The Dirichlet difference problem for Poisson's equation 247It is easily seen from ( 19) and ( 20) that no values ofμ( x) at the vertices
of the rectangle appear in this matter. This feature has some influence on a
proper choice of lh. For the third boundary-value problem and the scheme
of accuracy O(I h 14 ) (see Section 5) the boundary /h consists of all the
nodes on the boundary of the rectangle including its vertices.
To evaluate the accuracy of the difference scheme (19)-(20), we pass
to the difference z = y-u, where y is a solution of problem (19)-(20) and
u is a solution of problem (1'). Substituting y = z + u into (1') we set up
the problem for the error z(21) Az = -1/J, x E wh, z = 0, x E ih,
where 1/J =Au+ f is the error of approximation of equation (1') by scheme
(19). Since Lu+ f = 0, we have1/J =Au+ f - Lu+ Lu= Au - Lii,
yielding 1/J = A1i - Lu. From (8) it follows that
for uEC(^4 l,
where the symbol over-bar designates that the values of the arguments are
taken at some intermediate points of the intervals (i: 1 -h 1 , x 2 ), (x 1 +h 1 , x 2 )
and (x 1 , x 2 - h 2 ), (x 1 , x 2 + h 2 ), respectively.
Wit. h' rn t e h notat10n. M 4 = rg.ax I ~ 84u J , we get
G,a uxa1 °'//^1 ·l<M -^4 ~. 12
The proof of convergence of scheme ( 19) reduces to the estimation
of a solution of problem (21) in terms of the approximation error. In the
sequel we obtain such estimates using the maximum principle for domains
of arbitrary shape 'and dimension. In an attempt to fill that gap, a non-
equidistant gridw h = {x· l = (x(ii) 1 l x(i 2 2 l) ) i O' = (^0) J (^1) ) '.' ) N O') x(O) O' = (^0) ) x(Na) CY = l Q') CY= 1 2} l
with steps h;ii) = x;ii) _x;ii-i) and h~i^2 ) = x~i^2 )-x~i^2 -i) can be introduced
in the rectangle in the usual way for later use of the difference operator ( 13).
Thus emerged instead of (19)-(20) the problem
(22) Ay = -f(x), YI 'Yh = μ(x) ·