1549301742-The_Theory_of_Difference_Schemes__Samarskii

The su1nmarized approximation rnethod 621

that a solution of the relevant difference proble1n supplied by the zero initial

condition approaches zero. Such a priori estimates hinging on the summa-

rized approxi1nation properties hold true for additive schemes associated

with systems of parabolic and hyperbolic equations.

vVe outline in what follows the general theory for an additive typical

sche1ne in a Hilbert space H h such as

( 45)

cr=l,2, ... ,p, j=U.l,.. ., -::^11 =0.

Theorem 3 If B = B* is a posit;ive definite constant operator and the

1natrix-operatol' A is non-negative A = ( A 0 ,μ) > U, that is, for any vectors

(,, ~ 13 EH

p

( 46) L (Aap~a,~μ)>0,

Cl:' 1 /3=I

then a solution of problem ( 45) satisfies the a priol'i estimate

The proof of this formula is omitted here. It should be noted that from
such reasoning it seems clear that due to the sum1narizecl approxin1ation
in the space H B-' the convergence occurs in the space H B. That is to say,
the conditions

( 48)

guarantee the convergence llzj llB ---+ 0 for all j = 1, 2, ... by observing
that estin1ate ( 4 7) is valid under rather n1ilcl conditions: B is a positive
definite self-adjoint op er a tor and A is a non-negative matrix-operator. But.
in a Banach space H h another method of further derivation of a pnon
estimates is employed for scheme ( 45).
In what follows scheme ( 45) is supposed to be stable so that
p
( 49) llzj llciJ < M max L ll7,V~llc 2 J,
O<k<J - a=l