2. 6 Examples

In the last section a calculus has been introduced which offers a large variety of different methods for the discrete specification of formulae. The following section provides examples in order to show the advantages of this calculus in comparison to the standard formulation, especially when using computers for the implementation.

The main aim of the following examples is to show the required steps. Even though some methods only use features of the calculus, they are all formulated in a discrete manner. Methods are shown which are of discrete nature and are not results of a discretization process of a continuous problem, but which are modeled in a discrete manner.

In the following examples different underlying features are combined in order to calculate the required data. In the first example only topological properties are necessary to determine the solution. In the second example the considerations rely on quantities and the topological structure. The third example shows how geometrical problems can be solved. In general the geometrical treatment relies on quantities, however, implicit information about the geometrical structure of the cells is used. The same holds true for the fourth example which uses implicit information on both, geometry of the cells and the shape of the functions.

Even though also algebraic methods can be described via this formalism, the introduction requires the use of linearized equations. An example for algebraic methods is therefore shown in Chapter 4.