1549301742-The_Theory_of_Difference_Schemes__Samarskii

The Dirichlet difference problem for Poisson's equation 257

quantities A(x) and B(x,~) are the available coefficients of the equation.

From (32) and (34) it is easily seen that

(36) A(x)>O, B(x,~)>0, L B(x,~)=A(x) forallxEwh.

E EPatt'(x)

Equation (35) is put together with the boundary condition

(37) YI -Yh =μ(x).

The Dirichlet difference problem is a special case of a more general

problem in which it is required to find a grid function y( x) defined on the

grid wh = wh + /h and satisfying on wh the equation

A(x) y(x) = B(x,~) y(~) + F(x),

(38) EEPatt'(x)

y(x) = μ(x), x E ih,

where

(39) A(x) > 0, B(x,O > 0, D(x) = A(x) - L B(x,~) > 0

EEPatt'(x)

for all x E w h.

Remark The third difference boundary-value problem for Poisson's equa-
tion can always be represented in the form (38), equation (38) being satisfied
for all x E wh and conditions (39) being valid. Here, in addition, D > f; > 0
on ih·
To prove the existence and uniqueness of a solution of problem (38)-
(39), it suffices to ~heck that the homogeneous equation

.c [y l = A ( x) y( x) - L B(x,~) y(~) = 0,

( 40) EEPatt'(x)

y(x) = 0, x E ih,

has only the trivial solution y( x) = 0 for x E wh. We will show in the sequel
that this fact follows immediately from the maximum principle, valid for
schemes (38)-(39).