1549301742-The_Theory_of_Difference_Schemes__Samarskii

708 Methods for Solving Grid Equations

This provides enough reason to conclude that in the case of any com-

plex d0111ain the number of the iterations depends only on the main step h

of the grid w" regardless of near-boundary nocle.s.

Applying ( 91) to the rnodel problem in a square of sides! 0 yields

In (2/c)

n 0 (c) - -~-===

• 3.54 Jhjt;, '

whence it follows that the number of the iterations in an arbitrary domain

G is being increased in 3.54/3.4 = 1.04 times, that is, in 4% in comparison

with those performed in a square, whose sides are equal to its cliamete1·,

thus causing one-two iterations in practical implementations for [ = 10-^4

and h = 1/100.

Summarizing, the number of the iterations required during the course

of MATM in an arbitrary complex domain is close to the number of the

iterations performed for the same Dirichlet problem in a minirnal rectangle

containing the domain G and numerical realizations confirm this statement.

10. On solving difference equations for problen1s with variable coefficients.
    In the preceding sections this trend of research was due to serious develop-
    ments of the Russian and western scientists. Specifically, the method for
    solving difference equations approximating an elliptic equation with vari-
    able coefficients in complex domains G of arbitrary shape and configuration
    is available in Section 8 with placing special emphasis on real advantages
    of MATM in the numerical solution of the difference Dirichlet problem for
    Poisson's equation in Section 9.
    One of the most important issues is concerned with a smaller number
    of the iterations performed in the numerical solution of equations with
    variable coefficients. It was shown in Section 7 that the nmnber of the
    iterations required during the course of ATM is proportional to vz:r;:;,
    where c 1 and c 2 are the smallest and the greatest values of coefficients,
    respectively. The operator R in question can be put in correspondence
    with the operator A with variable coefficients such that

One way of covering this is connected with further intervention of the dif-
o 0
ference Laplace operator A : R = - A. A key role of R owes a debt to the
structure of the factorized operator B: