1549301742-The_Theory_of_Difference_Schemes__Samarskii

412 Stability Theory of Difference Schemes

If, in addition, the operator B is self-adjoint, then

n

II Yn+I lls < II Yo lls + L T II <fk lls-^1 •

k=O

What is more, a priori estimates (3) and ( 4) are valid with the mem-

bers II <p 11(2) = ll<fllA-^1 and II <p 11(2) = II <p II·

Theorem 5 Under the condition B > ~rA, scheme (1) from the primary

family of schemes is stable with respect to the right-hand side and for a

solution of problem (1) the a priori estimate holds:

n

(37) II Yn+I llA <II Yo llA +II <fo llA-^1 +LT II <ff,k llA-^1 •

k=l

Proof In preparation for this, a solution of problem (1) can be arranged

as a sun1

(38)

where wn is a solution of the equation (of the so-called "stationary" prob-

lem)

(39) Awn= <fn-1' n= 1,2, ... ,

Upon substituting (38) and (39) into equation (I) a new problem arises for

( 40) B vt +Av= rp,

where 1Pn = -( B - T A)wt,n and rp 0 = 0. In the estimation of v Theorem 4
gives

n

( 41) II Vn+l llA < II Vo llA +LT II B-lr:pk llA ·

k=O

A simple observation that

Wt = A-I <ff'