1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular n1ethod 693

Knowing / 1 and / 2 , it remains to find the iteration parameters { r;} with

a simple observation that formulas ( 42)-( 43) are still valid with another

k

member F:

( 47)

k k k

F = F (38) + <I> I

k

where F ( 38 ) is given by formula (38). The total number of the iterations is

equal to

where n~ (o:} is the total number of the iterations for the Dirichlet problem

associated with Poisson's equation ( 35). The total number of the iterations

required in ATM for a higher-order scheme is being increased in Jf5 ~ 1. 22

times as compared with a scheme of second-order accuracy.

In the two-dimensional case the iterative alternating direction method

or the direct decomposition method turns out to be more economical, but

for the multidimensional Dirichlet problem ATM is the most economical

one among other available methods. This advantage is stipulated by the

special structure of the operator A' (see Chapter 4, Section 5):

p p

( 48) A'y= LACY IJ (E+x{3Af3),

CY=! {3=1, {3;i:CY

h2

x - _p_

p - 12 I ACY Y = y,. ''-'0" x· Q^1

assuring the operator inequalities

( 49) (

2)p-l

• 3 R<A<R, - -

where

p

Ay=-A'y, Ry=-Ay=-L:ACYy, yEH,

CY= 1

and n~ (c) is the total number of the iterations required in connection with
solving the equation Ry = f, not depending, in fact, on the number of
observations p.