1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative~triangular n1ethod 695

a) An equation of tlie elliptic type with 1nixed derivatives. In that

case we are agree to consider

( 51)

(52)

p

L 1l = L La(31l ,

a,{3=1

p p p

c 1 L ~~ < L kn{3(x)(~~{3 < C 2 L ~~,

a=l °' ,/3= l

where~= (~ 1 ,~ 2 ,, .. '~P) is an arbitrary vector and c 1 > 0 and c 2 > 0 are

constants.

In dealing with difference operators on the grid wh at hand such as

p

(53) Ay= L Aa(JY, Aa{JY = ~[(ka(3Yx 13 )x 0 + (kn{3Yx,JxJ'

n,/3=1

we may set up in conformity with problem (50) the difference Dirichlet

problem

(54) Ay=-<p(x), xEwh, y=μ(x), XE/h,

0

and follow established practice: introduce in the space H = SI the operators

A and R acting in accordance with the rules

Ay=-Ay,

0

p

Ry= -A Y = - L Yx 0 x 0 '

a= l

where A is a difference (2p + 1 )-point Laplace operator, and accept instead
of (54) the governing equation Ay = <p, where <p f:- <p only at the near-
boundary nodes. Recall that

(55)

where c 1 and c 2 are constants arising from the ellipticity condition (52).
By analogy with Section 5 the operators Rl and R 2 are specified by
the formulas

p

R IY-_'""""' ~ Yxh^0 ,

a=l 'a

p

R2y = - L

a=l