1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular inethod 697

The calculations start from the corner i 1 = l, i 2 = 1, ,. , , iP = l, of the

parallelepiped of interest for which all of the adjacent nodes xC -l,, l E ih

fall into the boundary ones at which the values yC-l,,) = μ are already

known. With knowledge of y at that node we fix i 2 = 1, , , , , iP = 1

and change i 1 = l, 2,,. , , N 1 - 1. After that, continuing this process for

i 1 = 1, 2,, .. , N 1 - 1 and for fixed i 2 = 2, etc. we find at all the nodes

x E wh that

(

P 1 k)( w~)-1

y = w ~ h; yC -1,,) + F 1 + 4

Along these lines, the starting nodes (~ = N ex - 1, Ct = 1, 2, ... , p, suit us

per f ect 1 ly ' or d etermmat1on '. o f k+l y on t e h gn 'd wh b y t e h rue^1

k+l y = ( w LP - 1 k+l(+l) y G +Y -) ( l+-w~)-1

ex=l h2 ex^4

b) A system of elliptic equations. Let u = (u^1 ,u^2 ,, .. ,um^0 ) be a

vector and let a block p x p-matrix k = ( k~';I) with blocks of size m 0 x m 0

will be so chosen as to obtain some suitable matrix kexf3 = (k~';I) of size

n1 0 x 1n 0 for later use. The Dirichlet problem for a system of equations is

first considered in the parallelepiped G:

(61) Lus = -Js' x E G, H s = μ' s x Er, s = 1, 2, ... , m 0 ,

where

(62) L 'tl s =

P mo ,-:i ,-:i m

'\"'""' '\"'""' u ( k srn u H )

L.., L.., o:c u/^3 oa: '

u,/-1=1 m=l Ct i3

In such a setting the condition of ellipticity becomes

P mo P mo P mo

( 63) cl L L (~~)^2 < L L k;;3(x) ~~ ~~' < C2 L L (~~)2,

ex,{3=1 s,m=l ex=! s=l

where eex = ((;,(~,. '' '~~~'o), (\' = 1, 2,'.' ,p, are arbitrary vectors and
c 1 > 0, c 2 > 0 are constants. A reasonable form of the difference operator
IS

(64) Ays

p nio

LL

ex,{3=1 m=l

Asm exp Ym