1549301742-The_Theory_of_Difference_Schemes__Samarskii

238 Difference Schen1es for Elliptic Equations

solving the Poisson equation

p u ,:i2 'U

/:,.u = 2= ax2 = -t(x),

a=l ex

(1) x E G,

subject to the boundary condition

where x = (x 1 , x 2 , , • , xp), G is a p-dimensional finite domain with the

boundary f,

1. The difference approximation of the Laplace operator. We begin by
   defining a difference analog of the Laplace operator in the plane x =
   (xi' X2):

(2) o:=l,2.

In this direction the operator L1 1l =^88

2

xi t or L2 1l =^828 x2 ~ is approximated at

a point x = (x 1 , x 2 ) by the three-point operator A 1 or A 2 , respectively:

(3) L 1 v ~ A1 v = vx,x, = : 2 (v(x 1 + h 1 , x 2 )

I

(4)

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

L2 v ~ A2 v = V;c 2 x 2 =

1

1

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

'2

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

where approximation is denoted by the symbol ~ and h 1 > 0, h 2 > 0 are

the steps along the axes x 1 and x 2 , respectively.

The operator:o· A 1 and A2 are specified on the regular three-point pat-

terns

and

respectively, Taking into account (3) and ( 4), we replace the Laplace oper-
ator (2) by the difference operator

(5) AV = A 1 V + A 2 V = Vx- 1 ·'-' ~ 1 + Vx• ' 2 x· 2 ,