TABLE OF CONTENTS Partial differential equations (PDEs) constitute by far the biggest source of sparse matrix problems. The typical way to solve such equations is to discretize them, i.e., to approximate them by equations that involve a finite number of unknowns. The matrix problems that arise from these discretizations are generally large and sparse; i.e., they have very few nonzero entries. There are several different ways to discretize a PDE. The simplest method uses finite difference approximations for the partial differential operators. The finite element method replaces the original function with a function that has some degree of smoothness over the global domain but is piecewise polynomial on simple cells, such as small triangles or rectangles. This method is probably the most general and well understood discretization technique available. In between these two methods, there are a few conservative schemes called finite volume methods, which attempt to emulate continuous conservation laws of physics. This chapter introduces these three different discretization methods.
Physical phenomena are often modeled by equations that relate several partial derivatives of physical quantities, such as forces, momentums, velocities, energy, temperature, etc. These equations rarely have a closed-form (explicit) solution. In this chapter, a few types of PDEs are introduced, which will serve as models throughout the book. Only one- and two-dimensional problems are considered, and the space variables are denoted by x in the case of one-dimensional problems and x 1 and x 2 for two-dimensional problems.
