1549301742-The_Theory_of_Difference_Schemes__Samarskii

Two-layer iteration sche1nes

The con di ti on p;' < c is ensured if

which is certainly true for

( 36)

n > In (1/s)

In (1/ Po) '

(

ln(l/o-)

n 0 s) =.

2~

665

5. A model problem. Comparison of1nethods. Further comparison of vari-
   ous iterative methods will be conducted by having recourse to the Dirichlet
   problem associated with Poisson's equation in the square { 0 < x 1 < 1,
   0 < x 2 < 1} of the unit sides! 1 =! 2 = 1 and posed on a square grid wh
   with steps h 1 = h 2 = h. As a special case of problem (2) in Section 2, the
   problem of interest is characterized by the grid equations

(37) Ay=A1y+A2y=-f(x), xEwh, Yl 1 h=0,

where

cx=l,2.

The syste1n of equations (37) can be recast in the operator forn1 as

0

Ay = f, where Ay = -Ay in the space H =rt of all grid functions given

on the grid wh and vanishing on its boundary rh. An inner produc(. and

associated norm in that space Ii are defined by

(y, u) = I: y(x) v(x) h^2 , II Y II= VCY:Yf.

xEwh

We spoke about the operator A a lot of time in Chapter 4: it is self-

adjoint, positive and possesses the eigenvalues

k°' = 1,2,. .. ,N-1, cx=l,2,

so that

(38)
8 2 7r h
12 =max >-,,1 k2 = h2 cos 2.