Loss of uniqueness property in difference approximation of a Dirichlet problem

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I V (1970) ROCZNIKI POLSKIEGO TO W AR ZYSTW A MATEMATYCZNEGO

Séria I : PRACE MATEMATYCZNE X I V (1970)

A. P l i s (Krakow)

Loss of uniqueness property in difference approximation of a Dirichlet problem

In this note we give an example of a Dirichlet problem for a (smooth) elliptic differential equation possessing a nniqne solution and admitting a difference approximation without uniqueness property. The differential equation is of the form

(

) a ( x , у)ихх+Ъ(х, y)uxy+ c { x , y)uyy+ d { x , y)u =

and satisfies the assumption of elipticity

(

) 4a(x, y)c(x, у) — Ъ{х, у)* >

as well as uniqueness conditions

(3) d ( x , y ) ^ 0, a ( x , y ) > 0.

Inequalities (2), (3) are classical conditions implying the uniqueness of solutions for any Dirichlet problem for (1).

Eeplacing derivatives in (

) by suitable difference quotients gives a difference equation possessing non-vanishing solution of homogeneous Dirichlet problem. This example shows that the translation of uniqueness theorems from differential to difference equations need not be straight­

forward.

We shall start with construction of a solution of the difference equation, then we define the difference equation and from it we obtain the differential equation.

N o t a t i o n s . We shall consider a set Z of nodal points Z =

i , j integers,

< i < 3, \j\ < n}, where integer n >

will be determined

later (to be sufficiently large). On the set of interior nodal points

8 = {(i, j ): i

or

, j integer |j| < n} the difference equation will

88 A. Plis

be satisfied. Difference quotients corresponding to the derivatives uxx, uxy, uyy will be denoted by ul{, u\{, u%

= ui+1,i — 2iUij' + ,

— \{ux+l,j+l — u% ~1,i+1— ui+1,j~1 + ,

= ui’j+1- 2 u iJ + uiJ- 1, for (i,j)€S.

Now we define a solution of difference equation. We define uli

for (i, j ) e Z \ S ,

u 1* = + |j| < n

n U21 = \ + r,

W11 = r for |jj < n, j Ф

, where r = (w-j-l)~3.

Obviously

(4) 0 < uv < 1 for ( i , j ) e S

and therefore (5)

W e have

Nix|, N

1 < 2 for

(6) i

till = %r for 0 < 1 j| < n,

(7) 10 2 2

0 > ul

= --- \-2r > --- ,

n n

(8) < = h

(9) < = l - 1

--- Ы + ^{(j2—n2—2)r > 2r for} \ j \ < n . j Ф 1 , n

(10) 21 1

Щ] — _и ^— ₉ — — — (n2-\-l)r > 2r for large n.

n

Now we define coefficients of the difference equation. We put a

—

, Ъ

= — 9^ i — %nul 22 , c

= n, d

—

. It is easy to see that for {i, j) = ( 1 , 0 ) the difference equation

(

) + + + =

is satisfied and in virtue of (5) and (7) we have the inequality

(

) âaijcij— (bi])2 >

,

Difference approximation o f \DiricMet problem

89

for large n, as well as the inequalities

(13) 0,

(14) aij > 0.

For i = 1, 0 < \j\ < n we put

a 11 = r , b1} = 0 , clj =

, d 11 — — (ги]^-\- u^2) fu^.

Conditions (

), (

) and (14) are evident and (13) results from (5), (

) and (4).

For i — 2, \j\ < n we put

a 21 —

, b2j =

, c 21 — r, d2] = — (%]{-{-ruli) I u2\

Similarly as in the previous case conditions (11), (

) and (14) are evident. Condition (13) results from (9) or (10), (5) and (4).

From now on we consider integer n to he sufficiently large for (12),.

to be satisfied, and fixed.

We have constructed the coefficients al\ b%i, cli, dlJ satisfying (12), (13) and (14) for (i , j ) e S . Equation (11) has the non-zero solution ulf defined on Z and vanishing on Z \ 8 .

Now we define coefficients of (1) on nodal points. We put a { i , j ) = aij', b( i , j ) = Ьц , c { i , j ) = cij, d { i , j ) = dij

for i = 1 or 2, j being integer and |j| < n. Inequalities (

) and (3) result from (12), (13) and (14). It is easy to see that the definition of functions a(æ, y), b(æ, y), c(æ, y) and d ( x ,y ) can be extended in a smooth manner on the whole rectangle R = {(x , y):

< x < 3, \y\ ^ n} in such a way that inequalities (2) and (3) are satisfied on R.

If we suitably replace derivatives in (1) by difference quotients u\{) Uui u 22 and function u ( x , y ) as well as coefficients of (

) by their values at the nodal points (x, y) eR, x, y being integers, we obtain back the difference equation (

), and therefore equation (

) has the claimed properties.

INSTITUTE OF MATHEMATICS OF THE POLISH ACADEMY OF SCIENCES INSTYTUT M ATEM ATYCZNY POLSKIEJ A K A D E M II N A U K