Example Solutions of Elliptic Grid Equations

Abstract

In Section 14.1 we consider certain ways of constructing implicit iterative schemes, in particular ones based on the extraction of a regularizer. In Section 14.2 we construct direct and iterative methods for solving difference problems in the multi-dimensional case. We consider an iterative method for solving a boundary-value problem of the third kind for an elliptic equation with mixed derivatives in a rectangle. Methods for solving systems of elliptic equations axe discussed in Section 14.3. Here we consider the application of the general theory to the solution of certain problems in elasticity theory. In Section 14.4 we illustrate methods for solving a Dirichlet difference problem for Poisson’s equation in a region of complex form, based on the reduction of the given problem to the solution of a series of problems in a rectangle.