MIXED FINITE ELEMENTS FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS

MIXED FINITE ELEMENTS FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS

1997,17(3):319-329 MIXED FINITE ELEMENTS'FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS 1 Sun Pengtao ( :r'J. ~i4 ) Ineiiiutc oj Mathematics, Academi...

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1997,17(3):319-329

MIXED FINITE ELEMENTS'FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS 1 Sun Pengtao ( :r'J. ~i4

)

Ineiiiutc oj Mathematics, Academia Sinica, Beijing 100080, China

Abstract In this paper, we study mixed finite elements for parabolic integro-differential s 2 equations, and introduce a kind of nonclassical mixed projection, its optimal L and hestimates are obtained. We define semi-discrete and full-discrete mixed finite elements for the equations, and obtain the, optimal L 2 error estimates. Key words Integro-differential Equations, Mixed Finite Element, Error Estimates

1 Introduction We consider the following nonlinear parabolic integra-differential equations:

== V . {a (U ) (\7U + J~ V u (z , T) dT )} + f (U ), (x, t) (b) u( z , 0) == Uo( x ), x En, (c) U ( z , t) == 0, (x, t) E an x (0,T]. (a)

where

Ut

nc

R 2 is a bounded domain with smooth boundary

E

n x (0, T],

(1.1)

an, T > 0, a, Uo and f are known

functions. Problem (1.1) can arise from many physical processes such as S0111e gas diffusion problems, heat transfer problems with memory, and etc. (see [1]). For approximating the solution u, the classical finite element methods have been considered by several authors [2J,[3J,[4J,[5J in recent years, and the author [6]'[7J,[8J,[9] has studied classical and nonclassical finite element methods for integra-differential equations of evolution. But the mixed finite elements for problem (1.1) have not been considered by any authors. In this paper, we introduce a nonclassical mixed projection: Volterra-type mixed projection from which optimal £2 error estimates can be derived for semi-discrete and full-discrete mxed finite elements for problem (1.1). This paper is organized in the following way.

In §2 we give mixed finite element

formulations and some necessary preparations. The Volterra-type mxed projection will be introduced and studied in §3. Section 4 and 5 contain the error estimates for the semidiscrete and full-discrete mixed finite element aproximations, respectively for problem (1.1). 1 Received

Mar.5,1995; revised Jan.2,1996.

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2 Mixed Finite Element Formulations Let wp,q(O) be the usual Sobolev spaces on 0 with norm

.ro

II . IIp,q,

where

II . lip

denotes

the HP norm on n,o :::; p < 00,1 :::; q :::; 00. Let ( 0, such that

o< ao :::; a(u) :::; al, u E R. We also assume that the functions a(u) and feu) are smooth and bounded together with their derivatives. The global nature of our assumptions ,with respect to u(x, t) does not constitute any serious restriction. In fact, the approximation solutions to be considered will be shown to be uniformly close to the exact solution u(x, t) and thus depend only on the nature of a(u) and f(u) in a neighborhood of the range of u(x, t). Let v == a(u)(Vu+ J~Vu(x,T)dT),b(u)== a(u)-l. We note that

The weak form of (1.1) appropriate form mixed methods follows that, find a 111ap

(u, v) : [0, T] ---+ H x Z, such that (a) (Ut,
---+

Wh

X

Vh such that

(a) (Ut,
v« +

it

'ilu(x, T)dT

~ b(U)V = g(t),

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=

and obtain (Vu)t + Vu gt. Therefore the approximation of Vu can be obtained by working out this ordinary differential equation. Before defining the full-discrete approximation, we introduce some notations. Let N be a positive integer.

g(x, ti),

gi+1 /

2 = (gj+l

Let ~t == T/N,ti = j~t,j = O,l,···,N. We also define gi ==

+ gi)j2, Otgi+ 1 / 2 =

(gi+ 1

gj)j(b.t).

-

The full-discrete approximation is defined by the sequence ({Ui}f=o' {Vi}f=o) : [0,T] W h X Vh such that (a) (8 t Uj+l / 2, tp ) - (V· Vj+l/2,tp) == (!(Uj+l/2),tp),tp E Wh,

(b) (b(Uj+l )Vj+l,,,p)

j

+ (Uj+l, V . 7/)) + (b.t E

r=O

U r+ 1 / 2, \1 . "p) = O,,,p E Vh,

--+

(2.3)

(c) UO = W(O), V O = Y(O), where j = 0,1,"" N - 1, W(O) and V(O) are defined by (3.1). U sing arguments similar to those used to derive the existence and uniqueness of (2.2)

and Brown's fixed point theorem, we can easily obtain that (2.3) has a unique solution. Let X be a Banach space with norm

1I,B11l,2(x) =

r 11,B(t)lli-dt, Jo

11·llx and f3 : [0, T] =>

T

and

II,BIIL=(X) = ess

X. Define

sup

O~t~T

11,B(t)llx.

We will make extensive use of the z-type inequality: ab ~ Ea2 + (b2 j4E), E > O. And in what follows, G will be used to denote various positive constants that are independent of h.

3 Volterra-Type Mixed Projection We define the Volterra-type Mixed Projection of the solution u(x, t) to be a map (W, Y) : [0, T] --+ W h X Vil such that

(a) (\1. (Y - v),
(b) (b(u) Y, 7/,) + (VV, \1 . 7/,) + (J~ W (T)dT, V . "p) == O,,,p E Vh, ( 3.1 ) In order to prove the existence and uniqueness of (W, Y), we introduce two families of projections[lO]-[13]: ITh : Z --+ Vil and Ph : H --+ Wh such that

(a) (V· (q - llhq),cp) = O,q E Z,cp E W h , (b) (\1·"p,w - Phw) = O,W E H,,,p E Vh,

=

=

(c) (\1·)IT h Ph(V,) (or divIlj, Phdiv): Z --+ Who Then ITh and Ph have the following approximation properties:

(a) I/q - IThql/o :S Gllqllrhr,q E H T(0 )2, 1 :S r:S k + 1. (b) 11\1· (q - IT hq)lI-s :S GII\1 . qllrhT+ s, q E H r + 1 (0 )2, 0 :S r, s :S k + 1. (c) Ilw - PhW"-s~ Gllwllrhr+s,w E Hr(o),O ~ r,» ~ k + 1. Lemma 3.1 If (~Y) is defined by (3.1), then there exists a constant G

(3.3)

>

0 such

that

IIWllo :S GIIYllo. Proof We introduce an auxiliary problem, for ~a

(3.4)

f3 E L 2(0), let a be the solution of:

= j3 in O,a == 0 on

ao.

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Then we have a E H 2(O)

(W,f3)

n HJ(O),

II al12:S Gllf3llo.

and

Vol.17

It follows that

= (W, V . (Va)) = (W, V· (II h Va)) = -(b(u)Y, II h Va) - (J~ W(r)dr, V· (II h Va))

~ G(II Yllo + J~ IIW(r)llodr)llf3llo.

IIWllo::; C(llYllo +

Then

it

IIW(r)lIo dr).

Thus, by applying Gronwall's inequality, we have (3.4). . Theorem 3.1 Volterra-type mixed projection (3.1) has exactly one soluiton (W, Y) E W h X Vh • Proof In order to show existence it is clearly sufficient to prove uniqueness. Thus let v = 0 and set ep = W,,,p = Y in (3.1), we obtain

(b(u)Y, Y)

+

(it

W(r)dr, V · Y) = O.

J;

Then by setting ep = W(r)dr in (3.1a), we have IIY115/al ~ (b(u)Y, Y) = 0, so that Y = O. By Lemma 3.1 we conclude at once that W = 0 which completes the proof. In the error analysis of (W, Y) to (u, v), we introduce the following notations: dh = v - Y, eh = II h v - Y, Zh = PhU - w: We first substract (3.1b) from (2.1b) and use (3.2b), it follows with (3.1a) that (a) (V· d h , ep) = O,ep E W h , (b) (b(u) dh , "p) + (Zh' V . "p) Lemma 3.2 For s ~ 0,

+ (J~ Z h ( r )dr, V · "p) = O,,,p

E Vh.

( 3.5)

(3.6) Proof We introduce an auxiliary problem, for

V· (a(u)Va) =

I'

Then we have a E H s+2(O) n HJ(O), and It follows from (3.3a) and (3.3c) that

I'

E H*(O), let a be the solution of

in O,a = 0 on

Ilalls+2

ao.

(3.7)

~ Gllf3l1s.

(Zh,{3) = (Zh' V· (IIha(u)Va)) = (b(u)dh,a(u)Va - IIha(u)Va) +(V . dh, a - Pha) - (J~ zh(r)dr, V . (IIha(u)V'(a))

~

+ IIV· dhllohlnin(s+2,k+l)}llf3/1s + J~ /Izh (r)ll-s drllf3lls. G{lldhllohmin(s+l,k+l)

Then by using Gronwall's inequality, we have (3.6). Lemma 3.3 For s 2: 0,

(3.8)

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Proof Let w E H 2(O). Then, by (3.5a),

and (3.8) follows from (3.3c). Lemma 3.4 For s

Ildhll- s

::;

2: 0,

C{(lldhll o+

it

+(IIV· dhll o +

Ildh(r)llodr)hmin(s,k+l)

it

IIV· dh(r)lIodr)hmin(s+1,k+1)}.

(3.9)

Proof Let a E H s (O)2, it follows from (3.5a) and (3.2a) that

(b(u)d h, a) == (b(u)d h, IIha)

+ (b(u)d h, a -

~ -(Zh' V · Iha) I(b( u)d,,, a)11

Then

(it

::; (lI zh lI-s+1 +

IIha)

zh(r)dr, v . Iha) + (b(u)d,.. a - IIha).

it

Ilzh (r) lI-s+1 dr )llali s

+Clld h Iioliali s hmin(s,k+l). Thus we have (3.9) from (3.6). For s 2: O. Assume that u E Hs+2 (O), and let then we have error estimates:

Theorem 3.2

J~

Ilu(r)llpdr,

Ilulip

==

Clllulllrhr+s,O:::; s:::; k -1,2:::; r:::; k + 1,

Ilu -

WII-s :::;

{

Cillulllr_lhr+k,s == k, 1:::; r:::; k + 1,

Clllulllr+2hr+k+l, s == k + 1,0 :::; r

:::; k + 1;

Ilv _ YII-s :::; { Clllulllr+lhr+s,O:::; s :::; k, 1 :::; r :::; k + 1, CII'1Llllr+2 hr+k+ 1 , S == k + 1,0:::; r:::; k + 1;

IIV· (v -

Y)II-s

< Clllulllr+2hr+s, 0 ~ s, r

::; k

(3.10a)

+ 1.

(3.10b) (3.10c)

Proof The methods of proof are similar to those of [12],[13] except the analysis of the integral term. We take 'ljJ == eh in (3.5h) and use (3.3a) to have

(b(u)eh,eh) == -(V ·dh,Zh) - (b(u)(v - IIhv),eh) - (V .dh,J~ zh(r)dr) == -(b(u)(v - IIhv),eh). Then Ilehllo ~ Cllv - IIhvllo. The following arguments are similar to those of [12],[13], we can easily have

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It follows that

lIu - WI/-s :::; II z hll - s + Ilu - Phull-s :::; C{/Ivll ph min(p,k+l)+min(s+l,k+l)

+ IIV' . vllrhnlin(r,k+l)+min(s+2,k+l)

+ Iluli r hmin( r,k+ l)+min( s,k+1)},

where p

and

~

1.

Ilv - YII-s :::; C{llvllphmin(p,k+l)+min(s,k+l) +I/V' . vllrh min(r,k+l)+min(s+l,k+l)},

IIV'· (v - Y)II-s :::; CIIV' · v/lr hr+s. So we have

(3.10).

Theorem 3.3 For s ~ O. Assume that u, Ut E Hs+ 2(0), then we have error estimates:

lI(u - W)tll-s :::;

C(I/Iulllr + Ilutllr)hr+s,O:::; s:::; k -1,2:::; r:::; k + 1, C(lllulll r+l + IIUtllr+l)hr+k, s == k, 1 :::; r :::; k + 1, { C(IIlulllr+2 + Il utllr+2)h k+ 1 , s == k + 1,0 ::s r ::s k + 1;

(3.11a)

T+

II (v

_ Y) t 1/_ s

C(III u IIIr+1 + II Ut II r +1 ) hr+s , 0 < s < k, 1 < r ::s k + 1, C(llI u lllr+2 + Il u t llr+2)h r+k+1 , s == k + 1,0 :S r:S k + 1; s Y)t II-s :::; C(lllulll r+2 + II u t llr+2)h r+ , 0 ::s S, r ::s k + 1.

:::; {

IIV' . (v -

(3.11b)

(3.11e)

Proof Let dd h == (v - Y)t, eei, == (Ilh v - Y)t, ZZh == (PhU- W)t. We differentiate (3.1) to obtain (a)

(b)

(V' · dd h , VJ) == 0, VJ E W h , (b(u)ddh, 'l/J) + (bt(u)d h, 'l/J) + (ZZh' V' · 'l/J) + (Zh' V' . 'l/J) == 0, 'l/J EVil.

(3.12)

We still introduce the auxiliary problem (3.7). Then we have

(ZZh,{3) == (ZZh' V'. (Ilha(u)V'a))

== (b(u)ddh, a(u)V'a - Ilha(u)V'a) + (V' . dd h, Q

-

Pha)

-(bt(u)dh, Ilha(u)V'a) - (Zh, V'. (IIha(u)V'a)) < C{lIddhllohmin(s+l,k+l) + IIV'. ddhllohmin(s+2,k+l) +lId hll- s- 1

+ Il z hll - s }II,6lls.

It follows from (3.6) and (3.9) that

/Izzhll-s

::s C{(llddhll o + Ildhll o + J~ /Idh(r)llodr)hmill(s+l,k+l) +(IIV· ddhllo + IIV· dhllo + J~ IIV'· dh(r)llodr)hluin(s+2,k+l)}.

(3.13)

Let a E HS (0)2, then

+ (b(u)dd h, a - Ilha) == -(bt(u)dh, Ilha) - (ZZh' V'. Ilha) - (Zh' V'. IIha) + (b(u)dd h , a :S C{lIdh/l- s + II zhll-s+l + IIzzhll-s+l + Ilddhllohmin(s,k+l)}llalls.

(b(u)dd h, a) == (b(u)dd h, Ilha)

-

IIha)

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It follows from (3.6),(3.9) and (3.13) that

Ilddhll-s

Ildhll o+ J~ IIdh(r)lIodr)hmin(s,k+l) +(IIV· ddhll o+ IIV· dhll o+ J~ IIV· dh(r)llodr)hmin(s+l,k+l)}.

~ C{(ddhll o +

(3.14)

If we take 'ljJ == eeh in (3.12b) and use (3.3a), then we have

(b(u)eeh,eeh) == -(b(u)(v - IIhv)t,eeh) - (bt(u)dh,eeh)

-(V· dd h, ZZh) - (V . dd h, Zh) == -(b(u)(v - IIhv)t,eeh) - (bt(u)dh,eeh). Then, f1eehflo ~ C{II(v - IThv)tllo + Ildhllo}. So we can obtain II ddhllo ~ Ileehllo + II(v ~

IThv)tllo

C(llvllr + IIVtllr)hr, 1 ~ r

~ k

+

1.

Similarly as (3.8) was derived, we can easily.obtain from (3.12a) that

Next, note that

(V· eeh,
v· eeh == 0. r, ~ C(IIV· vll r + IIV· vtllr)h 1 ~ r

where we have used (3.2a) and (3.12a), so that

~ k + 1. Hence, \\V· ddhllo == IIV· (v - IIhv)tllo Using areuments similar to those used to derive Theorem 3.2, we have (3.11). Corollary 3.1 Assume that u, Ut E Hk+1(0) n Loo(O), k > 0, then there exists

constant C

== C( u) > 0, independent of h such that

IIWllo,oo + IIWtllo,oo < c. And if u,

Ut

E H

k+ 2(0)

(3.15)

n W1,ooCO), k > 0, then we have again

II Yllo,oo + IIYillo,oo

~ C.

(3.16)

Proof It is easy to obtain from the inverse inequality, triangle inequlity, (3.3), (3.10) and (3.11). We omit this proof here.

4 Semi-Discrete Mixed Finite Element Approximation In this seciton we will employ the new projection (W, Y) and derive the optimal L 2 error estimates for the semi-discrete mixed finite element approximation. Theorem 4.1 Let {U, V} be the solution of (2.2). If the solution u of (1.1) is such that u E L oo(H k +2 ) n LOO(W1,00), Ut E L 2(H k +2 ) n Loo(W1,00), where k is positive integer, then there exists a constant C > 0, independent of h such that

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~ Ch + { llu IILOO (Hk+2) + Ilu t IlL2(Hk+l )} . k

Proof Let

~

1

=U-

W,1]

=W

- u,(

=V -

Y. We substract (2.1a) from (2.2a) and

(2.1b) from (2.2b), it follows from (3.1) that (a) (et, ep) - (V· (, ep) == -(1]t, ep) + (f(U) - feu), ep), ep E W h , (b) (b(U) ( , 'ljJ) + (e, V . 'ljJ) + (J~ r )dr, V . 'ljJ) == ((b(u) - b(U) )Y, 'ljJ ), 1/, E

e(

Vit,

(4.2 )

We differentiate (4.2b) and take 1/J = ( to obtain (b(U)(t, ()

+ (bt(U)(, () + (~t, V

= (( b(u) - b(U))yt , ()

+ (e, V

. ()

+ ((b(u) - b(U) )t Y, o.

e and ep == et in (4.2a), respectively.

We take ep = have

. ()

(4.3)

Adding both of them to (4.3), then we

~ ~ {(b(U)(, o + 11~116} + II~tIl6

=

-~(bt(U)(,() - (17t,~) + (f(U) - f(u),~) + ((b(u) +((b( u) - b(U))t Y, () - (7]t, et)

+ (f(U)

b(U))Yf,() 7

- I( u), et) ==

LG

j .

(4.4)

j=1

If we integrate (4.4) with respect to t, it follows from Theorems 3.2-3.3, Corollary 3.1 and z-type inequality that

it where ~(o)

~ ~ {(b(U)(,() + 1I~1I6}dt ~ Co{II~1I6:- 1I(116},

= ((0) = O.

r

1

ft

i o c,« = -2 io (bs(U)(~t + wt)(, ()dt ::; C(lI(IILOO(LOO) + 1) it IICII6dt + e it II~tll6dt, where b, denotes one order derivative of b.

i t Gsdt = it {(bs(u)(Ut - ut}Y,() + ((bs(u) - bs(U))UtY,C)}dt

::; C(1I(IILOO(Loo) + 1){ h Zk+ Z i

t

(1IulI~+l + IIUtll~+l)dt

+ it (1I~1I6 + 1i(1I6)dt} + [; it lI~tIl6dt. Using arguments similar to those of G 1 and G 5 , we have

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We now make an inductive hypothesis: (4.5) where K

< 1 is a positive constant. They by taking

E

sufficiently sm all and applying

Gronwall's inequality, we have

To complete our argument, we must show that (4.5) is right for h sufficiently small. We use the inverse inequality and (4.6) to see that

It is well known that we can easily have (4.5) naturally for k > 0 and h sufficiently small through inductive reasoning. We see that Theorem 4.1 follows from (4.6),(3.10) and

(3.11).

5 Full-Discrete Mixed Finite Element Approximation In this section we will give the optimal £2 error estimates for the full-discrete mixed finite element approximation. Theorem 5.1 Let ({uj }f=o, {vj }~o) be the solution of (2.3). If the solution u of (1.1) is such that u, Ut E L OO(H k+2 ) n Loo(W1,OO), Utt, Uttt E L 2(L2 ), where k is positive

integer, and the following condition is satisfied: there exists a positive constant R such that (5.1) then we have sup {II(U - u)jllo O~j~N

+ II(V -

v)jllo + l::,.t

j

L 118 (U t

u)'-l/21Io}

1=1

(5.2) Proof We first take values at times of tj+1 and tj in (2.1a) , add both of them and

devide it by 2. In the second place we take value at time of tj+1 in (2.1b). Then we obtain (a) (u{ +1/ 2,
(b) (b(uj+ 1 ) , v j + 1 , 'l/;) + (ui+ 1 , V· 'l/;) + (J~j+l u(r)dr, V· 'l/;) = 0, 'l/; E Z. We subs tract (5.3a) from (2.3a) and (5.3b) from (2.3b) to obtain (a) (at~j+1/2,
(5.3)

(5.4)

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Let "p

Vol.17

= (j+l/2, it follows from (5.4b) that

+ (Otei+ 1 / 2 , \1. (i+1/ 2 ) + (e i+ 1/ 2 , \1. (i+ 1/ 2 ) = (b(Ui)-;tUi+1)(i, (i+1/ 2 ) + ((b(ui+ 1 ) _ b(Ui+ 1 ))Oty j+1/ 2 , (i+ 1 / 2 ) (b(U i+ 1)Ot(i+ 1 / 2 , (i+1 / 2 )

+(

(b(ui+l)-b(ui»)-(b(Ui+l)-b(Ui» yi ij+1/2) ~t

+(It

.r: W(r)dr - LltWi+ J

,~

1/ 2 } ,

(5.5)

\1. (i+ 11.2 ) .

j+ We take tp = e 1 / 2 , tp = Otej+ 1 / 2 and tp = Q = It{J/~+l W(r)dr - LltWj+1/2} in J (5.40a), respectively. If we add these equations to (5.5), then we have (b(U j+ 1 )Ot(i+ 1/ 2 , (j+l/2)

+ (Otej+ 1/ 2 , ej +1/ 2) + (Otej+ 1/2 , Otei+ 1/ 2 )

+( b(Uj)-;~Ui+l)(j, (i+1/2) + ((b(ui+1)

_ b(Ui+l n8tyi+1/2, (i+1/2)

+((b(ui+l)-b(U~))~~b(Ui+l)-b(Ui))yj, (j+1/2)

_ (Otej+1/2 , Q) -( Otrf+ 1 / 2 , Otej+ 1 / 2 + ej+ 1 / 2+ Q) + (u1+ 1/ 2 _ Ot uj+ 1 / 2, Otej+ 1/2 +e j+ 1/ 2 + Q) + (f(U j+ 1 / 2) - fj+1/2( u), Otej+ 1 / 2 + ej+1/ 2 + Q)

(5.6)

7

= L: K j . i=l

Now we multiply (5.6) by 2Llt and sum on j from 0 to M - 1, where M is positive integer such that 1 :::; M

< N. We often use s-type inequality, Taylor's formula, Theorems

3.2-3.3 and Corollary 3.1 in the following error estimates, where the methods of [15] are often used.

2Llt

M-1

L:

i=O

{(b(Ui+ 1)Ot(i+ 1 / 2, (i+ 1 / 2 )

2 Coll(MlI6+ IIe MI16 + 2Llt

+ (Otei+ 1/2, ej + 1 / 2 ) + (Otej+ 1/2, Otej+ 1 / 2 ) }

L: II Otej+1 / 2116

M-l

i=O

M-1

L: {K 1 + K 2 + K 3 + K 4 + K 6 } i=O M 2k 2 jI16)} :::; C{h + + (Llt)4 + Llt L: (II(jI16 + IIe j=O 2Llt

M-l

2Llt

L:

j=O

{K s + K 7 }

:::;

C(

sup Ilejllo,oo 05:j 5: Al -1

+Llt

M-1 + €Llt L: II 0tej +1/2116. j=O

+ 1){(h 2k+2 + (Llt)4

L: (lI(j116 + Ile j 115)} + ELlt L: IIOtej+1/2116. M

M-1

i=O

i=O

SlID: MIXED FINITE ELEMENTS FOR PARABOLIC EQUATIONS

No.3

If we take

E

329

sufficiently small, apply discrete Gronwall's inequality and make an induc-

tive hypothesis: sup O~j~M-l

II(i 110,00 < K < 1,

it follows that

(5.7)

M-l

sup O~i5:M k

:::; C{h +

{11(illo+ Ileilio + ~tL Il ote'+1/ 2 110}

1

1=0

(llu IIL OO (H k+ l ) + Il u t IIL oo(H k+l ) ) + (~t)2}.

(5.8)

Applying inductive reasoning[15], we can easily see that (5.7) is still right for M == N under the condition (5.1). So that (5.8) is right for M == N also. Thus, by combining (3.10),(3.11) with (5.8), we have proved Theorem 5.1. Acknowledgement The author would like to express his sincere thanks to Professor Yuan Yirang and Professor Liang Guoping for their elaborate instruction. References 1 London S, Staffans O. Volterra Equations. Lecture Notes in Math. Spinger- Verlag, Berlin, New York, 1979. 2 Cannon J R, Lin Y. A prior £2 error estimates for finite-element methods for nonlinear diffusion equations with memory. SIAM J. Numer. Anal., 1990,27:595-602. 3 Cannon J R, Lin Y. Non-classical HI projection and Galerkin methods for nonlinear parabolic integrodifferential equations. Calcolo, 1988 , 25: 187-201. 4 Lin Y. Galerkin methods for nonlinear parabolic integra-differential equations with nonlinear boudary conditions. SIAM J.Nurner. Anal., 1990,27:608-621. 5 Thomee V, Zhang NY. Error estimates for semidiscrete finite element methods for parabolic integrodifferential equations. Math. Comp., 1989,53: 121-139 6 Sun Pengtao. The finite element methods for nonlinear hyperbolic integra-differential equations J.Eng.Math. (in Chinese), 1994,11(2):76-82 7 Sun Pengtao. The interpolated postprocessing technique of the F .E.M. for nonlinear parabolic integrodifferential equations. J.Syst.Sci.& Math. (in Chinese),1996,16(2):159-171 8 Sun Pengtao. Elliptic H2_ Volterra projection and the H1-Galerkin methods for the iritegro-differeritial equations of evolution. J.Appl.l\1ath. of Chinese Univ.,1995,10(1):11-24 9 Sun Pengtao. The finite element method with moving grid for the nonlinear parabolic integrodifferential equations. J.Shandong Univ., 1995,30(1):43-51 10 Raviart P A, Thomas J M. A mixed finite element methods for 2-nd order elliptic prolems. Mathematical Aspects of Finite Element Methods, Lecture Notes In Math., Vol.606, Springer-Verlag. 1977.292315 11 Douglas Jr J, Roberts J E. Mixed finite element methods for second order elliptic problems. Mathernatica Applicada e Computacional, 1982,1: 91-103 12 Douglas -11' J. Global estimates for mixed methods for second order elliptic equations. Math. Comp, 1985, 44: 39-52 13 Brezzi F, Douglas Jr J, Marini L D. Two families of mixed finite elements for second order elliptic problems. Numer. Math., 1985,47: 217 -235 14 Johnson C, Thomee V. Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Model. Math.Anal.Numer., 1981,15:41-78 15 Wang Hong. The estimates of stability and convergence of fully discrete finite element methods for nonlinear hyperbolic equations. J.Cmpu. Math., 1987,2: 164-170(in Chinese) 16 Lin Y, Thomee V, Wahlbiu L. SIAM J. Numer. Anal., 1991,28: 1047-1070