1549301742-The_Theory_of_Difference_Schemes__Samarskii

640 Econorn.ical Difference Schernes for Multidi1nensional Proble1ns

It follows from the foregoing that

0

(A-y,y) = (A+y,y) = 0.5(Ay,y) > 0.5c 1 (Ay,y),

where

yielding

p p

(A y,y) = L (l,y;Ja > 8 L ,; llYll^2 ·

a=:J a=:l a

Thus, the operators A- and A+ so defined are positive definite:

p

A- 2 b E J A+ 2 b E J b = 4 cl L r;;^2.

a=l

Stability of scheme (86)-(87) with zero boundary conditions is asserted

by Theorem 3 in Section 11, due to which a solution of the auxiliary problem

a J'

Z[a = L A~(3 Z(!J) + w-; J L Atf3 z(f31) + ·iP-:; '

(3"' I (3=:a

z(a)=O, z(<>i)=O, z(x,0)=0 for XE/~,

where

satisfies the a priori estimate

p

II L ((w-;)j'+a/(2p) + (w-:;)j'+1-ca-1)/(2p)) II

(\'::: l

This supports the view that the additive scheme (86)-(88) converges in the
grid norm of the space £ 2 with the rate 0( fi + lhl^2 ).