Elementary Finite Elements

Abstract

A numerical approach for solving a differential equation problem is to dis-cretize this problem, which has infinitely many degrees of freedom, to produce a discrete problem, which has finitely many degrees of freedom and can be solved using a computer. Compared with the classical finite difference method, the introduction of the finite element method is relatively recent. The advantages of the finite element method over the finite difference method are that general boundary conditions, complex geometry, and variable material properties can be relatively easily handled. Also, the clear structure and versatility of the finite element method makes it possible to develop general purpose software for applications. Furthermore, it has a solid theoretical foundation that gives added reliability, and in many situations it is possible to obtain concrete error estimates in finite element solutions. The finite element method was first introduced by Courant in 1943 (Courant, 1943). From the 1950's to the 1970's, it was developed by engineers and mathematicians into a general method for the numerical solution of partial differential equations. In this chapter, we describe the finite element method. We first introduce this method for two simple model problems in Sect. 1.1. Then, in Sect. 1.2, we discuss the small fraction of Sobolev space theory that is sufficient for the foundation of the finite element method as studied in this book. In Sect. 1.3, we develop an abstract variational formulation for this method and give some examples. Section 1.4 is devoted to the construction of general finite element spaces. In Sects.