1549301742-The_Theory_of_Difference_Schemes__Samarskii

622 Econo1nical Difference Schemes for Multidimensional Problems

where II · 11(1 1 and II · l!r 31 are some suitable norms on the space Hh. The

usual trick we have encountered is to represent the residual 1/J 0 by

0

(50) 1/!cx = 1/! a + 1/J:, so that

By setting z)+cx/p = T/j+a/p + vj+cx/p, where T/j+cx/p is determined from the

conditions

T/j+cx/p ~ 17J+(a-l)/p o

B = ,1.J 'f/ Ci ' a:=l,2, .. .,p, 17°=0,

T

it is plain to show that

Ci

B T)j+cx/p = BIT + T L ·~ ~, B 17j+l = B T)j = · · · = B 17° = 0,

J=l

giving 17J = 0 and zj = vJ for all j = 1, 2, ... and

Ci

l]j+cx/p =TL B-1

0

·tf!/ =~r

/3=1

]J

2..::

0.

B-1 ,1,J

'f/ /3 ' a:=l,2, ... ,p~l.

In turn, vj+cr/p satisfies equation ( 45) with the right-hand side

Ci ]J

~~ = 1/J:j + T L Aap L B-^1 ~~

/3=1 /3^1 =/3+1

and the initial condition v^0 = 0. Having stipulated condition (49), the
following estimate is valid:

The reader is invited to prove this assertion on his/her own. The sumn1a-
rized approximation condition means that

1) tlw residual V'c, achnits representation (.50),

2) i!J~i!(2) ---t 0 as T ---t U and !hi ---t 0.