10
Methods for Solving Grid Equations
In this chapter econo1nical direct and iterative methods are designed for
nmnerical solution of difference elliptic equations.
In Section l we confine ourselves to direct economical methods avail-
able for solving boundary-value problems associated with Poisson's equa-
tion in a rectangle such as the decomposition inethod and the 111athod of
separation of variables.
The general theory of iterative methods is presented in the next sec-
tions with regard to an operator equation of the first kind Au = f, where
A is a self-adjoint operator in a finite-dimensional Euclidean space. The
applications of such theory to elliptic grid equations began to spread to
1nore and more branches a.s they took on an irnportant place in "real-life"
situations.
10.1 DIRECT METHODS- Direct and iterative n1ethods. Recall that the final results of the dif-
ference approxi1nation of boundary-value problems associated with elliptic
equations from Chapter 4 were various systerns of linear algebraic equa-
tions (difference or grid equations). The sizes of the appropriate matri-
ces are extra large and equal the total number N of the grid nodes. For
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