3. Introduction to the Finite Element Method

Normally the problems arising from mathematical physics are described by partial differential equation systems [13,14,15,16] defined in a given domain of interest. These boundary value problems represent the models of specific problems for further simulation and analysis [17] and are usually approximately solved by numerical procedures. One of these procedures is the finite element method. Originally this method has been successfully applied to mechanical problems [18,19]. Today, the finite element method is the general technique widely used for mathematical and engineering numeric analysis. A lot of books and scripts have been issued to help understanding and applying this interesting and useful topic [20,21,22,23,24,25,26,27,28]. The method is also well suited for object-oriented treatment [29], which is essential for software implementation. Further enhancement of the finite element method can be achieved by formulating self-adapting procedures and techniques to solve open region problems [30] and by the analysis of the error in the computed finite element solution [31,32]. There are two most widely used classical methods for approximation of boundary-value problems. One is the Ritz Method and the other is Galerkin's method [33,34]. The Ritz method^3.1 is a variational method. It formulates the boundary value problem in terms of a variational expression, called functional. Galerkin's method belongs to the family of weighted residual methods. These two methods build the foundation of modern finite element analysis. In this work Galerkin's approach is preferred for the introduction of the finite element method.

In general an area $\mathcal{A}$ is enclosed in its boundary $\partial\mathcal{A}$ which usually consists of several closed curves. Closed surfaces have no boundary ( $\partial\mathcal{A} = 0$ ). Analogously the boundary $\partial\mathcal{V}$ of a three-dimensional domain $\mathcal{V}$ can be represented by one or more closed surfaces. The unit normal vector to the boundary curve $\partial\mathcal{A}$ for the two-dimensional case or to the boundary surface $\partial\mathcal{V}$ for the three-dimensional case, respectively, is denoted as $\vec{n}$ . It has the characteristic length one and points outward.