CHAPTER 10 - The Use of Iterative Methods in the Solution of Partial Differential Equations

Abstract

This chapter discusses the use of iterative methods in the solution of partial differential equations. Most of the large sparse systems, which are solved by iterative methods, arise from discretizations of partial differential equations. The value of a particular generality is a function of the problem or class of problems to be solved. Three basic parts of a computer program to solve a boundary-value problem are the mesh generation, the discretization, and the solution of the matrix problem. The mesh generation and the discretization parts determine the accuracy or value of the numerical solution, while the matrix solution part determines most of the computer solution cost. These three parts are not independent of each other. Increased generality in the mesh generation and discretization parts can significantly increase the matrix solution cost. Thus, solution cost can only be given as a function of the mesh decomposition and discretization methods under consideration. The factors that mostly affect matrix solution costs are total arithmetic operations required, storage requirements, and overhead because of data transmission and logical operations associated with the implementation of the solution method. The most efficient solution procedures usually are those that minimize storage and arithmetic requirements.