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Finite element

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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.

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