ABSTRACT
The finite element method has emerged as an extremely powerful technique for solving systems of partial differential equations. As described in the nowclassic textbook by Strang and Fix [224], the method was originally developed on intuitive grounds by structural engineers. It was later put on a sound theoretical basis by numerical analysts and mathematicians once it was realized that it was, in fact, an instance of the Rayleigh-Ritz technique. It is similar to the finite difference method in that it is basically a technique for generating sparse matrix equations that describe a system. However, it provides a prescription for doing this that is in many ways more straightforward than for finite differences, especially when going to high-order approximations. We start with an example using the Ritz method in one dimension with low-order elements. In subsequent sections this is extended to higher-order elements, the Galerkin method, and multiple dimensions.
