1549301742-The_Theory_of_Difference_Schemes__Samarskii

706 Methods for Solving Grid Equations

k+l

For determination of y we must solve the equation

k+l k

(D+wA1)D-^1 (D+wA2) y = F

k k k k k k

with the right part F = B y+rk+ 1 (Av +<p) (v =yon wh and v = 11on11i)·

As usual, this a1nounts to successive solution of the following equations:

k

(D+wA1)Y=F, :iJl,,h = 0'

k+l

v I -y h = o.

In this regard, the more detailed forms of the members rnay be useful in

subsequent constructions:

y= - f{^1 [~ ~

CY::: 1

w aCY T;(-1"') + Fk l

fi CY h-CY ' '

(89)

k+l 1 [2=

2

1j -- w a+ L> k+l 1/ (+^1 ") +dy -]

' - f{ CY=l fi (y h+ (Y '. '

where

(90)

2 +

I\. -- G l + ~ 2 ~ ~ ':'!____ fi (ac, h+ + h-a,,).

a=l Ct' a O'

The sarne procedure is workable here as was done in Section 5 of the present
chapter.

9. The Dirichlet problem for Poisson's equation in an arbitrary complex do-
   main. The algorithm of MATM is demonstrated by appeal to the Dirichlet
   problem associated with Poisson's equation

EJ2u EJ2u

~ 1t = ;::i 2 + ~ = -f(x), x E G, u = μ(x), x E r.

vx 1 vX 2

In working on a square pattern