CHAPTER 5 - The Finite Element Method

Abstract

The term finite element method denotes the implementation of the Ritz–Galerkin method with finite element basis functions, that is, basis functions that are continuous, piecewise polynomials and that have local support in the sense that each function vanishes outside of a small subregion of the domain Ω. Such functions are defined with respect to some chosen subdivision of Ω into (normally) triangular or rectangular elements that, together with a choice of nodes, makes up a finite element mesh. The mesh is by no means required to be uniform, a fact that gives the finite element method great strength. This chapter discusses the fundamental properties of finite element meshes and basis functions. It describes three common types of basis functions: the piecewise linear and piecewise quadratic functions, both defined on a mesh of triangular elements, and the piecewise bilinear functions defined on a mesh of rectangular elements. The chapter discusses the computational details of constructing the Ritz–Galerkin system of equations for a boundary value problem, given a set of finite element basis functions. It also discusses how the class of basis functions can be expanded to include the so-called isoparametric basis functions. The advantage of the latter is that they provide even more flexibility in the choice of a mesh than was available previously. The chapter presents an error analysis of the finite element method and an analysis of the spectral condition number of the Ritz–Galerkin matrix in the context of the finite element method. It further reviews the effect of boundary corners and material interfaces on the smoothness of the exact solution of a boundary value problem. This is relevant to the effectiveness of the finite element method.