Overview

In this chapter, we will look more deeply into finite element methods for solving steady-state BVPs. Finite element methods form a vast area, and even by the end of this chapter, we will have only scratched the surface. Our goal is modest: We wish to give the reader a better idea of how finite element methods are implemented in practice, and also to give an overview of the convergence theory.

The main tasks involved in applying the finite element method are

• Defining a mesh on the computational domain.

• Computing the stiffness matrix K and the load vector f.

• Solving the finite element equation Ku = f.

We will mostly ignore the first question, except to provide examples for simple domains. Mesh generation is an area of study in its own right, and delving into this subject is beyond the scope of this book. We begin by addressing issues involved in computing the stiffness matrix and load vector, including data structures for representing and manipulating the mesh. We will concentrate on two-dimensional problems, triangular meshes, and piecewise linear finite elements, as these are sufficient to illustrate the main ideas. Next, in Section 10.2, we discuss methods for solving the finite element equation Ku = f, specifically, on algorithms for taking advantage of the sparsity of this system of equations. In Section 10.3, we provide a brief outline of the convergence theory for Galerkin finite element methods. Finally, we close the book with a discussion...