Mathematical Theory of the Finite Element Method - Some Introductory Aspects.

The mathematical prerequisites from functional analysis that enable the understanding of the mathematical theory of finite element are organized, presented, and explained. They begin with the definition of linear vector spaces and include all intermediate definitions up to the definition of Hilbert spaces. The Ritz approximate solution method for boundary value problems is developed so that the close similarity between it and the finite element method can be observed. Solutions given by the Ritz method are projections of the true solution vector onto a subspace defined by the governing differential equations and boundary conditions. Finite element shape functions are shown to be superior to classical Ritz functions as basis vectors in the Ritz process. The finite element and Ritz methods differ primarily in the choice of continuous functions for the basis vectors. The shape functions are therefore primarily responsible for the wide acceptance and popularity of the finite element method. Author