1549301742-The_Theory_of_Difference_Schemes__Samarskii

740 Methods for Solving Grid Equations

provided that conditions (34) hold. The number n of iterations for doing

so is no less than n 0 (c-):

n>n 0 (c):2'.ln-^1 I In-.^1

c p

vVhat can happen upon replacing the describing explicit sche1ne by an

in1plicit one'? Being concerned with one of the irnplicit schemes

( 40) B Yk+l - Yk + Ayk = f, k = 0, 1, 2, ... ,

Tk+l

Yo EH,

with a self-adjoint operator B = B* > 0, the intention is to employ the

explicit scheme

with the members

C = B-1/2 AB-1/2, cp = B-1/2 f

and rearrange conditions (34) with regard to the operator C as revised

conditions for the operators A and B:

( 41)

assurmg

and

in contrast to the preceding estin1ate just established.
The minimal residual n1ethocl can be e1nployed for the equation Av.=
f with a non-self-adjoint operator A, the convergence rate of which c01n-
cides with that of scheme (30) for T = f.
For the explicit scheme ( 13) we have

k = 0, 1,2,. .. Yo EH given,

where the para111eter Tk+l is chosen by the approved rule