Iterative methods for elliptic partial differential equations

Abstract

Iterative methods are just approximate methods, applied in the solution of partial differential equations (pdes) of elliptic, parabolic and hyperbolic types. In this paper, we analyze the basic theory, convergence, and other properties of iterative methods for elliptic pdes. We examine the three basic iterative methods, Jacobi, Gauss-Seidel and Successive Overrelaxation (SOR) methods, and perform numerical experiments with them, with a view to establishing the most efficient of the methods in terms of rate of convergence, simplicity, and ease of implementation on the computer. It was discovered that the SOR has the fastest convergence rate, followed by the Gauss-Seidel and then the Jacobi method. In terms of simplicity, however, the Jacobi method is far simpler than the more complicated Gauss-Seidel and SOR methods, in view of the fact that it involves lesser computational rigour than them.