# Finite Element Method (FEM)

In order to explain unsteady problems, the heat conductivity problem from chapter 3.1.3 can be reused as an example. This time the unsteady temperature distribution T(x,y,t) in a two-dimensional isotropic medium  with the boundary  is wanted. The differential equation for this problem is:

with the density , the specific heat C, the heat conductivity coefficient k in direction of the x and y-axis (isotropic medium) and the heat generation per volume unit Q(x,y,t) (source term). The boundary condition that is necessary to solve the differential equation is given by the initial solution at the time t=0.

To allow for time-depended problems to be solved completely with the finite element method, special elements are necessary, that can be integrated as well spatially as with respect to time. This means a lot of additional work, although the finite element method itself is very time-consuming. The result is that the already complex solving strategy becomes even more abstract and more difficult to understand. That is why a so-called semi-discrete method is used in many applications of the finite element method (e.g. in RMA2). In this case it is assumed that it is possible to separate the dependency on time from the spatial variation. In the semi-discrete method just like for the steady problem, the weak form is derived, which in our case can be expressed as:

The equation above makes the assumptions of the Galerkin method, i.e. approximation function = weighting function. The terms of Eq. 3-49 depend on the time as opposed to the weak form of the steady problem. No partial differentiation is done with respect to time, and the weighting function is assumed to be a function of only x and y.