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,*V*_{k}. The problem then becomes. Find *u*_{k} in
*V*_{k} such that

If we construct a set of basis functions, that spans the
space *V*_{k}, the problem can be rewritten as find such
that,

The FE method has six basic stages:

- 1.
- Construct a decomposition of the domain into elements.
- 2.
- Choose suitable basis functions. The basis functions are parameterized in terms of values at nodes in the elements. The nodes may be internal, on the edges, or at the vertices of an element. When using linear basis functions, as I do, the nodes are usually at the vertices of the elements.
- 3.
- Calculation of the Galerkin system.
- 4.
- Imposition of boundary conditions.
- 5.
- Solution of the Galerkin system.

12/18/1997