1549301742-The_Theory_of_Difference_Schemes__Samarskii

698 Methods for Solving Grid Equations

with

(65) Asm cx(3^1 j m = 2 ~ [(ksm cx(3 Yx~ m) :ra + (ksm cr(J Yx~ m) .r 0 ] '

making it possible to set up the difference Dirichlet problem

(66) Ays = -<p^8 , X E Wh, Y^8 = ll^8 , X E ih,

0

in the space H = Q of all grid functions under the inner product structure

mo

(y,v)=L(Y^8 ,V^8 ), (y^8 ,~/)= L ~/(a:)v^8 (x)h 1 h 2 ···hp,

s=l ~·Ewh

Y- ( 1 y,y,.^2. .,y,. s. .,y mo) , v- (· v,v, 1 , 2 ... ,v,. s ... 1. v m0 ) ,

y^8 EH, V^8 EH, s=l,2,.,.,rn 0 •

Being concerned with the operator Ays = -Ay^8 and the regularizer Rys =
0 0
-Ays = - L~=l Y~,,.>:a in the space H, where A is a (2p + 1)-point dif-
ference Laplace operator, we rely in the further derivation on the Green
formula and condition (63), whose con1bination gives the operator inequal-
ities (32). Having involved the same operator R as was done in problem
(53 )-(54), we obtain the constant w and the operator B in terms of known
members 6 and~. Just for this reason the same algorithm as in a) is work-
able for determination of the ( k + 1 )th iteration for either of the components
k+l
ys and so it is 0111itted here.
c) A system of equations in elasticity theory. The system of Laine 's
equations arose from the stationary elasticity theory:

(67) Lu=μ~ u + (.+μ)grad divu = -f(x)

with vectors u = (11^1 ,u^2 , ... ,11P) and f = (/^1 ,/^2 ,. .. ,JP) and Lame's
constants .\ > 0 and μ > 0. We may attempt the preceding system in the
form

(68)

Further comparison of this with (62) allows us to deduce that

(69)