1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular method 699

where f;ij is, as usual, Kronecker's delta.

We learn from Chapter 9, Section 2 that the constants c 1 and c 2

involved in inequalities (63) are equal to

C 2 =A+ 2 μ,

With these, the difference Dirichlet problem associated with (68) is de-

scribed by

(70) A^8 y^8 =-f^8 , xEwh, Y^8 =μ^8 , XE/h, s=l,2,. .. ,p,

where

]J ]J

As y^8 = p. L Aay^8 + (.\ + μ) L A(3sY{3 '

a=l f3=1

The above fra1nework necessitates specifying for any ys E H the operator

Ays = -As ys and the regularizer Ry^8 = - L~= 1 Aa ~/, there by clarifying

that the same operator R is adopted for these purposes as before.

By Green's formula it is straightforward to verify the inequalities

c 1 R < A < c 2 R with constants c 1 = p. and c 2 = ,\ + 2μ incorporated.

By exactly the same reasoning as in the preceding examples ATM requires

n 0 (c) iterations, where

(71)

and n~ (c) is the total number of the necessary iterations m solving the

Laplace equation.

8. ATM for solving grid elliptic equations in an arbitrary complex domain.
   Two lines of research in subsequent considerations of such equations are
   evident in available publications in this area over recent years. No 111uch
   is known in the case of an arbitrary complex domain. The first one is
   connected with nonequidistant grids, while the second one deals with some
   modifications of available methods. Even a constant step in each direction
   does not allow to overcome the difficulties in the near-boundary zones as a
   result of emerging non-uniform grids.
   It turns out that the convergence of the aforementioned methods be-
   come worse on account of widely varying bounds of the spectra of difference
   operators.