Estimates of the rate of convergence of difference schemes for fourth-order elliptic equations

U.S.S.R. Comput.Maths.Math.Phys.,Vol.23,No.2,pp.64-70,1983 Printed in Great Britain

OO41-5553/83 $iO.OO§ 9 1984 Pergamon Press Ltd.

ESTII~TES OF THE RATEOF CONVERGENCE.OFDIFFERENCESCHEMES FOR FOURTH-ORDERELLIPTIC EQUATIONS* I.P. GAVRILYUK, R.D. LAZAROV, V.L. MAKAROV

and S.I. PIRNAZAROV

The second boundary value problem is considered for two-dimensional linear and quasilinear fourth-order elliptic equations in a rectangle, when the solution belongs to classes IV~3+~ s=O,i. Using operators of exact difference schemes, schemes are constructed for which convergence-rate estimates of order O(lhl'+') in the mesh norm of IVz~(6))are established. Convergence-rite estimates have often been discussed for boundary value problems for higher-order equations. Increased demands have then been made on the smoothness of the problem, which are not satisfied in many important applied problems. For instance, in /1,2/ schemes for fourth-order equations are constructed, which are shown to be convergent in the mesh norm of |VzZwith a rate O(lh] z) on solutions of class C (~i. These studies do not embrace e.g., the important practical case when the right-hand side of the equation is a delta function (the case of concentrated forces in the theory of elasticity). Hence the need to construct and study the rate of convergence of difference schemes under natural conditions on the smoothness of the solution. In /3/, using exact difference-scheme operators difference schemes were constructed for the second boundary value problem for the biharmonic equation in a rectangle, and their convergence was proved in the mesh norm of L: at a rat e D(lh [,+0i), provided that the solution of the differential problem belongs to class [|r%§ ~ s = U, I. These studies were continued in /4/, where estimates of the same order, but in a stronger norm (the mesh analogue of the norm in |V22) , where obtained for the first and second boundary value problems. The convergence-rate estimates of difference schemes in which the smoothness of the required solution and the rate of convergence are matched are of considerable interest. More precisely, a matched estimate is defined as an estimate of the type

Iiy "-ull.-..(~'~Mlhl'-'llul!,,.,.(o

,,

k>s,

where k, s are integers, u is the solution o f the initial differential problem, y is the approximate solution, and ]Vz'(o), |V,h(~)are Sobolev spaces in the set of functions of discrete and continuous arguments respectively. Notice that, for some s and k, estimates of this type are typical for variational difference methods /5--7/. The present paper is closely linked with /3,4/, and is devoted to obtaining matched estimates of the difference schemes of /3,4/, for the second boundary value problem, for the biharmonic equation in a rectangle, and to extending these results to equations with variable coefficients and quasi-linear equations. It proved possible to obtain such results by means of special continuation of the solution of the initial differential problem into an extended domain. This device is at the same time a theoretical justification for using difference schemes with contour nodes. i.

Consider the problem

~ O~tt+.___ O~a + Ota AZtt Ox," 2 axi~ OXz "-~q_, axz = / ( x ) , O'u(x)

u(x) =0,

where

On"---z - = O,

x ~ .0.,

(i)

xel',

(2)

~ = { x = ( x , , x=): O~
to the boundary

F.

We shall use the following semi-norms and norms:

exj o

IluIh,-,.(,~,=

,

i+j~s

t htl..v(,~,,

Ilulh,-..c=~=vrai z~tl

s-o

max d+j,~s

I ~

.

Let E={x=(z=, X2): --l~
~ = {p(x) : p(x)--- Z

a~jx,'x2'}.

We shall later need the following lemma, which is a particular case of the Bramble-Hilbert lemma /8/. *Zh.vychisl.Mat.mat.Fiz.,23,2,355-365,!983 64

65 L e m m a i. Let the linear functional l(u) be bounded in ]V~ +I (E), i.e. , [l(u)[<~Mlluli,~,~.,c~, and let it vanish in the set nk. Then a constant ~ exists, independent of tt and such that 2 . We cover the plane (z,, x:) by a mesh Q,.= {x= (x,, x:)" xa=iJz~, i~=0, -'|..... h~=l=/N=, ~=|,2}. Denote by ~ the subset of nodes lying in the closed domain -Q0|', and by o the subset of nodes lying in the open domain Q, ~=~--6) is the set of boundary nodes, ~ • cos(za, n)=~l}, ~ = d,2. We shall also use the notation

y=y(z)=y(x,, x~),

y~• (x)=y(x,•

, z~),

V ":',~ (z) = y (z,, z~-4-h~),

y~=(!t(+'"--y)/h,

y~=(y--y(-'")/h,, y ~ , = (V~+'#-- 2g + #-'i))lhi 2,

Vii.: (V<+'O -- y(-td)/2hi,

(v,v)=(V,v).= ~,v(z)v(z)hd,..,

~t~a, x l f a :

Ilvll~..,=tlvll=@,v) 'r

x,=h, ~:~h t

IIV vll'=ll vJ'+II~.ilL IIA~vlI'=II~,~,II'+II~,~II'+2 Z ilt'll~(.~=maxlv(z) l,

IIvIh'= IIvll~,,-er

~,r,hthz,

IMh'~llvll~,,r162

IIvIl~',-,,r

',

A~vlI'.

Using the relevant results of /9, p.292, p.3Ol, p.309/, we can prove: L e m m a 2. (see /4/) Given any mesh function specified on the mesh ~ and vanishing on the boundary have the inequalities

IIV vll'~
IIvlV,

y,

we

e>0,

c,=32(l,-:+lz-:)z+1281,-H2 -:.

IIA~vll~c, llvll~,

By Lemma 2, we see the equivalence of the norms:ilA~yll and llyll:for functions vanishing on the boundary, i.e., [IAhyII~[[y!!-.~ c2[[Ahy][, where the constant Cz "is independent of h. 3. We consider the resolving operators of the exact difference relation /3, iO/

schemes,

defined by the

r,u (z) = ~ ( 1+s ) u (z,+ (2 - 0 sh,, z~+ (i- 1) sh~) ds+ -!

(l--s) u (x,+ ( 2 - 0 sh Iv x,+ (i-- 1) sh,) ds. 0

It is easily shown that a2ll

Y~,~xj=U~,,~,(x), Applying operator

T,T: to Eq. (i), for

x~o

x~(,), o:=1,2,

uaIV2Z(.O.).

we o b t a i n t h e

identity

azu 2 )\ ~ , = T,Y2/(x). Tz ~azu \) r,=,+2u;,.,r~,+ t~ T , -~-~x

We s h a l l

approximate

problem

(Z),

(2) b y t h e d i f f e r e n c e

scheme

A:y=---y~,.,~,~,+2yrs,~:+yr~,z~:=Y,T~], y(x)=O,

x~'~,

y r : .: ( x ) : O,

(3)

xE'f•

x~o, a=i,2.

{4) (5)

Notice that the mesh function y is defined,not only on the mesh ~, but also on the nodes of the mesh .O.h, located outside ~ at a distance h, or h_, from 4. Hence, in order to study the error z=y--u, we have to continue the solution outside the domain ~ while preserving the smoothness (we shall assume that the solution of problem (i) , (2) belongs to one of classes

IVan,2 ~(.o.), s = O, 1). Denote by ~ the domain ~={x=(xl, xz):--l=<~x~<~31=, ~=I, 2}. Further, let Q, be the mirror reflection of domain Q relative to the stright line z,=0, and let ~: be the mirror reflection of domain .QU.Q, relative to the line x._=0, while ~-)sis the ~eflection of ~.UQ.,U.O.2 relative to x2=Iz, and .Q, is the reflection of Q.U~tU-Q20-Qs relative to z,=ll.

66 Q~, then from We continue the function ](x) oddly from ~ through the line Xt=0 into .O.U.Q, through xz=0 into O.~, then from Q.U~-,UQz through xz=l, into Q=, and finally, from .o.u~suQ.zUQs through xt=l, into Q~. Denote the resulting function by ~, and the boundary of t h e domain ~ by P. Let ~(x) be the solution of the problem

A'~=/(x), (z)

(6)

z~,

O:ri(z) On~

O,

x~r.

(7)

It is easily shown that the contraction of the function ~(x) onto Q is the same as the solution u(x) of problem (i) , (2) , while ~(x) is the odd continuation of the function u(x) through the boundary F. For s let ](x)~Lz(~). We shall only consider the case of continuation into the domain Qa (the other continuations are considered in the same way). Obviously, f~L2(~). Denote by i}'=4(.Q) the class of functions u(z)~IV2~(.Q), satisfying conditions (2) on r. Then the solution of problem (6) , (7) in the domain .O.O.O.,,which we denote by ut(x), is defined as a function of space I],=~(.QU.Qa),satisfying the identity

j~U~a t O : u ,

O'-Vdx+2 ~

cr

02u,

Ozv

ox, Ox, oz,

030t

J r,e

ca)

~UQ=

for a n y v~Wz=(Qu-Q,). This solution u,(x)~I~==(.O.UQ,)exists and is unique. Since [ is oddly continued through the line xs=0, the function u,(x,, x,)+u,~--x,, x2) satisfies in ~UQ, the homogeneous equation (6) and the homogeneous boundary conditions (7), whence it follows thatu,(x,, xa)+u,(--x,,x2)~O. This means that the function ui(x) is odd with respect to x,=0. Further, we have

~-10'u, O:v Ox=z

!

i

--O:u,z Ox=: O~u. 0.'/)Ox~d x O'v dx+2 S "O'-~ Ox= Ox, ~Vt Ox~

[r

O'zt, OZV(--X,,Xz) O:ut

~

w=v(x,,

x2)--v(--xl,x2),

69'tt, O'v(--x,,x~)dx=

S0,u,

~,=s Ozj ~xJ dx+2 where

Ozu, O'u Ox, Oxa Ox, Oxz

and w~I~2t(.Q).

O"w

Ox, Ox+.Ox, Ox~dz' Since

lv dx= I lw dx, then, since w~I~=t(.O.) is arbitrary, we see that ut(z), xE~, is the solution of problem (1) , (2) , and since the solution is unique, is the same as u(x)in the domain -Q. If [(x)~|/2-s(Q.), then, in the above arguments, we have to replace the right-hand side in (8) by the expression /ii/ aU

a

where

v~

(-o-U-o.,), /=Vo+ ) " a-~2.

We thus arrive at: L e m m a 3. Let /(x)~|[2', s~0,o--1. Then the solution of problem (6), (7) exists and is unique, and belongs to the class TV~+'(~), s=0, I, while it has the following properties: i) the function ~(x)is odd with respect to the sides of the rectangle Q; 2) ~(x)=u(x), x~: where u(x) is the solution of problem (i), (2). Let us now study the rate of convergence of difference scheme (4), (5). For the function z=y--~ we obtain the problem

A*z=r z(x)=O,

x~'~,

x~o), zr~%(x)=O, x~'f+~, a = i , 2 ,

(9) (10)

where #(x)=TITd(x)--A'~ is the approximation error of the difference scheme on the solution of problem (i), (2). Notice that the homogeneity of the second of conditions (10) is a consequence of Lemma 3 and plays an important role in future. Using identity (3), we can transform

4(z)

to

a~|

*azu

a=+

67 We have: Lamina

4.

Let

zt(x)~|V~ ~s (Q), S=0, |. Then, Iq~(z) I
where the constant 31 is independent of h and

x~6),

u, while

s = 0 , 1, e=e (x) = ( z , - h , , x,+h,) X (x~--h,, x~+h,).

Proof. we map the rectangle e onto the square E = { s = (sl, s,):-l~
I+=.+.

0 ~ ~< M T,-=O--~C=,

t

this functional vanishes on polynomials

II~lh+=,(+,,

of

a=t,2.

.~s. By Lemma l, we have

I ~ ( x ) l < i l l l~l,~7.(~,<~Mlhl,§ ha z

this proves the lemma. An a priori estimate for the solution of problem Lamina

5.

Proof.

Let z be a solution of problem

Multiplying Eq. (9) by

(9),

(9), (iO) is given by:

(iO).

Then we have [IAhzII~]lq,[[+][q,[l.

z, we obtain z

(A;z, z ) . = ~

( ~ = . = , z)

Ul

==l

u s i n g summation by p a r t s , the l e f t - h a n d s i d e of t h i s e q u a t i o n i s e a s i l y z).=llA~zll ~. N o t i n g t h a t z(x)=0, x ~ , and t h a t , by r e=aa 3, the r e l a t i o n ,

,

~

O's(z)

-

x~T•

where

~ ( x ) = 0 , x~•

(A=z,

,,

-- UZ=:,=iX)

11~iX) = I~-~ ~ is defined for

t r a n s f o r m e d to

we obtain

(n~r=~, z).= (n=, z;=~=) + (n~=z-n~zr=) k=,+=-

(,l==z-q.z==)l=~, ==(,l~,zr===)., On then applying the Cauchy inequality, From Lemmas 4 and 5 we now have:

a=l,2.

we obtain the lemma.

Theorem i. Let the solution u of problem (i), (2) belong to the space ]V~§ (~)),s=0, I. Then the difference scheme (4), (5) is convergent in the mesh norm of |Vz'(~), while

Ily-ulh,-,.(.~<~Mlhl'+sluhr,..ca.

s=0, 1.

(ii)

Notice that schemes somewhat similar in the structure of the approximation are obtained by the finite-element method, if, as the coordinate functions, we use the tensor products of one-dimensional parabolic and cubic splines (see e.g., /7/). In this case the convergencerate estimates are similar to (ii). 4. Consider the problem

A'-u=/(x, a),

0'u U=

x~Q,

(12)

_

O--~z = O ,

XEF,

(13)

I/(x,'u,)--/(x, u~)]~-KIz:t--,2[, the condition of assuming that / satisfies a Lipschitz condition strict monotonicity [[(z, ut)--/(z, Uz)] (ut --us)<0, and t'.~|V32 +s (~), s=0, i. Applying to (12) the operator T,Tz, we obtain the identity

02u

(r,_=-?7) =-t \

We approximate

problem

(12),

@x=

+2u~+,~,=r,r,/(.,~), ~. ~===

(13) by the difference

scheme

A=y~y;,=,~:.,+2y+,~,~,+yn~,nn= T, T:/(., Ly),

(14)

y(x)=O, x~'t,

(15)

y;~.=(x)=0,

xET•

Ly=y(x)+(~t-x,)yr,+(h--x=)y+,, The unique solvability of difference /2/. We have:

scheme

(14),

a=l,2, (~,, ~z)~e(x), x~os.

(15) is proved in a similar way to that in

Lamina 6. Let the solution u of problem (12) , (13) belong to the space ~V~+s (~-),s=0, I. Then a function if(x), x ~ exists, such that i) ~(z)E~V32+s (~.), $=0, I; 2) ~(x)=u(z), x~.o.; 3) ~(z)

68 is an odd function with respect to the sides of the rectangle ~.. For t h e p r o o f we only need to note that, obviously, there is a linear problem of type (i), (2), whose solution is identical with the solution of problem (12), (13), in order to state constructively a problem of type (12) , (13) for 3, while for x~.Q, we have to put /(x, ~ ) = --](--x,,x,, --if) , and using the same scheme, continue it onto the domain .(.). The error z=y--ff is the solution of the problem A*z=~+~', z(x)=0, where

x~,

xe~,

(16)

z;=,=(x)=0, x~-~,

a=l,2,

(17)

z ~-.,

OXa

r u=u(x)

=

~a*a

~-,

Lu)-/(z, u)).

We shall denote by the function of a continuous argument, specified on by uh=ua(/), x ~ , the sage function, evaluated at a node of the mesh (~. Noting that we obtai~

It is easily seen that the

Ir =1T,r,[l(=, n(~+z))-l(x, u)]l~< Kr, n(ILu.--N)+KIL:I. functional Luh--u vanishes in nt, and

e(x), and y=z+ff,

hence, by Lemma i,

i L.~-~I
(see /iO/)

c.'= ~--E[a(l,'+l~')+ (l,'+z/)' 2Z,%']j'

II=llo
ILzl
for ~" we have

I*'l
(18) Ly=y

Note. In the case is sufficient to put in (14). We multiply (16) scalarly by z. After summation by parts and using the Cauchy inequality, we obtain by means of the =esh imbedding theorems, 2

2

a-!

a-i

whence z

ltA~zll< ~ , IIn~ll+c,ll~'ll. a.t

From (18) we find that

( l-Sc, c,K) [IAszll~~--~,I[n~ll+MlhPIlulh~. r

From Lemma 4 we now have: Theorem 2. Let the solutlonu of problem (12) , (13) belong to the class |V~+' (.Q),s=0, I. Then, for fairly small K , the difference scheme (14), (15) is convergent in the mesh norm of ~V2*((0), while

I[y-tdh~,,~.~
s=O, t.

5. The above results can easily be extended to the case of linear and quasi-linear equations of type (12), (13), if, instead of operator A', we consider a fourth-order elliptic operator with variable coefficients. Consider, for example the equation O'

,

0',,. \

(19) I,J~ !

w i t h boundary conditions (2). We assume t h a t the coefficients a~(x)~IV.'(~)filViz§ (~), s=0, I. Let a0(/), x ~ be functions obtained by continuation of the functions a~j(x) through the sides of the'rectangle ~ evenly into the domain ~ according to the scheme of Sect.3, and let [ be the continuation of the function [ into the domain ~ likewise according to Sect.3. In addition, let the ellipticity condition hold for the differential operator. Then, the results of Lemma 6 hold for Eq. (19). As an approximation of (19) we take the flnite-difference equation

~ [T,-,(a,j)yrj.~b,.,=T,T,(1), z~o. The approximation error is transformed with the aid of the identity

~[T,-, I,i--I

to the form

O:u

(20)

69

where

~l,(x) =gT,_,[ ao(z) We rewrite the expression for z

/ O'u

l],(x)as

Ozu

/

Ozu

/

T,(ai,)Ts-,t~--ur,=,) } ,

' = 1 , 2.

For convenience, the variable with respect to which the operator T= acts will be denoted by

~ , ~ = 1 , 2.

Consider,

for

example,

the estimation

of the term

Using the representation of the operators T= and the Cauchy inequality, we obtain condition r 1 6 2

(with

the

xl

|l

[

=*

o,.(~,~,)a~ + fo,.(~,,.,) a,,,'~lll

Tl

~

O'c

'3

O~Ov

~,+at

~<

/l/I

=,+a,

xt--h I

Xl--~ t

x~§

xz§

='+~' O'u(~,h)

='+~: Oa,,(h,v)

dr)

zz--hz

xt--h t

xz+hz

~t ( ]z=+' ~ *Oal,(~[ .%.)OV [ dr) ~t ( ~ [ Ol~(~.'%')O~.IzOy dy)] ~. .~/]~ll'lvt,(t)[u[ir**(,, D x,-a z

=z-hz

where the constant Jl is independent of For the term

h, a;~, and

it.

~],~==T,[a,,(~,,x=)]T=[ O'u(x''~') u~,.,(x,,x:)] Oxt = we have, by L e n a

i,

[*l,=l~M[ T, [a,, (~,, z=) ] I lhl (h,hD-'H ul.-,,c,~<

f o~,,~,,,) tt

(h,h=)-'H uh,-e(.,<~M max {lla,,ll w,,t.. Ila.lh,-..t.,} Ihl (h,hD-'hlul.-,,~.~. Similar inequalities hold for the other terms. Tn the case a,,~W.'fllV=',u~IV,'(~) , we have the same estimates, except for using the norms of the coefficients and functions in spaces |VzZ(Q), |V~'(~) |Vz'(~) respectively. It is easily shown that, for the error z=y--~t, we have the a priori estimate IIAhzII~ Ll/(lllbn+l]q,u), where the c o n s t a n [ M i s independent of h,ao,u. Combining the above estimates, we obtain : Theorem 3. Let the solution u of problem (19) , (2) belong to the class |V *, (-q),a0~ ]V.s(.Q)0IV (.Q), s=0, I, and let the ellipticity condition be satisfied. Then the difference scheme (20), (5) is convergent in the mesh norm of |V2(o), while

Ily-ull,,-.,c.)<~MI hi '+" max max{lla~ll,,,,=.o~=,, IIa~,[[~..~o,} Ilull ~Tt~

g,J

6. The results of Sect.4 can be extended to quasi-linear equations = / 0u Ou~

a

(21)

70 We assume that

[l(x, u,, u~, u,)--l(z, u:, u,, u,)] (u,--u,)
II(x, u,, u~, u~)-l(z, ~,, ~,, g,) I<_2 K,I u,-i~J, We replace problem

(211,

(2) by the difference

x~T,

3+s

(~),

s=O,l.

scheme

A'y=T,T,[(x, L,y, L2y, Lsy),

y(x)=O,

tt~IV,

(22a)

xE~,

yr~,=~,(x)=O, x~l•

, a=l,2,

(22b)

where

L,y=y (x) + (~,-x,)

y;, (x) + (~,-x2) yr,(x),

L2y = y~, (x) + (~-I- - x,) y~,~, (x) + (~. -- z2) y~&, (z), t3y = L';~,(*) + (h - - zO Y~,t:, (x) + (~.2 -- z~) y;.~, Cz), The unique solvability of difference scheme (22) is proved in the same way as in /2/. For the error z=U--~ we obtain problem (16), (17), where

r

i u,--au ,--Oa\1 o:,

9

substituting

y=~+Z,

we obtain

,,~.l <.T~T, [ K~i L~uh_u, + Kz l The functional

o=31"

L.o _art I +K..

~

[i~--Liu^--u vanishes in the set xi. Hence, by Lemma l, we have the e situate II,l< I,=L~uh--Oa/Ox,, I,=L,u,--Ou/Oz2 we have ll.I
MIhl'(h,h,)-'~'lul,v.:<,,For Now, using Friedrichs'

inequality,

it is easily shown that

II~'ll~,c.,<~c,Kllzll~zc.)+Mlhl'+'lul,~;..c~,, s=0,1,

K = m a x Kt. Furthermore, in the same way as in Sect.4, we find that, for fairly small for scheme (22). The authors thanks A.A. Samarskii for valuable discussions

K, Theorem 2 holds

and advice.

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