2. 4 Topological Mappings of Shape Functions

In this section the function space introducted in the former sections is founded on the definition of the cell complex $\mathcal{C}$ . Once the cell complex is established and available within the framework of the computer, weighting coefficients can be stored in association with the underlying cell complex and a function (which is an element of a predefined function space) is established.

Again, it has to be stated that such an interpretation of a function stored in a computer completely differs from the function by point interpretation, which is used in most methods based on finite differences [57] or finite volumes [58].

A basic data structural requirement for the specification of the function space based on a cell complex is that data can be associated with single elements of the cell complex. In general, one or more mappings between elements of the basis $\mathcal{H}$ of the cell complex $\mathcal{C}$ and some numeric data are used. Such a function might be defined as follows:

$$\forall \mathbf{e} \in \mathcal{H} : f\left(\mathbf{e}\right) \mathrm{is} \; \mathrm{defined}$$ (2.27)

A function which assigns an element of the cell complex a numerical value is called a quantity. It is also possible that such a function is only defined on a partial set of the cell complex. The domain of definition has to be given explicitly in order not to obtain invalid function values (see Fig. 2.6).

$$\forall \mathbf{e} \in \mathcal{K} \wedge \mathcal{K} \subset \mathcal{H}: f\left(\mathbf{e}\right) \mathrm{is} \; \mathrm{defined}$$ (2.28)

Figure 2.6: Quantities defined on vertices, edges and cells.