Finite element

The finite element method belongs to the class of weighted residual methods. It is a very powerful method, yet its basic principle is simple and interesting. The differential equation governing the transport of a scalar <math> \phi</math> is first written as <br>
<math> L(\phi) = 0</math> <br>
We then assume an approximate solution <math> \bar{\phi} </math> of the form <br>
<math> \bar{\phi} = a_0 + a_1x + a_2x^2+ ... + a_mx^m</math> <br>
where the a's are unknown coefficients that are to be determined. For an initial value, it is clear that <math> \bar\phi</math> does not satisfy the governing PDE, therfore leaving a residual R defined as <br>
<math> R = L(\bar{\phi})</math> <br>
The idea is to drive the residual to zero by performing the convolution of R with a certain weight function <math> W </math> <br>
<math> \int W R dx = 0 </math> <br>
By choosing a succession of weight functions, one can generate as many equations as there are unknowns (the a's) thus yielding a albegraic system of equations.

The method can be viewed as a minimization problem. In fact, most discretization methods have this concept built in which is a fundamental principle in the theory of iterative methods.

== References ==
#{{reference-book|author=Donea, Jean and Huerta, Antonio|year=2003|title=Finite Element Methods for Flow Problems|rest=ISBN 0471496669, Wiley, GB}}