SOME RECENT ADVANCES IN THE MATHEMATICS OF FINITE ELEMENTS

Abstract

This chapter discusses the mathematical theory of finite element methods. It also discusses the Sobolev spaces and variational formulation of elliptic boundary value problems, nodal finite element method, abstract finite element method, and nonlinear elliptic problems and time dependent problems. It is well known that the finite element method is a special case of the Ritz–Galerkin method. The classical Ritz approach has two great shortcomings: (1) in practice, construction of the basis functions is only possible for some special domains: (2) the corresponding Ritz matrices are full matrices, and are very often, for simple problems, catastrophically ill-conditioned. The crucial difference between the finite element method and the classical Ritz–Galerkin technique lie in the construction of the basis functions. In the finite element method, the basis functions for general domains can easily be computed. The main feature of these basis functions is that they vanish over all but a fixed number of the elements into which the given domain is divided. This property causes the Ritz matrices to be sparse band matrices, and the resulting Ritz process is stable. There are in addition to the Ritz–Galerkin method many other direct variational methods, and the differences among these stem from the variational principles used.