1549301742-The_Theory_of_Difference_Schemes__Samarskii

Iterative alternating direction rnethods 725

Remark 1 Frmn such reasoning it seems clear that ADM with variable

parameters is equivalent to the two-layer scheme (26) with iteration parain-

eter T = Tk = Tk (1 ) + T~ 2) and factoriied operator

\^1 Ve will not pursue analysis of this: the ideas needed to do so have been

covered.

Remark 2 If A is a sum of p > 2 pairwise c01n1nutative operators such

that

p

A = ~ Aa, A~ = Aa > 0, baE < Aa < l::laE, b0: > 0,

<>=1

CY= 1, 2, ... ,p, AaAp = AμAa, o:,(3 = 1, 2, ... ,p,

direct applications of ADJ\!I described by ( 17) is i1npossible in principle,

so there is a real need for constructing sche111e ( 26) with the factorized

operator

p

Bk= II (E+r~"')Aa),

<>=1

In this regard, we are unaware of any exact solution to minimax proble1n

and the so-called cyclic set of the ensuing para1neters may be of help in

achieving the final aims.

The 1nain idea behind this approach is connected with the equation

related to a new iteration Yk+l:

vvhich reduces to successive solution of p equations

( E + T k (I) A 1 ) yl I ) -- F k J (E., + T(u)A k u. ) 1/(u) = .,/(o-1) < , a:=,^2 ... ,]!,

with Yk+l = ylP) incorporated.
The difference Dirichlet problem for Poisson's equation in a p-di1nen-
sional unit cube such as

AuY = Yx• • Cl:'' .,. l" '