# 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 $\phi$ is first written as
$L(\phi) = 0$
We then assume an approximate solution $\bar{\phi}$ of the form
$\bar{\phi} = a_0 + a_1x + a_2x^2+ ... + a_mx^m$
where the a's are unknown coefficients that are to be determined. For an initial value, it is clear that $\bar\phi$ does not satisfy the governing PDE, therfore leaving a residual R defined as
$R = L(\bar{\phi})$
The idea is to drive the residual to zero by performing the convolution of R with a certain weight function $W$
$\int W R dx = 0$
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

1. Donea, Jean and Huerta, Antonio (2003), Finite Element Methods for Flow Problems, ISBN 0471496669, Wiley, GB.