tip(t). Now replacing E(t) by —TDtE(t) by parts, we have E*K*Bf

= f

f TDsE(t

= T [ K(tJo

- s)K{s -

r)Bf(r)dr

r)Bf{r)drds

and using integration

T)dsBf{r)dT

-TE*

(K)' * Bf,

where K(0) = 0 is used. Similarly, E*K*Bf

= E*Bf*K

= T[

K{t-

s)Bf{s)ds

-TE*K*

B{f)''.

Jo

Hence it leads to a useful expansion tGf = E*K*Bf-TE*{k'*Bf

+ K*Bf')+T{K*Bf

+ K*Bf).

(7.26)

For the most troublesome case that (3 — 1, ||£G/||3 can be estimated by (7.26) and Lemma 6.3,

PG/Us < C f [\t Jo Jo

T)1-V\S

+C f\(t - ry-WBfh Jo

- r)-a{\\Bf\\ + \\Bf\\)dTds + (t- T)-a\\Bf\\i)dT

f\t-T?l2-"{\\f{T)\\x+T\\f{T)h)dT


+C [\(t - T ) 1 - + (t- T)-aT]\\f(T)\\2dT. Jo

7.4. Cases with Nonsmooth Data

135

Noting that (t - r ) 1 " " + (t - r)-ar = t(t - r)~a, the proof of the lemma is complete. Remark. If F = BEUQ we have similarly, K*F

= F*K

= BTDsE{s)u0

* K = BT{u0K{t)

- Eu0 * {K)')

and hence TK * BEu0 = TBT(u0K(t)

- K' * Eu0).

Because T — A~x is an operator of order —2, and B is a differential operator of order /?, thus TBT is a bounded operator in H4~@. The estimation of V(t) = GEu0 = E*K*BEu0 is more complicated. For example, if 0 = 2 and 7 = 4 - /? = 2, then ||S£?i60|| < C ^ I K H and K * iT 1 is divergent. To overcome this obstacle, Thomee—Zhang's technique can be applied to this term in the form BEUQ * K. Below, for simplicity, we assume that B is independent of t and s. Lemma 7.9. For F(t) == BEu0 * K there is \\F{t)\\i < Ct^-V+WWuol

0
Proof. Denoting <j> = ^T(^)^ = K, from (6.18) for I + /? < 2 we have directly ||F(t)||« < C / ||£(t-s)noif(s)|| z + / 5 ds Jo < C A t - 8)-W23-a\\uo\\da

Ct1-"-(l+«/2||iio||,

<

JO

and, from (6.2) for I + (3 = 2,

||F(t)||2_^ < Ct"1 y*(t - af-VWKM

+ |^(s)|)||tio||
a

+ct- K||
ITOIU < Cf>-a\\uol

0 < A < 4 - /?,

p=(4-p-

in particular, p - a = (4 - P - 2a)/2 - A/2 > - A / 2 .

A)/2,

Chapter 7. Cases with Weakly Singular Kernels

136

Proof. For A = 0 or 1 we directly have by Lemmas 6.3 and 7.9 with / - BEuQ \\V(t)\\x = \\E * (K * / ) | | A
* f\\ds

Jo

f ( t - 5 ) - A / 2 s l - « - / 3 / 2 | | ^ o | | < Ct0-*\\U0\\. Jo In order to discuss the case 2 < A < 4 — (3, we recall the modified formula (7.26),
tV(t) = E*K*f-TE*{K'*f

+ K*f')+

T(K * / + K * / ) .

Noting that /'(£) = BE(t)u0 + tBE'(t)uQ and ||/'|| < C\\{tE' + E)u0\\p < Ct~P/2\\uo\\, we have, by Lemma 6.3 for A = p + q and q = 2 — /?, that

*||V(«)IU < C

f\t-8)1-^-^2s-a\\u0\\

Jo

+c||K*/-j-K*/|u_ 2
K0IU
for 0 < A < 2 ,

(7.27)

then

\\Mt)h < ct- 3 / 2 ||uo||. 2

Proof. Noting that t = (t - s)2 + 2(t - s)s + s2 we have from (7.16), t2ut+tzAu

= \ /V (foBu = Jo

+ faBus +

KBus2)ds,

Jo

where (j)2 — {t — s)2K(t — s) and 4>\ = 2(t — s)K(t — s). Denoting z = t2ut and differentiating the above equation in t, we get / KBzds-2tAu+ / (
±DtA(z,z)

+ \\Az\\2 < f K(Bz(s),Az(t))ds + j\(t

- s)1-" + (t-

+

c(t\\Au\\

s)-<*s}\\Bu(s)\\ds) \\Az(t)\\.

7.4. Cases with Nonsmooth Data

137

By dominateness of A and Young inequality, the integration in t leads to ||*(*)ll? + f\\Az\\2ds< Jo

f\s-T)-a\\Az(T)\\\\Az(s)\\dsdt

f Jo Jo

+C I (s\\u\\2 + s f\s

-

r)-a\\Bu{T)\\dr)2ds

and hence, by Lemma 7.2 and the assumption (7.27), IK*)lli + f \\M?ds Jo

< C f\s\\u\\2 Jo
f

+ a f\s Jo

(s • 5" 1 + s f\s

r)-a\\Bu(r)\\drfds

-

r)-ar-^2dryds\\uo\\

Ct\\u0l

and the lemma follows. Proof of Theorem 7.4. Construct {wn}^, a sequence of approximate solution to w = Gu, which satisfies the integral equation w = WQ ■+- Gw, where wo = V(t) = GEu0, wi = w0 + Gw0l

wn = wn-i -f Gwn_i =w0 + ] P Gnw0, ra = 1,2, • • •. For 0 < /3 < 2 it is known, by Lemma 7.10, that I K I U = II^IU < Ct"-a\\uo\\,

p

= (4 - /3 - A)/2 > 0,

and, by Lemma 7.8, \\Gwoh w in C(J; H2) a s n - > o o and 00

\\w(t)\\2 < \\w0\\2 + J2 \\Gnw0\\2 < Ct-a\\u0\\. n=l

(7.28)

Chapter 7. Cases with Weakly Singular Kernels

138

Further we shall consider sharper regularity estimates. If (5 = 0 or 2, we have by (7.22), ||GHU <

CtP

~"

* H h < CtP-"\\u0l

p = (4 - /J - A)/2 > 0.

If (5 = 1 and 0 < A < 2, again by Lemma 7.8, \\Gw\\x < Ctp'-« * \\w\\f3 < Ct?'-a * \\w\\2 < Ctp'~a * t-a\\u0\\ < C ^ - ' I K I I ,

P' = 1 - A/2.

In these cases we get

IMU < Mwoiu + \\GMU < ct"'-a\\u0\\ and hence (in particular, (7.27) holds)

||«IU < H^olU + IMU < c(t~x/2 +t?'-a)\\u0\\ < ct-^2\\u0\\. It remains to discuss the difficult case (3 — 1 and A = 3. By our definition, w = Gu, from Lemmas 7.8 and 7.11 we can get IM|3 = HG«||3 < Ct-1

Jo

[\t-T)l-a(\\u(T)h+T\MT)h)dT

pt


/ Jo

{t-r^^r^^drWuoW^Ct^WuoW

and Il«ll3 < ||£(*)«0||3 + M b < C{t-Z'2

+ t-a)\\u0\\

< Ci- 3 / 2 ||« 0 ||.

These estimates complete the proof. Concerning the discretization of PIDE for both weakly singular kernel and nonsmooth data, not much work has been done so far. For a special case a— 1/2, D. Xu [165] considered some time discretizations and obtained \\Un-u(tn)\\
for

0<

7


Chapter 8

Long-time Estimates Consider t h e following initial value problem ut + Au=

K(t-s)Bu{s)ds

+f(t)

for

t G R+ = (0, oo),

/g-x

u(0) = u0, where A is a self-adjoint strictly positive-definite linear operator with compact inverse in a real Hilbert space if, K(i) is a scalar function on R+ and B is an operator with D(B) D D(A) such t h a t , with the inner product (•, •) and the norms || • || in H and \\v\\j = \\Aj/2v\\, \(Bv,w)\

< C0H1HI1

with

Co>0.

(8.2)

We know t h a t , under appropriate assumptions on i^o and / , the problem (8.1) has a unique solution on R+, but in general a priori estimate obtained by the energy m e t h o d will depend on the Gronwall lemma, and the bound for u(t) will grow exponentially with t. Based on t h e research of Thomee and Wahlbin [155], we shall consider the case t h a t K is exponentially decreasing and t h a t the memory term can be dominated by t h e elliptic term, i.e. for some / and a > 0 it satisfies /»oo

(i)

\K(t)\

< le~at

and (ii)

0=

\K(t)\dt Jo

< 1/C0.

(8. 3)

T h e long-time estimates for purely parabolic case have been studied by S. Larsson [91] and others. 139

140

8.1

Chapter 8. Long-time

Estimates

Continuous Problems

First we consider a simple case where B = 0. Let Ai > 0 be the smallest eigenvalue of A, i.e. „ (Av.v) inf V T ^ > AI > 0. \\v\\2

v£H

From (8.1),

-Dt(u,v)

+ A(u,v) =

(f,v),

and then

AIM| + Ai|H<||/||. Multiplying by e

Xyt

and integrating in t, we get the following estimate

Ht)||<e- Alt K||+ f e - A l M \\m\\ds, ./o

which means that the influence of the initial value will exponentially tend to 0 as t —* oo and if J0°° ||/(s)||ds < oo, then

f

Jo

e-W-)\\f(s)\\ds

<eAlt/2 /

\\f(s)\\ds+

J0

||/(a)||ds->0

ast-^oo.

Jt/2

We now turn to the general case. Lemma 8.1. Let E(t)uo be the solution of the homogeneous case with / = 0 in (8.1), then u(t), the solution of (8.1), is given by u(t) = E{t)u0 + / E(tJo

s)f(s)ds.

Proof. By linearity and uniqueness, it suffices to show that the second term on the right-hand side, denoted by u(t), satisfies (8.1) with UQ = 0. From / K(t-s)Bu(s)ds= Jo

[ K(t-s)B Jo

f Jo

E(s-a)f(a)dads

= I [ K(t - s)BE(s Jo Jo /ȣ

a)f{a)dsda

pt — S

= / / Jo Jo

K(t~s-a)BE(a)f(s)d(rds,

8.1. Continuous

Problems

141

hence ut + Au-

/ K(t-s)Bu(s)ds

= /

\Et(t - s) + AE(t - s)

- f

K(t-s-

a)BE(a)da\

f(s)ds

Theorem 8.1. Let u be the solution of (8.1) and assume that the as­ sumptions (8.2) and (8.3) hold, then \\u(t)\\ < e - ^ K U + f e-^-^\\f(s)\\ds for t G fl+, Jo with some 7 > 0. Proof. Consider first the case of / = 0. Denoting w(t) = e7t?x(£), from (8.1) there is wt - -yw + Aw = / K7(t-s)Bw{s)ds Jo Now choose 7 > 0 so small that

where

K^(t) = ellK(t).

(8.4)

/•OO

/ \K7(s)\ds < (1 - 2 7 /A 1 ) 1 / 2 /C 0 . (8.5) Jo From (ii) of (8.3), /?7 —> /3 as 7 —» 0, this can be verified by the dominated convergence theorem. Taking the inner product of (8.4) with 2w(t) and using Cauchy—Schwarz inequality and the Young's inequality, we have /J7 -

AlH|2-27|M|2+2|M|? =

2

/ K^(t Jo

s)(Bw{s),w(t))ds

< \\w(t)f1+^C20 /V 7 (t- a )||Ka)||Sd8. Jo

2

Since Ai||it|| < ||w||i, after integrating in £ and interchanging the order of integration in the resulting double integral, it becomes

HI 2 +(1 - 27/A0 f \\w\\ids Jo

< \\U0\\2 + /?7Cg' / / \K^(S - <7)HH<7)||2d<7 Jo Jo

<||«o||2+)37Cg/"|K(r)||?da. Jo

Chapter 8. Long-time

142

Estimates

Further by (8.5) there is \\w(t)|| < ||u 0 ||,i.e. ||£(£)u 0 || < e-^||u 0 ||. The general case now follows from Lemma 8.1. Conclude this section by elucidating the conditions (8.3) in the case of B = A (and hence Co = 1 in (8.2)), K(t) = le~at and / = 0. Consider a Fourier component u\(t) = (u(t),ifx)j with respect to the orthonormal eigensystem {A, if\} of A, and setting v(t) = eatu\(i), we have vt + (X — a)v = IX Jo

v(s)ds,

or, after differentiation, vu + (A — a)vt = IXv, with initial data v(Q) — u\(0) and vt(0) = (a — j)u\(0). This equation has the solution of the form e r ± t , and hence u\ is a combination of e^r±~a^t, where A-j-a

r±-a =

( X+ a

\i

2

— ± ( ( - y - ) + (Z ~ a ) A ) •

In general, with the above initial conditions, both r+ and r_ are always real and will be present in the solution. In order that u\ decreases exponentially in £, one must have r + - a = (Z - a)A/ [-^— + ( ( - £ - ) 2 + (J - a)A)*J < 0, i.e. /? = J0°° \K{s)\ds = l/a < 1. If I > a, the blow-up might occur.

8.2

Backward Euler Scheme for B = A

For the time discretization of (8.1), we shall consider the following backward Euler scheme (BES), dtUn + AUn = Qn(KnBU)

+ fn

for

n > 1,

U° = u0,

(8.6)

where Kn = if (£n — s), fn = f(tn), and the following quadrature formula with non-negative coefficients o; nj = Ljnj(k) Qn(y) = Y<"njy{tj)^

y(s)ds.

(8.7)

For example, one can refer the rectangular rule with wnj = k in Chapter 4 and the modified trapezoidal rule with sparse weights cunj in Chapter 5. Note that

8.2. Backward Euler Scheme for B = A

143

Un does not appear on the right-hand side of (8.6), this would be convenient for implementation. In Chapters 4 and 5 we have derived by Gronwall lemma the error esti­ mates for Un — u(tn) with constants C(tn). Obviously, these estimates depend on t n , the upper limit of time, and will grow exponentially with tn. This kind of error estimates will become useless for large tn in numerical computation. The goal of this chapter is to establish similar estimates with constant C independent of tn, hence they can serve as long-time estimates. To achieve this, obviously one has to make some restrictions on the kernel K and the operator B in (8.2) and (8.3). First discuss the boundedness of \\Un\\ in the case that / = 0. Multiplying (8.6) by 2C/n, we have n-l

dt\\U"\\2 + 2 | | £ H ? <

2^njKn^{BU^Un) 3=0


Again, multiplying this inequality by k, summing up with respect to n and applying Cauchy—Schwarz inequality to the last term, under the following assumption n-l

Qn(\Kn\) = ^2unj\K(tn^)\

<0,

P<(3< 1/Co,

3=0

there is

\\uN\\2 + kjr\\u'\\\ j=l

< \\v°\\2 + pc%kJ2Yt<>>nj\Kn-A\\ui\\l 71=1 j = 0

N ~ Changing the order of summation and denoting LJJ = J2 bJnj\Kn-j\,

it follows

n=j

that

H^^ll2 -H A^XI ll^-7!!? < ll^°l[2 -K^OgA;XZZ^||t7-^[|2. 3=1

(8.8)

3=0

If Qn takes the rectangular rule with LJUJ = k for j — 0,1, • • •, n - 1, and then Wj < (3 for small /c, by a proof similar to Theorem 8.1, it can be proved that

Chapter 8. Long-time

144

Estimates

the coefficients in the sum on the right-hand side of (8.8) are bounded by those in the sum on the left-hand side, and then \\Un\\ < \\U°\\. However, let Qn be sparse, with u)nj = 0 for many j and LJUJ for some j are larger than k, the cancellation may be prohibited. For instance, using the modified trapezoidal rule, LUJ is of order & _1//2 for certain jf, and thus the coefficients of ||E/J'||i on the right is of order /c 1 / 2 > k. To continue our discussion we first consider a special case where B = A. Theorem 8.2. Assume that there are positive constants /?,7 and A: such that for K7(t) = e^K(t) and Klyn{s) = tf7(£n - *), P1,n = Qn(\K1,n\)
for

7

<7, n>l

and k < k.

Then there are positive constants &o and 70 such that, for the solution of (8.6) (with B = A), n

\\Un\\ < e-Ttn\\U°\\+k'52e-tt"->\\fi\\

for

n > 0, k < k0.

Proof. By linearity one may express the solution of (8.6) as Un = UQ + t/f H (-[/£, each U? satisfies dtU? + AUr = Qn(KnAUi)

+ 6nifn

for n > 0, i > 1,

,

.

(8 9)

Uf = si0u°.

-

The solution of (8.9) is obviously zero for 0 < n < i and U\ = (I + kA)~1kf% for n = i > 1. For n > z one may consider (8.9) as a homogeneous equation dtU? + AU? = Qn(KnAUi)

for n > i,

l

U\ = {I + kA)' kf\

i> 1.

If one can show that ||tfril<e~ 7 t "-'l|tfill

for

n>i,

(8.10)

in view of the choice of U\ for i > 0 , the theorem follows immediately. To show (8.10), with {A,
+ k\Qn(Klind)

for

n > i.

Thus we find that K l < (l + k7)- 1 (e 7 f c + Afc/3) max \dj\ < max |eP|,

8.2. Backward Euler Scheme for B = A

145

provided that 7 and k are so small that elk — 1 < \\k(l — /?), where Ai is the smallest eigenvalue of A. Since dj = 0 for j < z, we have \dn\ < \d%\ and by Parseval identity,

\\e^un\\2 = J2 \dn\2 = Yl ^2 = W^^f, x

x

from which (8.10) follows. We now turn to the error estimate. Assume that the quadrature rule is such that, for some integer g, \qn(y)\ = \Qn(y) - [ " y(s)ds\ < CV'm Jo

f " \y(s)\^ds Jo

for n > 1, j =

l,m,

(8.11) where

i»(t)iW) = X>fr(t)i. For example, for the modified trapezoidal rule, there is ftm

fin

\qn(y)\ < Ck\ \

\ytt\ds + Ck /

JO

ptn

\yt\ds < Ck /

Jtm

|y|<2>ds,

j = m = 2.

JO

We further assume that < Ce~at.

\K(t)\^

(8.12)

Lemma 8.2. Under the conditions (8.3), (8.11) and (8.12), for any 0 with (3 < (3 < 1 there are positive 70 and ko such that (3^n = Qn(\K^n\)<(3
for

7<7o,

k < k0 ,

n>l

(8.13)

holds. Proof. Note that Dt\y\ = (Dty)sgn y, using (8.11) with j = 1 and setting v = 1/ra,

Jo /*oo

v


\ Jo

roo

\K{s)\Me^sds+

/ Jo

\K(s)\ elsds =

I1+I2.

Here, by (8.12), I\ is bounded if 7 < a and tends to zero with k. Further, by the dominated convergence theorem, 72 approaches (3 = J0°° \K(s)\ds < 1 as 7->0.

Chapter 8. Long-time

146

Estimates

Theorem 8.3. Assume that (8.3) (with c0 = 1), (8.11) and (8.12) hold. Then there are positive C, 6 and /CQ such that, for the solutions of (8.1) and (8.6) (with B = A), \\Un-u{tn)\\

e-6^-s\\\utt{s)\\-^\\Au{s)\\^)ds

< Ck / Jo

for n > 0, k < k0. (8.14)

n

n

Proof. Setting e = U — u(tn), we have dten + Aen = Qn(KnAe)

+ qn(KnAu)

- rn

for

n > 1,

e° = 0, where r n = dtu{tn kY^z~ltll~j\\rj\\

t(tn).

< kJ2^~ltll-j

) —u

It is easy to see that \\utt\\ds < ke^k /

/

e- -77( ^t »--^*|)||u t t | | d s .

Further, by (8.11) and (8.12), Wq^KjAu^l


f3 \\K(tj - s)Au(s)\\^ds Jo

< Ck [ * Jo

\\Au\\^ds

and thus, with 6 < 8Q = min(o:, 7), n

n

„tj

J = l ^°

3= 1


[n(tn-s Jo


I" Jo

{n +

i)ke-6o^-s^\\Au\\^ds k)e-6o^-s)\\Au\\Wds

e-^-^WAuW^ds.

The desired result is just a consequence of applying Theorem 8.2 to (8.15). Remark that the right-hand side of (8.14) may be estimated in terms of data / and -uo with the techniques used in Theorem 8.1 and note that the time derivatives of u satisfy integro differential equations of the same type as those of u.

8.3. Second Order Backward

8.3

Difference

Method

for B = A

147

Second Order Backward Difference Method for B = A

We now deal with t h e second order backward difference time-discretization of t h e following t y p e (dt + ^d2t^Un+AUn

= Qn(KnAU)

+ fn

for

n>2

(8.16)

+ f1.

(8.17)

with starting values U° and U1 given by 0 * = uo,

and

dtU1 + AU1 =Q1(K1AU)

Introduce a quantity A by A = sup A(i/), where

and r j - = (2 ± \ / l — 2i/)/(3 + 2v) are the roots of the following polynomial Mr) = r 2 —

— r -\

— (r — r+)(r — r _ ) .

T h o m e e and Wahlbin showed t h a t A < 1.1 (their computation experience suggested t h a t 1.04 < A < 1.05). For v < 1/2 t h e roots are real, r + > r_ and hence A(z^) = 1. T h e o r e m 8 . 4 . Assume t h a t (8.3) (with C 0 = 1) and (8.13) hold, and also A/? < 1. T h e n there exist positive C, 7 and &o such t h a t , for the solution of (8.16), n

ll^ll^Ce-^CII^II + II^IIJ+Cfc^e-^^ll/^ll

for

n > 2, k < k0.

P r o o f . We shall proceed along t h e line of the proof in Theorem 8.2, and write Un = UK + U? + • • • + L/£, where (9t + ^ t ) ^ i l + ^ ^ = Q n ( ^ n A C / i ) + « n i / n [7™ = (5 mi C/ m , m = 0 , 1 , i = 0 , 1 , . . , n.

for

n>2,

We wish t o show t h a t HUTU < C e - ^ - l t t f U

for

n>i.

(8.18)

Chapter 8. Long-time

148

Estimates

Since U\ = (§J + kA^kp for i > 2, the proof will follow from (8.18). First consider the case 7 = 0 and for fixed i and v = A:A, the Fourier coefficients dn = d™ = (U^^ipx) satisfy 2dn~1 + ^dn~2 = uQn{Knd)

(1 + v\dn-

for

n > max(z + 1, 2),

where 1 and d£ = 0) are given. Its characteristic polynomial is i\) and its solution may be represented as n-i+l _

n-i+1

d» = -± r . — r_ = ^

9

n

n-j+1


_

n-j+1

= r+— r_ ^

QJ(KJd) = Ji + •*»,

(8.19) for n > i, with the first term on the right replaced by —r_|_r_(r™ x — r™ 1 )d° /(r+ — r_) and the summation extended from j = 2 when 2■ = 0, and with the obvious modification when r + = r_, i.e. i/ = 1/2. It is easy to see that | r + | < 1 and |r_| < 1/2, and hence | j x \ = |I±

ZLz

d*| < | r n - i + . . . + r r ' | | < f | < 2|d*|.

For J2 we have IJ2I < A(i/) max 10^(1^^)1 < A/3 max |d'|, and hence |d n | < 2|d*| + A/3 max \dj\

for

n > max(z + 1, 2).

j
If A/3 < 1 we conclude that \dn\ < C\dl\. From this and the Parseval identity, when 7 = 0, (8.18) follows. We next show the inequality (8.18) for some positive 7. For n > max(i -f 1,2) and dn = (e^Uf^x) there is dn

- sT^ 7 ^ - 1 + JTT^kdn'2 =

zT2~^K^

where KljTl{s) = e 7 ^ Tt ~ s ^K(t n — s). Replace r± by p± = elkr±, the analogue of (8.19) follows. Since the minimal eigenvalue Ai of A is positive and | r + ( ^ ) | < |r + (*;Ai)| < e~ckXo for k sufficiently small, we have | p + | < e-Ko\i-i) < 1 for 7 < CAi. Similarly, 1/(1 - |yo_|) < C, and hence \JX\ < C\d%

8.3. Second Order Backward Difference Method for B — A

149

It remains to investigate the quantities A7(z/), which is obtained from replacing r ± by p± in A(v). For v < 1/2 and small 7A;, one can estimate 2v A » = 3 + 2v

2v e~2^k 3 + 2^(e~^)

1 (l-/o + )(l-p_)

2l/ < 1 + C ^ < 1 + C7 < A(i/) + C7, 3 + 2i/ - 4e^fc + e2^fc ~ 1/

which is uniform for k < k0 and 1/ < 1/2. For v > 1/2, since £ = (3-f 2z/) - 1 / 2 < 1/2, we have A

» -^

2^

°°

-'

-'

e 1)7 v °" = TZK7, E( * - "r+-r_ ^ 3 + 2z/^ 00

< 7^ ^ i 2 ( e 7 f c / 2 ) - 7 " 1 < C7A: for small k. 3=1

Hence, for small k and 7, we have |J 2 | < A/3 max \d3\ and |d n | < C|d l |, and i
(8.18) with a positive 7 is obtained. This completes the proof of Theorem 8.4. In order to derive an error estimate, we shall use the following quadrature rule of second order accuracy,

\qn(y)\ = \Qn(y)- I" v{s)ds\ < Ck2 f" \y\(mUs+Ck JO

JO

[" |y|<m>ds, (8.20) Jtn-i

and the integral Jt n_ y(s)ds will always be approximated by the left rectan­ gular rule as ky(tn-i). Theorem 8.5. Assume that (8.3) (with C 0 = 1), (8.11), (8.12) and (8.20) hold, and further that A/? < 1. Then there exist positive C, 6 and k0 such that for the solutions of (8.1)(with B = A), (8.16) and (8.17), we have, for n > 0 and k < fc0, \\Un - u(tn)\\ < Ck2(e-6t"\\utt(0)\\

+J

n

e-6^-s\\\uttt\\ + \\Au\\^)ds).

Proof. Let en = Un — u(tn), we have (dt + ^d2t)en + Aen = Qn{KnAe)

+ en(KnAu)-rn

for n > 2,

150

Chapter 8. Long-time

Estimates

where rn = (dt 4- %d2)u(tn) - ut{tn). Here e° = 0 and, with U1 given in (8.17), the argument in Theorem 8.3 immediately gives He1!! < Ck2(

maxJu t t (s)\\ + [ \0<s
< Ck2(\\utt(0)\\+j

\\Au\\™da)

JQ

J

(HutttH +

\\Au\\W)ds).

As in Theorem 8.3 but now using (8.20) with S < min(a, 7), we have

'

»

t

+Ck2 V e - 1 ' - ' f ' \\Au\\Wds


[U

e-*(t"-*>U^ll(mW

Finally, by expanding rn at tn, we have k V e - ^ " - | | r " | | < fcVfc [ " e- 1 '< t »-'>||u m ||ds ) J ° j=2 and by Theorem 8.4 and Lemma 8.2, the theorem follows.

8.4

The General Case for B

In this section we shall consider the backward Euler scheme (8.6) in the case that the operator B is different from the positive definite operator A and derive the discrete stability estimate in the form of Theorem 8.2. Assume that |K(t)| < I and instead of (8.2), IIA-^H < b,

(8.21)

and assume further that the quadrature weights are dominated in the following sense cjnj < ujj for n > j + 1. (8.22) Let u'nj = ^2^Zj us' For the rectangular rule, modified trapezoidal rule, mod­ ified Simpson rule, and etc., u'n- < C^tn-j 4- C3 will hold. Sloan and Thomee [147] obtained an estimate for a finite time interval, n

||tr,||<eIc,w^(||l70||+ib5^||/J'||) 3=1

for

n > 0.

8.4. The General Case for B

151

Thomee and Wahlbin further sharpened the techniques to improve this result as follows. Theorem 8.6. Assume that (i) in (8.3), (8.21) and (8.22) hold, and let Ai be the minimal eigenvalue of A, 7 = f min(a, Ai). There exists &o > 0 such that, for the solution of (8.6), n ||[/»|| <

e^nO-7tn||yO||

+

fc^e'^i-ytn-ill^U

for

fe

< ^ .

( 8 > 2 3)

3=1

Remark. For some quadrature rule mentioned above, (8.23) gives long­ time stability provided that lb is sufficiently small comparing with a and Ai. Proof of Theorem 8.6. With the notation and argument used in the proof of Theorem 8.2, it is sufficient to show that \\U?\\ < e Z 6 w » * - 7 t ^ | | t / ? | |

for n>i.

(8.24)

With Ek = (l + kA)'1 we have n

jjn

= En-ijji +k Y,

E J

r ^QJ(KJBUj)

f

°r

U

>L

(8'25)

.7=1+1

Here

k ]T ErJ+1Qj(KjBUi) = k J2 E%-i+1YiwjsKi-.aBU'i =* E I£ s=i+l

^sKj-sErJ+1A]A-1BUts.

j=s+l

Assume for the moment that n

\\k Y, ^jsK3-sE^-j+1A\\

< Icjse-^-*

for n > s.

.7=5 + 1

We have, from (8.25) and (8.21) for k small enough,

wn <

e-^-^WUtW+lb^w^^-WUtl s=i

tn

n

n

Multiplying by e^ ~% with y = e^\\U \\,

we find for 7 < Ai/2,

n-l

yn < V1 ^-IbY^^sV8

for n > i.

(8.26)

Chapter 8. Long-time

152

Estimates

By discrete Gronwall lemma, (8.24) follows. We now prove (8.26). By the spectral representation it can be reduced to show n

\k ] T ujjsKj-.sr(kX)n-j+1X\

< e~^-»

where

r(t) =

,

or, using (8.22) and (8.3), n

fcA ] T r(fcA) n - J ' +1 e- ttt »-^ < e - 7 t » - , i.e. after changing the variables, n

Q = Q(A,/c,n) = k\Y^r(k\y<e-<**»-j+i

< e-7t.

(327)

i=i

Consider first A > a and show that (8.27) holds with 7 = 0.4a. Replacing ak by k, A by A/a and 7 by 7 / a , we may assume a = 1. Then we show that for any A0 < 0.6e(« 1.63), there is a ko such that (8.27) holds with 7 = 0.4 for 1 < A < Ao and k < ko. In fact, since r(kX) is decreasing, we have for these A, with any 6 < 1 and small fc, r(kX) < r(k) < e~6k, and hence, using xe~px < (pe)-1 for x > 0, n

Q < \k^r,e~Sjke~St"-j+i

< Xkne~st"

< Ao^e-* 6 - 0 - 4 )*'^ 0 - 4 *" < X0((S - 0.4)e)- 1 e-°- 4 t - < e" 0 4 4 ", if Ao < (<5 — 0.4)e, which holds for 6 suitably close to 1. We must show that (8.27) holds also for A > Ao > 1. Summing up the finite power series in (8.27), Q can be rewritten as follows, Q = \k(rn - e'tn)/{ek

- (1 + Afc))

or e 0.4t„Q =

Afc ( e -0.6t„

= \k{e~0M''

_ (e0.4*r)n)/(1

+

Afc

_

^

- e- ( y -°- 4 ) '")/(l + Afc - ek),

8.4. The General Case for B

153

where we have set r = e yk with y = y(\, k) = -\nr(Xk)/k. Note that y(\, k) is monotonic increasing in A and l + o < e a , thus y(\,k) < A. Further, y(Xo,k) = A0 4- 0(k). An elementary investigation shows that the function \e~at — e~ 6t |, with a, b > 0 and 0 < t < oo, has a maximum b _b_

b—a

)a-6.

(

a

a

With a = y — 0.4 and 6 = 0.6, we hence have n At ^

AA;

,

y — 1

0.6

,

x

. Q.6

For k small enough, since y < A, 1

< A - A - 1 + 0{k)

X

A-l A - 0.4 ~

+

0.4 A - 0.4

+

O(fc) A- 1

< l + - ^ y ( 0 . 4 + O(*)), and thus

Further, since y(A, fc) > y(Ao, A;) = Ao + 0(&), lnJ

0.6

y - 0.4 ^

0.6

rAo-04

^ * - ^ T T l n - b ^ - * -^TT l n (-o-6— + ° W ) '

and hence ln(eo.4t„g)

_0.6 \2 , ^ „ _ , , ,3A < - £ L [ | + O(fc) - l n ( ^0 - | + V

0(k)j\.

Since 5Ao/3 is arbitrarily close to e and | — ln(e — | ) « —0.05 < 0, one may first choose AQ and then /CQ SO that the right-hand side is negative for k < /co, which shows (8.27). It remains to consider the case that Ai < A < a, where Ai is the smallest eigenvalue of A. Since y(\, k) = A 4- 0(k) holds uniformly for these A, instead of (8.27) we consider an alternative n

Qi(A,fc,n) = XkJ2e~Xtje~atn~J+1

•.

n

= -[a*JIe~atie"atn-i+1]-

Comparing the expression inside the square bracket with the previous Q(A, k, n), we see that the roles of A and a are reversed, and since A/a < 1, the analysis

Chapter 8. Long-time

154

Estimates

above is clearly valid with appropriate modifications. Taking now 7 = 0.4Ai, the proof of Theorem 8.6 is complete. As shown earlier, the stability implies an error estimate. We shall omit the details here.

8.5

Application to Finite Element Discretiza­ tion in Space

In this section we shall briefly discuss applications of the above results to the complete discretization of an integrodifferential equation, i.e. the following backward Euler scheme dtUn + AhUn = Qn(KnAhU)

+ P h fn

for n > 1,

U° = uoh.

(8.28)

We shall use the result on stability in Theorem 8.2 to obtain an error estimate. Checking the proof of the theorem, one sees that the properties of the operator A is contained only in the condition elk — 1 < X\k(l — /?), where Ai is the smallest eigenvalue. Since the smallest eignevalue Ai^ of Ah is bounded below by Ai, we may apply Theorem 8.3 to (8.28) with constants that are independent of ft. By using the Ritz-projection Rh,u(t) € 5^, rewrite Un - u(tn) = (Un - Rhu(tn))

+ (Rhu{t) - u(tn)) = f + pn,

where ||p n || < Cft r ||^(t n )|| r . By our definition, AhRhV = PhAv for v G H2 n # o > t h e n 0n e Sr satisfies dt0n+Ah0n

= Qn(KnAh9)

+ Ph(dtpn + rn + qn(KnAu))

for

n > 1,

and assuming for simplicity that uoh = Rh^o, we have 6° = 0. Here, rn = du(tn) — ut(tn). As remarked above, we may apply Theorem 8.2 (with H = S^ equipped with the L 2 -inner product) so that, since ||i\v|| < ||v||,

\\0n\\ < kj^e-^iWW

+ \\rj\\ + hHKjAu)^,

where Wdtp'W^Chrk-1

f'

\\ut\\rds

and

||rJ'|| < f

\\utt\\ds.

8.5. Application to Finite Element Discretization in Space

155

Thus the following theorem is immediately derived. T h e o r e m 8.7. Assume that (8.2) (with C 0 = 1), (8.11) and (8.12) hold, and let UQ^ = RHUQ. Then there exist positive C, 8 and ko such that for the solutions of (8.1) (with B = A) and (8.28), \\Un - u(tn)\\ < Chr(\\u(tn)\\r

+ /

n

e-W^WutWrds)

Jo

+Ck [ n e-6^-s\\\utt\\ Jo

+ \\u\\M)ds

for n > 1, k < k0, q

where the constant C is independent of £ n , and ||w||^ = ]T) ||Z)^.Aw||. j=o

A similar result also holds for the second order backward difference scheme, with k replaced by k2. Finally, to obtain an error bound in the case B ^ A, rather than compar­ ing Un to Rhu{tn), one may conveniently compare it with the Ritz—Volterra projection Vhu(tn) of u mentioned before. We conclude by remarking that even in the case that u(t) approaches its limit UOQ "exponentially fast" as t —► oo, Theorem 8.3 (and its analogue for the second order backward difference scheme) merely asserts that Un is uniformly within O(k) (and 0(k2), respectively) of u(tn). In fact, we cannot assert in general that Un has a limit as n —> -foo for a fixed k. For instance, in the case of the modified Simpson's rule, from computer experiments it appears that Un approaches a periodic limit cycle with period determined by periodic changes in the quadrature formulas. It seems, though, that the periodic changes are very small compared to the main part of the error. An important topic in connection with the long-time estimate is the asymptotic property of the solution. Engler [47] considered the following semilinear PIDE ut-Au+ K(t- s)F(u(s))ds / Jo | ^ = 0 on cft7 x (0, oo), ^ u(0) = UQ

in

=f

in

fix(O.oo),

QJ.

Assume that F{u) is non-decreasing, f(x,t) = / ( # ) , and that the kernel sat­ isfies K(t) > 0, K'{t) < 0, K" > 0 and sup{e\K"(t) + eK'(t) > 0 for every t > 0} = e > 0, then, for A = min(2Ai, e), \\ut(t)\\2 < C(uoJ)e~xt

as

t -> -foo,

Chapter 8. Long-time

156

Estimates

and u(x, i) has a limit u(x) as t —> H-oo, which is the solution of the following stationary problem /•OO

-Au + /

K(s)dsF(u)

=7,

w = 0 on d a

In order to calculate the solution u(x, t) on the whole domain Q x (0, oo), we suggest that the graded grid {nk)v, z / > l , n = l , 2 , 3 , - - - , starting from some time t > 0, will be useful. These problems haven't been discussed yet.

Chapter 9

Maximum Norm Estimates In this chapter we shall turn to the maximum norm estimates in two dimen­ sional case which are important in the study of both superconvergence (Chap­ ter 10) and nonlinear problems (Chapter 11). We begin with L°°-estimates of Ritz—Volterra projection Vh,u(t), which were formulated in Chapter 3, but here we only give a simplified proof in the weaker form.

9.1

L°°-Estimates of Ritz—Volterra Projection

Consider a Volterra integrodifferential operator Lu = A(t)u(t) - / B(t, s)u(s)ds Jo and its adjoint operator L*u = A(t)u(t) + /

5 * (5, t)u(s)ds.

The corresponding bilinear forms are respectively r* L(u, v) = A(u, v) + / B(t, s; Jo /o

u(s),v(t))ds

and L*(u,v) = A(u,v)+

/ 157

B*(s,t;u(s),v(t))ds.

158

Chapter 9. Maximum Norm Estimates

By using the Dirichlet formula / [ f(t,s)dsdt= JO JO we have an identity

[ [ f(t,s)dtds= Jo Js

[ f Jo Jt

fT fT / L(u,v)dt = / L*(viu)dt, Jo Jo Define the weight = (X,Z) = (\X-Z\2+UJ2)1/2,

IML^ = ( £ \oc\<m

[

f{s,t)dsdt,

u,veHQ.

u = jh,

7 » 1 ,

l\Dau(x)\2d8)V2,

Jn

respectively. We know that ( C/(2 -I), l<2 ~lds<< C|lnw|, 1 = 2,LJO<1, J n { Cw a -7(Z - 2), l>2, \Dm
m>0,

and weighted interpolation estimates

llu - ihu\y + h\\D(u - inU)y < chr\\Druy, where I is any real number and the constant C is independent of z and h. In this section we shall use the following notations: Mm,0'

= IK*)llm,^ +

/ JO

\\u(s)\\m4>ids,

a(t) = cr(t) + /
Jo

Mm,*1 = IK*)ILf*' + / \\u{8)\\mt4tids, a*{t) =
cr(s)ds.

9.1. L°°-Estimates

of Ritz—Volterra

Projection

159

Obviously, by Dirichlet formula, / u(t)a*(t)dt= Jo

/ Jo

u(t)a(t)dt.

Lemma 9.1. Let fibea plane domain and the regularity assumption A | in (3.3) holds. Then the solution w(t) G HQ of the problem Lw = f(t) in ft x J has the weighted estimate

Proof. Follow Nitsche, letting U{ — (x{ — Zi)w(t) G HQ, i = 1,2 and noting that D2xUi — (xi — Zi)D2w + 2Dx(xi -

Zi)Dxw,

Aui = (xi — Zi)Aw + 0(Dw), we have, by the assumption i4|, that

< C ( | | D 2 U l | | 2 + \\D2u2\\2 + w2||£>2tu||2 + 2

< C(\\AUl\\

2

2

2

+ \\Au2\\ + u \\Aw\\

+

\\Dwf)

\\Dwf)


I \\DMs)h***

+ C\\*>\\i-

Jo

Thus the lemma follows by Gronwall lemma. To define the Green function, we introduce a discrete delta function 8h — 8^{x) which satisfies (6h,v) = v(z), v£S*. There are two ways to define 8h- The first definition requires 8h G 5^, then Dz8h G S£. Delseux (1972), and Douglas, Dupont and Wahlbin [42] proved that |£>m<^(*)| < C / i - 2 - m e - C l l x - ^ / ' 1 , d > 0, m = 0,1. While another definition requires that 8^ G HQ (O) does not belong to S£ but it has small support, for example, in some element r(z) including z. Obviously, in the latter case, | D m ^ ( x ) | < Ch~2~rn in r(z). According to the above two definitions, we have, for any real /, \\Dm8h\\^
m = 0,l-

Chapter 9. Maximum Norm Estimates

160

Now following Y.P. Lin [95] and T. Zhang [170], we define a regularized Green function g(t) € HQ and its finite element approximation gh{t) € S^ such that L*g(t) = 6h
(9.1)

and £ * ( 0 f c ( * ) - 0 ( * ) , * ) = O,

vGS*,

(9.2)

where cr(t) = (Te(t) is a given function satisfying 0,

cr{t)eC^{J),

f

(j(t)dt = \

Jo

and / ( t 0 ) = el i m /

-*°Jo

f(t)a€(t)dt

for

/ <E C(J),

t0 £ J.

For example, we can take a mollifier function „

m

_ _ r,

M

_ / cxe- 1 e x p ( - l / ( l - (t - t0)2/e2)),

\t - t0\ < e,

where Cf 1 = /

exp

^|t|
—dt. 1-^

Similarly, we define a gradient type Green function G(t) £ HQ and its finite element approximation Gh(t) £ S% such that L*G(t) = -Dx6ha(t)

in ft x J,

(9.3)

and L*(G h (t)-G(«),t;) = 0,

.G5rft

Lemma 9.2. For g(t) defined by (9.1) we have \\D2g{t)\\*> <

C\\nh\^<j\t)

and ||fl(t)||2,i <

C\lnh\a*(t).

Proof. By Lemma 9.1, we have \\&9(t)\U*6h\W(t)
+ \\g(t)\\i C\\g(tm.

(9.4)

9.1. L°°-Estimates

of Ritz—Volterra

161

Projection

It remains to estimate ||g(£)lli- By the coerciveness of A, "IMOII? < A{g,g) = L*(g,g) - j = Phg{t, z)a{t) - j

B*(g(s),g(t))ds

B*(g(s),

g(t))ds


+£

\\9(s)\\ids)\\g(t)h.

Eliminating the common factor and using Gronwall lemma we get \\g(t)h
< Ch**(t).

We shall further show the following result. T h e o r e m 9 . 1 . Assume that g(t) G HQ and gh(t) € S^ (9.1) and (9.2) respectively. Then

are

defined by

\\g(t)-gh(t)\\ltl2 < / =

(t>2{aijDieDje-{-{aQ-\-v)e2)dx A(e^l+e\\e\\l^+C(e)\\ef

= L*(e, V - M)

- j

B*(e(s), ^{t))ds + e||e||^ 2 + C(e)||e|| 2

Chapter 9. Maximum Norm Estimates

162

+ C ( | l l e ^ H ^ d ^ + Cllell2 and then rp

\\e\\lt
j

\\e(S)\\1>2ds + C\\e\\,

where Wh,*-*

< CiWDeU* + ||e||)

and HV-^lli,*-* < Ch\\D2n^ < Ch{\\D2g\\^ + \\De\\ +

\\e\\^)

< C\h\ In hi1 '2 + h)
and the theorem follows by the estimate l l e l l i . ^ H ^ l l l l e l l x ^ ^ C f t l In%*(*)• R e m a r k . In the proof of Theorem 9.1, D2gh = 0 in each element r is required, i.e. r = 2, however we believe that the estimate is valid for r > 2. On the other hand, when gh is used, for example, in the proof of Theorem 9.4, we can always consider gh G 5£ e v e n if r > 2. We now turn to the gradient type Green function. Theorem 9.2. For G(t) G Hi and Gh{t) G S* defined by (9.3) and (9.4) respectively, we have ||G(t)lli,^ + \\Gh(t)\\i,^

< Clln/^VW

and \\G{t)\\ltl +

\\Gh{t)\\i,i
Proof. Here we shall only discuss G, similarly for Gh- Denoting ip = c/)2G and using the coerciveness of A one can calculate HIGH? ** < [ 2{*ijDiGDjG + (a 0 + v)G2)dx Jo.

9.1. L°°-Estimates

of Ritz—Volterm Projection
163

e\\G\\l^+C(e)\\G\\2

+

= L*(G,V)- /

B(G{s),iP)ds

+e\\G\\l^+C\\G\\2 = (D6h
WGWW^dsWmh,*-* 2

+e\\G\\l,+C\\G\\ <W2DSh\\\\G\\*(t)

+

2e\\G\\l^+C\\G\\2

+C(Jt WGMU^ds)2. Taking e — z//4, it leads to ||G||i,^
\\G{8)\\li4fld3.

By Gronwall inequality we get ||G||i^
= I (Lw,G)dt= Jo

[ Jo

= [ (Dw,6h)a{t)dt
[

(w,L*G)dt [ Jo

\\Dw\\0,p\\Sh\\o, p>a{t)dt

\\w\\2a(t)dt

Jo

< C\lnh\^2

[

[G{t)a{t)](j{t)dt

Jo

< C\\nh\^2

[

\\G(t)<j{t)\\a*(t)dt.

Jo

From the arbitrariness of cr(t), it leads to

HGWH^CIln^/VW.

Chapter 9. Maximum Norm Estimates

164

The theorem follows from the above two estimates. The main aim in this section is to discuss Ritz—Volterra projection Vh,u(i) £ Si? of u(t) G HQ which satisfies L(Vhu(t) - u{t), v) = 0,

v e S*.

(9.5)

A simplified proof for the following weaker results will be given as follows. Theorem 9.3. The Ritz—Volterra projection opreator Vh : HQ -* S^ is almost bounded in W1'00, \\Vhu{t)\\lt00
< C\\nh\ inf [u(t)-v]lf00

< C^lln/ilM*)]^,

1 < I < r.

Proof. Let e — G — Gh- By the definitions of V^u and G, and by Lemma 9.2, we have, for v € S*, / DzPh(u-Vhu)a{t)dt= Jo

I Jo

L{u-Vhu,G)dt

fT = / L*(u — v, e)dt Jo
Hii-vlli.oolleWIII.idt

< C\\nh\ = C\ In h\

f Jo Jo

\\u-v\\lt00
v]lt00cr(t)dt,

and thus \Ph(u - Vhu)(t)\\hoo

< C\ lnh| inf [u - v] l f 0 veS'J:

Finally, the theorem can be derived from the triangle inequality and H-P^Hi.oo < C||w||i |0 o,

\\u - Phu\\it00 < C inf \\u - v||i | 0 0 . ves>ri

Theorem 9.4. The Ritz—Volterra projection operator Vh ' Co(^) —> S^ is almost bounded in L°°(0), ||^^)llo,oc
9.1. L°°-Estimates

of Ritz—Volterra

Projection

165

and \\Vhu(t)-u(t)\\0too

< C|lnfc| inf [u(t)-v]0tOO vesp

< Chl\ Inh\[u(t)]li00,

1 < I < r.

Proof. By integration by parts we can rewrite (9.5) in the form L(Vhu,v)

(u(t)Fi(v)

= J2f

+ /

u(s)F2(v)ds)da.

dr

(9.6) dx*

v G S„ ,

where Fi(v) = a^DiV cos nxj and F2{v) = b^D^ cos nxj. The equality is valid for u(t) G C(Q.) and determines the projection operator Vh : Co(0) —> 5y • Noting that, for g G H2, ] T y (u(t)Fi((7) + J T

u(s)F2(g)d8yjdl

= J

(u(t)F1(g) + J

u(s)F2(0)cfa)
dr

= o, and using the trace estimates for e = (^ — # and the inverse property, l l e l l L L a r ^ C H e l l a ^ + C^llell!,!,,., l|e||2>i,T < hh - hgh,i,r

+ ||//,
9h,i,r

< Ch^iWgn - fflln.r + ||Jhff - s||i,i, T ) + C|| 5 || 2 , 1)T

T + C|M| 2lllT , from (9.6) with v = gn and by Dirichlet formula and Theorem 9.1, we have that / Vhu(t,z)a(t)dt= °

j

^iu(t)(i?,1(e)+ / r dr

F2(e)ds)dldt

+ / 5 Z f u(t){L*e + 6ha(t))dxdt JO


Jo

T

JT

fT\\u{t)\\o,JY l\\e(t)\\i,i,9r v T

+ £HCll2.1.T+ " ( * ) ) *

166

Chapter

9. Maximum

Norm

Estimates

rT


\Ht)\\0tOO**{t)dt [u(t)]0,co
From this t h e desired bound of Vh in L°° follows. Finally, L°°-error estimate can be derived by t h e triangle inequality a n d t h e b o u n d of V^, \\VhU - U|| 0j oo < \\Vh(u - U)||o,oo + \\V - ^||o,oo < C\]nh\[u-v]oiOQ

+ ||U-TA||O,OO <

C\]nh\[u-v]0tOO.

T h e theorem is proved.

9.2

Estimates for Semidiscretization

Consider now t h e m a x i m u m norm estimates of semidiscrete finite element a p ­ proximation for parabolic integro differential equations. Following t h e proof suggested by Bramble, Schatz, Thomee a n d Wahlbin [9] in purely parabolic case, we can begin by defining a semidiscrete Green function gh(t) € S^ satis­ fying f (ght,v) + L(gh,v)=0, veS^ (9.7) \ gh(0) = &h in ft. L e m m a 9 . 3 . T h e r e are t h e following estimates:

WgnW + WlghlWi^Ch-1 and

\\\gh\\\
^Dt\\gh\\2 + A(gh,gh) = J

B(gh{s),gh{t))ds

and, after integration in £,

||0h(t)||2 + 2./||M||? < ||flh(0)||2+C /

f Wg^hdsWg^Widt

Jo Jo


+ v\\\gh\\\l+C

[ \\\gh\\\idt. Jo

167

9.2. Estimates for Semidiscretization

The first conclusion follows immediately from Gronwall lemma. To estimate | | | ^ | | | , introduce an auxiliary function w^t) G 5^ satisfying (v,wht) -L*(v,wh)

= (v,gh),

v G S£,

(

wh{T) = 0 in n,

v*'

^ 0)

there is a priori estimate

I K W I I i < CHICHITaking v — gn in (9.8) and using the definition of gn, it is easy to obtain that \\9h\\2

= (gh,Wht)-

= Dt{gh,wh)

L*(gh,wh)

- ((ght,wh)

+L*(gh,wh))

=

Dt(gh,wh)

and, by integrating in t, lllfffclll2 = (9»,
<

\\Sh\\0,i\\wh(0)\\0tOO

< C|K(0)||0,oo

the lemma is proved. Lemma 9.4. There are ||5h(*)l|o,i
(/

WgnWlidt)1'2 + J

\\gh\\idt
Proof. Taking the weight = (\x — z\2 -\- cu2)1/2, u = jh, denoting ^ = 4>2gh, ¥h — Ph^P and noting (Ph