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MATHEMATICS OF COMPUTATION VOLUME 45. NUMBER 172 OCTOBER 1985, PAGES 319-327

Finite Difference Approximations of Generalized Solutions By Endre Siili, Bosko Jovanovic and Lav Ivanovic Abstract. We consider finite difference schemes approximating the Dirichlet problem for the Poisson equation. We provide scales of error estimates in discrete Sobolev-like norms assuming that the generalized solution belongs to a nonnegative order Sobolev space.

1. Introduction. Recently, there have been many theoretical advances in constructing finite difference schemes approximating boundary value problems for partial differential equations with generalized solutions belonging to Sobolev spaces. For example, Lazarov [4] presents a finite difference approximation of the Dirichlet problem for the Poisson equation with a generalized solution belonging to the Sobolev space Wk-2 of integer order k = 2, 3 using the so-called Bramble-Hilbert lemma [1]. Unfortunately, the Bramble-Hilbert lemma is stated only for integer-order Sobolev spaces. Recently, Dupont and Scott [3] gave a constructive proof of this lemma using an averaged Taylor series and extended it to fractional-order Sobolev spaces. In this paper a basic framework is given which allows the application of the finite difference method in order to approximate generalized solutions belonging to Sobolev spaces Ws-p, 0 < í < 4, 1 < p < co (Theorems 1 and 3). Proofs are based on the Dupont-Scott approximation theorem. We shall prove a discrete interpolation inequality (Lemma 2) which will enable us to derive several scales of error estimates (Theorems 2 and 4). For simplicity, the analysis in this paper only deals with the Dirichlet problem for the Poisson equation in rectangular domains. Extensions to other elliptic boundary value problems in less special domains or to nonlinear problems are possible.

2. Preliminaries and Notations. Let si be an open rectangle in two-dimensional Euclidean space R2 and 1 < p < oo. Throughout the paper Ws-p(s4) is the Sobolev space of order s ^ 0 (cf. [8]) equipped with the Sobolev norm II Ir0 -VI ¿-i \u\k,p,jtr f \\u\\s.p.^~

with I \P

\U\k.p.j/ =

\~'

Il

a

\\P

¿j \\D°'U\\l>'(j*')> |«|-*

Received May 16, 1983; revised September 14, 1984. 1980 Mathematics Subject Classification. Primary 65N05, 65N10. ©1985 American

Mathematical

Society

0025-5718/85 $1.00 + $.25 per page

319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

320

ENDRE SÜLL BOSKO JOVANOVIC AND LAV IVANOVIC

if s is integer, and m \\p

m n''

i \p

\\"h,p,*=\M[s],p,s/+mi,p,J*, if s = [s] + a, with [s] = integral part of s, 0 < a < 1 and \Dau(x) \u\s,p,jf

^[s]^L

- Dau(y)\p

\x-y\2

+°p

N will stand for the set of nonnegative integers. P'(s/) will denote the set of polynomials in two variables of degree < / over the set sé, for any / e N. The next lemma is an easy consequence of the Dupont-Scott approximation theorem [3] (the case a = 1, p = 2 follows from the Bramble-Hilbert lemma [1]). Lemma 1. Suppose s = I + a, where 0 < o < 1 and / e N. Let rj be a bounded linear functional on Ws-p(sé) such that P'(sé) c kernel(rj). There exists a positive constant C (depending on sé\s,p) such that for any u e Ws'p( sé)

h(«)l< cHs.p,*?Remark 1. Lemma 1 also follows from the Tartar lemma [2]. Remark 2. If tj(m) = 0 for some polynomials of degree > /, then an analogous estimate is valid, containing only a part of the seminorm |w| ^(cf. Lazarov [4],

s e JV,p = 2). Let S¿'(&) denote the space of distributions on 0, for any open set & c R2. Define the differential operator A on 3>'(0) by 32w

32w

Au = —- + —-. 3xj dx2

Let us assume, for the sake of simplicity, that ß is an open rectangle in R2 with boundary 3Í2, and consider the Dirichlet problem

(1)

Au= -f

in £2,

(2)

u= 0

on 3ß.

By changing rectangle is

variables, we may suppose, without

loss of generality,

that the

S2= (0,tt) x(0,7t). Throughout the paper we assume that (1) has a unique generalized (distributional) solution in Ws'p(iï), 0 = u* G„ will be called mollifier. Remark 4. Since Gv is a distribution

with compact support, the convolution

u * Gv

is well defined. For h > 0 and v = (»»,,p2), we set £2„ = {x = (xj,x2)

g R2: /j^

< x, < 7T- /u-,/2, / = 1,2).

Let u g 3>'(ü) and w* g 3'(R2) be any extension of u. Tvu will denote the restriction of Tvu* to Í2„. Remark 5. Let us observe that Tvu is well defined since it does not depend on u*. For simplicity, we shall write Tv „ instead of T(v v ,. 4. Construction

of Difference Schemes. Pick a nonnegative

integer N > 2 and let

/¡ = 7T/./V.We define the following grids

R2,,= [x = (x{">, x2'2>) G R2: x^

= ij ■h, ¡i} < ooj = 1,2J,

uh = £2 n R2,

yÄ = 3Í2 n R2A,

w* = "/,UyA,

YÍ = y, n({0,W} x(0,tt)),

Y,2= Y,n((0,7r)x{0,^}), YA3= Y„n({0}x(0,77)u(0,7r)x{0}),

uh = "a u Ya-

For v, a function of discrete arguments, defined on R2,, set v(x + e,h) - v(x) (VjV)(x) = ^->-±—^, t;(x)-t;(x-e7/,)

(Y/i;)(x) =-r-,

y = 1,2, .

y = L2,

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322

ENDRE SÜLL BOSKO lOVANOVIC AND LAV IVANOVIC

with ex = (1,0), e2 = (0,1), and define Ahv = VxVxv + V2V2v.

A function v of discrete arguments defined on uh (or on üh and equal to zero on yh) is said to belong to Lp(uh), 1 < p < oo, if there exists a positive constant M,

such that

IHU=(ä2 E Hx)f) ">2E H*)f Let us suppose that v is defined on ah (or on wAand equal to zero on yh). The discrete Fourier transform v of v is given by vk=

E

A2tf(-*) siniA:, • x,) sin(&2 • x2),

k = (kx, k2), x = (xx, x2).

The inverse discrete Fourier transform of v is defined by v(x) =

withKh = {k = (kx,k2)e

—

E

u/t sini/c, • xx) sin(A:2 • x2),

N x N: 0 < kjh < 0. Let w = I_aJlv. It follows that

\H\p-a,p,h=\\4i!.p,h and HHI^^HMLp,/.. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

FINITE DIFFERENCE APPROXIMATIONS OF GENERALIZED SOLUTIONS

323

Moreover, II II W\\r-a.p.h

^ r\\ \\l~H f< Q\w\\p,h\M\ß-a,p,h

and the desired inequality follows immediately. Consider the finite difference scheme

(3)

Ahz = VxVxi)i + V2V2r)2, z(x)

(4)

= 0,

x

x G yh,

with ijy defined on uh U y/ and equal to zero on y¿,j = 1,2. An easy argument based on the discrete multiplicator techniques [6] shows that

(5)

Vh,P,h < CdlViV^iUp,*+||V2V2t)2|Ua

(6)

\\A\\,p,h< C(|[V,lhL.A +|[V2TÍ2L,a),

(7)

lklU* 2/p,

1 < p < oo, we may associate with (1), (2) the finite difference

(10)

Ahv= -(T22f)(x),

(11)

v(x) = 0,

scheme

X '

x

Yh-

Error estimates will be given in Section 5. Let us turn to the case when u, the solution of boundary value problem (1), (2), belongs to Ws-p(ti), 0 < s < 1 + 1/p, 1 < p < oo. Define

W'-p(Q) =

»"■'(0),

0.î2,2/p < 5 < 4. Likewise, |v2V2ih|L>.A < c/iî_2|«|îi/)iB,

2/p < S < 4,

and that completes the proof for k = 2. (b) Let k = 1. By (6) it suffices to estimate |[VyT)7||p,A,j = 1,2. In the same manner as in (a) we conclude that

vMiih, i2h) = J{ñ(l,0) - 6(0,0) -/^ is a bounded

linear functional

on Ws-p(E),

82(s)(ü(l,s) - 6(0, j)) ds} s > 2/p,

with a kernel

D P2(£).

Therefore,

ItViTjjII/,^< chs~l\u\StpM, 2/p < s < 3, and, similarly,

Itv^zllp.A< ch'-^ul^Q,

2/p < s < 3.

That completes the proof for k = 1. (c) Finally, let & = 0. Let us estimate \\7]j\\ph,j = 1,2. Since

rjiO'j/J,í2/j) = 6(0,0) - f is a bounded linear functional on Wsp(E),

02(s)«(O,j) ds

s > 2/p, with a kernel 3 Px(£), thanks

to Lemma 1,

hiWp.h< chsWl,p,Q> 2/p < 5 < 2, and, similarly,

hiWp.h< chs\uYp,a,

2/p < s < 2.

By (7) we obtain the desired error estimate. Lemma 2 enables us to derive scales of error estimates. Theorem

2. Let u be the solution of boundary value problem (1), (2) and v the

solution of discrete problem (10), (11). If u G Ws-p(Q), 2/p < s < 2 and 0 < r < 2, or 2/p

< 5 < 3 a«d1->u* g WJ,/,(ß*) is continuous for 0 < s < 1 + 1/p, s + 1/p, 1 < p < 00. Finally,

Itv^t.A

< Chs~l[u\s_pM,

0 < i < 1 + 1/p,5 * 1/p.

Similarly,

|[v27J2ILa< Chs-l[u\s%pM,

Q.A