1549301742-The_Theory_of_Difference_Schemes__Samarskii

682 Methods for Solving Grid Equations

Theore1n 2 Let conditions (12) hold. Then an optimal set of parameters

{ rk} specified b.r formulas (27)-(29) exists a.nd ma.r be of assistance for

solving problem (6), providing the validit.Y of llie estiniate

(18)

with

( 1 g) q11

2 p~'

1 + p;n

1 - /[

l+v'['

<.,-c - -fj

~( 2

Recall that Theorem 2 has been proved in Section 2 placing a spe-

cial emphasis on one particular case of explicit schemes with the identity

operator B = E involved. To make our exposition more transparent, the

implicit scheme transforms into the explicit sche1ne ( 17). Having stipulated

the conditions /i E < C < / 2 E, the estimate

(20)

is an immediate implication of estimate (31). The forthcoming substitutions

<p = B-^112 f, B-^1 l^2 x,, = Yn and C = B-^1!^2 AB-^1!^2 may be useful when

providing current manipulations:

II c xn - <p 112 = (C .'C,, - <p, c .r:11 - <p)

= llAy,, - fll~-1 ·

By inserting llAy,, - JllB-1 in (20) in place of the norm II Cxn - <p II we
arrive at inequality (18), thereby completing the proof of the theorem.
In concluding this discussion it is worth noting that the type of the
original equation Au = f and the operator B have no influence on a univer-
sal method of numbering the parameters T 1 , •.• , r,, that can be obtained
through the use of the ordered set M~ of zeroes of Chebyshev 's polynomial
of degree n, whose description and composition were n1ade in Section 2 of
the present chapter.