1549301742-The_Theory_of_Difference_Schemes__Samarskii

The Dirichlet difference problen:1 for Poisson's equation 243

domain. We follow the second way of approximating the Laplace operator

on an irregular pattern (see Fig. 8). The expression

* _ 2_ [ v( +^1 ") - v _ v - v( -l.,) l

(16) Aa v - h (\' h (\' +. h (\' - ,

reveals A~ instead of formula (14) as

A* a v = !i(\' h Vx;,,;:.,.

(\'

We claim that in this case the operator A~ generates the local approxima-

tion of zero order

1/Ja = A~u -Aau = 0(1).

Indeed, taking into account (15) we arrive at the chain of the relations

We clarify the situation, in which approximation (16) is quite applicable,
on the basis of one possible example.

Example For the boundary-value problem

u" = - f(x), O<x<l, u(O)=O, u(l)=O,

we form the grid

which is everywhere equidistant, but near the boundary h 1 < h, h 2 < h,
h 1 + h 2 + (N - l)h = 1. At the regular nodes xi, 1 < i < N, we find that

ui+1 - 2ui + ui-1

h2