# Finite Element Method (FEM)

## The weak form

The weak form of a differential equation is a weighted integral similar as in Eq. 3-6 and it satisfies as well the differential equation it is based on as the natural boundary conditions it is connected with. There is a weak form of each differential equation of the order , because only then the order of the derivative of the variable can be reduced by partial differentiation (e.g. Gauss theorem). Metaphorically speaking, the differentiation of the variable is transferred to the weighting function. At the same time, the natural boundary conditions are integrated into the weighted integral expression by means of the boundary integrals.

A simple example shall clarify the derivation of the weak form. The goal is the steady temperature distribution T(x,y) in a two-dimensional isotropic medium with the boundary . The following equation (Poisson equation) can be stated to describe this problem:

 where: k is the heat conduction coefficient in direction of the x and y-axis and Q(x,y) is the provided heat generation per volume unit (source term).