M-Matrices and Discretization Methods

Abstract

There are two different views of looking at discretization methods for the numerical solution of boundary and initial boundary value problems. The first one focusses attention or the convergence analysis of the methods used, the second primarily investigates how the applied methods reflect basic properties of continuous problems in discrete approximations. For second order linear elliptic and parabolic problems, which we shall consider in the following, these properties are, for instance, inverse monotonicity, nonnegativity and monotonicity of solutions, maximum principles and conservation laws. Of course, there exist close relationships between these two approaches to discretization methods.