Overview

Most practical applications involve multiple spatial dimensions, leading to partial differential equations involving two to four independent variables: x ₁, x ₂ or x ₁, x ₂, t or x ₁, x ₂, x ₃, t. The purpose of this chapter is to extend the techniques of the last three chapters Fourier series and finite elements to PDEs involving two or more spatial dimensions.

We begin by developing the fundamental physical models in two and three dimensions. We then present Fourier series methods; as we saw earlier, these techniques are applicable only in the case of constant-coefficient differential equations. Moreover, in higher dimensions, we can only find the eigenfunctions explicitly when the computational domain is simple; we treat the case of a rectangle and a circular disk in two dimensions.

We then turn to finite element methods, focusing on two-dimensional problems and piecewise linear finite elements defined on a triangulation of the computational domain.

8.1 Physical models in two or three spatial dimensions

The derivation of the physical models in Chapter 2 depended on the fundamental theorem of calculus. Using this result, we were able to relate a quantity defined on the boundary of an interval (force acting on a cross-section or heat energy flowing across a cross-section, for example) to a related quantity in the interior of the interval.

In higher dimensions, the analogue of the fundamental theorem of calculus is the divergence theorem, which relates a vector field acting...