Stability and convergence of the Dirichlet difference problem 271
- The uniform convergence and the order of accuracy of a difference
scheme. In the study of convergence and accuracy of scheme (2) we begin
by placing the problem for the error
z=y-u,
where y is a solution of problem ( 1) and u = u( x) is a solution of problem
(1) arising from Section 1. Substituting y = z + u into (1) or (2) yields
(15) Az = -1/J(x), xEw, zl,,=0,
where 1/J( x) = Au+ \0( x) is the residual.
We stated in Section 1 that
1/J(x) = O(I h 12 ) = O(h^2 )
1/J(x) = 0(1)
or, more specifically,
l·'·l<M4lhl2< '// - 12 - p M4h2 12
11/J I< p M2
where
Mk= max
xEG
l~a~pk = 2,3,4,.. .,
at the regular nodes,at the irregular nodes,at the regular nodes,at the irregular nodes,p
lhl^2 = 2= h!,
a=lAlso, Theorem 1 of Section 1 asserts that estimate (14) is of the form
R2
II z lie< 2 P 111/J lie + h2
II1/J lie·.Using the estimates of 11/J I obtained above we arrive at the relation
(16)
making it possible to formulate the following statement.
Theorem 2 If u(x) E C^4 (G), that is, a solution possesses continuous
derivatives in c = G + r of the first four orders, then the difference scheme
converges uniformly with the rate O(h^2 ), that is, it is of second-order ac-
curacy, so that estimate ( 16) is valid.