## Introduction (2)

One of the main disadvantages of the variation approach becomes clear when applying it to complex geometric problems – and most problems in practice are of this type. It is difficult to find an appropriate approximation functions here, because these functions depend on the area geometry, among other dependencies, which will be shown in "The approximation functions"*.*

Now the finite element principle comes into play. The solution area is **divided into a finite number of small sub areas (finite elements)**, as can be seen in the figure to the right. This is also called a finite element discretization of the model area. The geometry of the sub areas is selected to be simple (triangular or rectangular), so that the approximation functions that are necessary for the variation approach can be generated systematically and after a certain pattern.

The solving of the differential equation is then related to vertices of the finite elements; if square approximation functions are used, additional vertices are generated on the midpoints of edges and maybe in the centre of elements.

In the following a short introduction to the finite element method shall be given. It does not claim to be a complete and exhaustive explanation of the method however. A number of excellent standard works like for example [*1] and [*2] exist that can be suggested for further reading.

The introduction is rather meant to be a repetition of known material that is supposed to emphasize the main characteristics of the finite element method and that hopefully has an “oh-that-ishow-it-was”-effect for our advanced students. The access to the finite element form of the depthaveraged shallow water equation will probably be easier with this jump start.