Since the FE method has not been discussed much in SEP reports, I will now attempt to give a general introduction to the method. If you are a FE expert the generalizations and omissions in this description may offend you so you should skip to the next section.
The FE method for solving a partial differential equation starts by casting the P.D.E. as a variational problem. The differential problem
This is a variational problem in an infinite space, V. The approximate solution is obtained by restricting the search to a finite space of dimension k,Vk. The problem then becomes. Find uk in Vk such that
If we construct a set of basis functions, that spans the space Vk, the problem can be rewritten as find such that,
This system of equations is know as the Galerkin system. If the basis functions are chosen to be orthogonal in the metric defined by the function , then the matrix A is diagonal. Unfortunately, for irregular problems, such functions are difficult to construct. In the FE method the functions are chosen to be functions with local support so that most elements of the matrix A are zero. The non-zero elements of A correspond to basis functions whose regions of support overlap. The basis functions are defined in terms of nodes of the elements.
The FE method has six basic stages: