 # Finite Element Method (FEM)

## Method of weighted residuals -> Finite element form

This section does not apply to the FEM in general, but only to the specific method of weighted residuals with application of the Galerkin method. First we have a look at the general differential equation Eq. 3-2, for which we want to find the solution u.

D is a linear operator here, in this case a differential operator, and q is some kind of outer load. If we substitute the approximation uN from Eq. 3-1 into Eq. 3-2, the initial equation is not exactly satisfied anymore, and a remainder, also called residual, is generated.

Assuming u to be a function of only x and y (i.e. a two-dimensional, steady problem), the residual R is also a function of x and y, but also of cj. With help of the method of weighted residuals, the parameters cj are chosen so that the residual R approaches zero. The weighted integral below has to be solved.

 Integration is over the area (two-dimensional area) and are the weighting functions, that are principally different from the approximation functions . Only for the Galerkin method and are set equal.  