1549301742-The_Theory_of_Difference_Schemes__Samarskii

The Dirichlet difference problem for Poisson's equation 241

2

3 0 1

4

Figure.^8. Th e. uregu^1 ar " cross " pattern

where h 1 ± > 0, h 2 ± > 0 and hex+ f hex- at least for one value of CY (Fig. 8).

We approximate either of the operators L l and L 2 at three points

(x 1 -h 1 -,x 2 ), (x 1 +h 1 +,x 2 ), (x 1 ,x 2 )

(x1 'X2 - h2-), (x1 'X2 + h2+ ), (x1 'x2)

(the points 3, 1, 0),

(the points 4, 2, 0),

respectively. These approximations can be a.nanged via transformations

(Chapter 2, Section 1) such as

(12)

[

v(x 1 +h 1 +,x 2 )-v(x 1 ,x 2 )

hi+

_ V ( X 1 , x 2 ) - V ( X 1 - h1 - , x 2 ) l

h1- '

[

v (x 1 , x 2 + h 2 +) - v (x 1 , x 2 )

h2+

_ v(x 1 ,x 2 )-v(x 1 ,x 2 -hr)]

h2- '

where na = ~(hex- + hu+) for CY = 1, 2. On any irregular pattern the
difference Laplace operator takes the form

(13) Av=A 1 v+A~v=v"' ,;., "-' 1. 1 x·· +v"'_,,, .2 1 ' -•. ' 2