Analysis of Convergence of Numerical Methods

Abstract

In the previous chapter the basic principles of discretization of problems involving partial differential operators were outlined. The objective of this chapter is to describe the essential aspects of the theory of convergence of the resulting schemes. Just as discretization methods were roughly divided into two basic categories as those based on the replacment of the partial differential operators by partial difference operators (the finite difference approach), and those based on a certain characterization of the sollution of the problm (the Galerkin approach), the corresponding convergence analyses possess different features, even though both approaches eventually lead to (finite) systems of equations which are similar, or even identical. The finite difference approach may lead to discrete systems with drastically different properties even when these systems originate from the same problem. For this reason, each discrete system requires special treatment, and the analysis may be much more demanding than that of the original problem! This is why, to date, the discrete counterparts of elliptic boundary-value problems have not been treated satisfactorily (in more than one variable!), even though the theory of the original equations is a show-piece of mathematical achievement. On the other hand, the analysis of approximation schemes based on what we chose to label rather imprecisely as the Galerkin approach is relatively straightforward, due to the fact that for a given problem one has a limited number of (practical) characterizations of the solution.