1549301742-The_Theory_of_Difference_Schemes__Samarskii

The Dirichlet difference problem for Poisson's equation 239

2

3 0 1

4

Figure 6. The regular "cross"-pattern

which is defined on the five-point pattern consisting of the nodes (x 1 ±

h 1 , x 2 ), (x 1 , x 2 ), (x 1 , x 2 ± h 2 ). Any such regular pattern is called a "cross"

pattern and is depicted in Fig. 6.

Here the symbol 0 corresponds to the point (x 1 , x 2 ), while the symbol

1 corresponds to the point (x 1 + h 1 , x 2 ), etc.

From formulae (3)-(5) and Fig. 6 it follows that

1 1

A v 0 = h 2 ( v 1 - 2 v 0 + v 3 ) + - 1 2 ( v 2 - 2 v 0 + v 4 ).

l^12

(6)

In particular, for h 1 = h 2 = h (on any square pattern) we thus have

(7)

The next step is to calculate the error of approximation of the Laplace
operator (2) by the difference operator (5). Since for o: = 1, 2

(8)

EJ2v. h2 EJ4v 4 h2 ., 4

A 0 v = ~ux 2 + _Q_ 12 ~ux 4 +O(h) (\' =Lav+ _Q_ 12 L;v+O(h) (\'

O' Lt

(see Chapter 2, Section 1), it is plain to derive the expression

This provides support for the view that

Av-6.v=O(lhl^2 ),