Journal of Mathematical Sciences, Vol. 72, No. 3, 1994

CONVERGENCE BOUNDS OF DIFFERENCE SCHEMES FOR SECONDORDER ELLIPTICAL EQUATIONS IN DOMAINS OF ARBITRARY SHAPE

S. A. Voitsekhovskii and V. N. Noviehenko

UDC518:517,944/947

The fictitious domain method is applied to construct a difference scheme for the first boundary-value problem for elliptical equations of second order in domains of arbitrary shape. The rate of convergence bound 1

I/b?- 7~I!ue~a o~~ ~lh5- Itf IIL:~.~,

is proved, where ~] is the polylinear extension of the solution of the difference problem, -u is the solution of the original problem continued as zero in ~1, and ~I is the complement of the domain ~2 to the rectangle 9 o.

The Dirichlet problem for the second-order elliptical equation in domains of arbitrary shape is solved in [1] by a difference scheme based on the fictitious domain method and exact difference scheme operators. The rate of convergence of this scheme was estimated in [I] as O(h 1/4) in the W21 norm. In this paper we show that the difference scheme proposed in [1] in fact satisfies the rate of convergence bound l

] z I[w~(~ol~ Mha [][ IIt,~,. 1. Let ~ be a bounded domain in the plane of the variables x = (Xl, x2) with the boundary 17 E C2. We consider the boundary-value problem in f~ 2

-~

k~(x)~ +q(x)u=I(x), xEn;

(1)

~l

u(x)=0,

zEE.

(2)

Assume that fix) E L2(f~), the function q(x) E Lo~(f]) and is nomaegative. Also assume that the functions k~(x) E Wo~1(9) and satisfy the condition kc,(x)~>ks>0,

cz= 1, 2.

(3)

The problem (1)-(3) has a unique generalized solution in the class WzZ(f]), which satisfies the bound [3] [Iu I]~g~a~~ M fl[ [[rz,a). By M we denote constants that are independent off, e, and h. We solve the problem (1)-(3) by the fictitious domain method, completing the initial domain f] to the rectangle f~0 = {x = (x I, x2): 0 < x,~ < l,~, c~ = 1, 2} with the boundary 170. In the rectangle f~o we consider the following Dirichlet problem:

Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 20-25, 1991. Original article submitted November 14, 1989. 1072-3374/94/7203-3061512.50

©1994 Plenum Publishing Corporation

3061

0

e

-

o-V)

+ q

x E 9--o;

=

(4)

u~(x) = O, xEro,

[°%1 [--8~jr

[Udr = O,

(5)

= O,

where

~(x)=

q(x),

xEf2,

o,

xE~,

x

/ k~ (x),

x E f~,

e -~,

xEf2~,

Qi = 2

ON -- ~

xEQ,

o,

xE~.

?(x)=

k~( ) = ~ 0%

{ f(x),

a=

1, 2,

~o\~,

e

Oue

~ (X) ~

COS(n, X~,),

~1

n is the outer normal to P, e is a sufficiently small positive number, [u] r is the discontinuity of the function u(x) on the curve F. The following results were established in [1]. L E M M A . The solution of the problem (4), (5) is from the class W23/2(f2o) and it satisfies the a priori bound

IIu~ II _~

~ M I1f IIL=(~,.

~v~ (a,)

(6)

T H E O R E M 1. The solution of problem (4), (5) converges as e --, 0 to the solution of problem (1)-(3) with the rate of convergence bound

I[u - u~ll~,, < a./~ I[t [[,,(~).

(7)

2. In the rectangle fl0 we define the uniform grid & = ~ U "~, where w is the set of interior nodes and 3~ is the set of boundary nodes. Denote o~1+ = o~1 U v + 1, O~x+ = o~2 O.y+2,.y+~ = {x E V: cos (x a, n) = 1}, ot = 1, 2. We approximate problem (4), (5) by the following difference scheme: ')

_A

TxT~ (qy) = % x E o;

- - ~ (a~y~c,),~~ + o..~--1

y (x) = O,

x E ~',

(9)

where as(x ) = PaT3_a(kaE), ~ = 1, 2; 9 = T1T2(f); T~ are exact difference scheme operators [4] 1

(1 - - I t l ) u ( x ~ +

T~u (x) =

( 2 - - ~) thx, xi + (~x - - 1) ths) dt,

--1

~=1,

2,

Pa are the averaging operators [1] 0

P~u(x) = S u(xl + ( 2 - - a ) th 1, x2+(~z--

l) th~)dt, o~ ~ 1 , 2,

--1

= y (x) + ( ~ - - x~) y~, (x) + ( ~ - - x~) YL (x) + + ( ~ - - x~) (~2 - - x~) YLL (x), 3062

= (~1, ~) E e (x),

(8)

x E co,

e (x) = (xl - - hl, xO X (x2 - - h2, x2).

Applying the method of energy inequalities to solve the difference problem (8), (9), we obtain the a priori bound The proof relies on the inequality (T1T2(c;0)), y)L2(oD >_ O, which follows from the relationship

lly IIw~,(~) -< MllfllL~(m.

(T~T: (qy), y) = J q (y)= dx. The difference problem (8), (9) is thus uniquely solvable. 3. Let us investigate the rate of convergence of the difference problem (8), (9). For the error z = y - u e we have the problem 2

qx=-r ~ ,

=

¢~-~I

x~o;

(10)

,7=1

z(x) = O, xE?,

(11)

where

,,__,

=

:<

,.

=

'1o (x) = TaT~ (q(.) (u (.) - - u"~)). Scalar-multiplying Eq. (10) by z and using the summation by parts formula, we obtain

UV~=~, i~.(.~×.~ + ~V'a=--*~.I1"~.,~#×..,+ 5 77(;) = dx =

= i#". zO<¢×.. + in", zO<.,<4 + 5 ?i.. -a.)~ dx.

(12)

rio

Using the Cauchy-Bunyakovskii inequality, we obtain from equality (12)

Vm, z~, fL,,~?×o,, + !1V~,~, L
,(I),

_*<~4. i~.,.+×,~.,ll V~z~, I!L.(~+×o., + U,., u~,<,.+×.,/ x .

"

fi

=~ I

>/~v a,z~, fl,.,,<~+×o>,,+ tl l/-~ (u:- ud lls.:,,~o,II ] / q ~L,,,.,, where r/*(~) = ~7(a)/x/'~, a = 1, 2, Hence we obtain the bound

Uv lz., k.<~,+,,,o., + itV~z;, k.,,~#×.,, + IJl/~;.IL,,,°, <~ M [ ., _. . m , ,~~,(o>+ × ~,, + I}n "(~>L,,~ + × o,,, +

flV~q ( u~ -

(13)

;,~) LI~,,~.,] .

Using the bound 3

Ul/-q iu~ - ;,~)ilL~,.o,~< MhV IIu~llw2:, (~ we obtain from (13) .3

W~2 (f~°j

(14)

where h = max(h 1, h2). 3063

Let us estimate the norm

tl~l*a)la~(~+xo /

. The norm

Jl~l*(~)/Ir~(o+x,o,; is estimated similarly.

Consider the linear functional

[~{ ou~ --u;~)) ,v< ~, (~) =

y ~

el(x) = {( = ((:, ( 2 ) : x : - h : < (1 < xt, x2-h2 < ~2 < x2 + h2} is mapped by the linear transformation s~ = (~=-xc~)/hc~)/hc,, a = 1, 2 on the domain E 1 = {s = (s 1, s 2 ) : - 1 < Sl < 0, - 1 < s 2 < 1}. Let u~(( 1, (2) = ue(x1 + qh 1, x2 + s2h 2) = u~(sl, s2) = fte(s). The functional n*O) is transformed to the form The domain e 1 =

-1 -1 kj ~'-8~--~ -- [u~ (o, o) -- u~ (-- 1, 0)1 ds

:

,~*(1)

hi

V

(15)

0 1

S S?~ d~

--1 --I

Clearly, if

ue E W23/2(e1), then tie E W23/2(E1). The functional r/1 *(1) is linear m a E. Using the Cauchy-Bunyakovskii

inequality, we obtain from (15) 1

I n'("t ~< W

-, -, k, ~,-N-~) ds

+ 2 tlu~llc,:,, V:

~<

M W2 ¢EO

This functional vanishes on first-degree polynomials. Applying the procedure of [5], we obtain M

In *")1<~ V~V: Ilud 3

W: (e1,

Hence (16) W: (rid

Similarly,

II,I ,L.,o,~×,,,,, <~ M ~

llu d ±

.

(17)

W2 (~0)

Substituting the bounds (16), (17) in inequality (14) and using the bound (6), we obtain

L--¢-2- + Since ~ E W21(flo) and

Ilztlw,~(~°)~ < Mllzllw~(~) [21, we

(18)

have from (18)

It; "=ll"m, <~ M ~v ~ IIf il~,(.).

(19)

The bound (7), the inequality

and (19) lead to (20) where u(x) is the solution of problem (1)-(3) continued as zero in ill. Setting e = h a, we obtain from (20) 3064

The maximum rate of convergence is obviously achieved for = = 1/3. Taking e =

h 1/3, we obtain the final bound

1

[]u -- g Itw~(ao ~

Mh-g- I]f ]]c,(a).

THEOREM 2. The extension ))(x) of the grid function y(x) solving the difference problem (8), (9) (e = h 1/3) converges as h --, 0 to the function ~(x), which vanishes in ~1 and is identical with the generalized solution of problem (1)-(3) in f~. The rate of convergence of the process satisfies the bound 1

REFERENCES 1.

2. 3. 4.

5.

S. A. Voitsekhovskii, "Approximate solution of the Dirichlet problem for second-order elliptical equation in domains of arbitrary shape," Vychisl. PriN. Mat., No. 54, 26-31 (1984). O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Moscow (1973). O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptical Type [in Russian], Moscow (1973). V. L. Makarov and A. A. Samarskii, "Application of exact difference schemes to estimate the rate of convergence of the method of lines," Zh. Vychisl. Mat. Mat. Fiz., 20, No. 2, 371-387 (1980). A. A. Samarskii, R. D. Lazarov, and V. L. Makarov, Difference Schemes for Differential Equations with Generalized Solutions [in Russian], Moscow (1987).

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