Finite-Element Methods: Spaces and Properties

Abstract

Finite-element method(s) (FEM) were first introduced by R. Courant [56] in 1943. Since the 1950s, the method has found wide applications in civil, mechanical and aerospace engineering. A rigorous mathematical theory also began to emerge in the late 1960s. Many software packages for applying these methods to various types of partial differential equations have been developed by scientists and engineers. Today, the FEM are probably the most important and powerful tools for obtaining numerical solutions of linear and nonlinear partial differential equations; see § 1.3. The main references for this chapter are [173, 49]. In discretizing boundary integral equations, finite elements are used to collocate the equations at nodal points. In this chapter, we will introduce the commonly used finite-elements in one, two and three space dimensions, and prove some of the basic properties of finiteelement approximation spaces, which are essential in understanding the collocation error estimates for BEM in Chapter 10.