2. 3 Topological Structures

In the last section it was shown that different collocation points result in different local support of basis functions of a function space $\mathcal{F}'$ which are constructed via tesselation in archetypical elements and a subsequent attaching of local shape functions.

The question arises if such a behavior can be specified independently from the geometrical properties, because relying on geometrical features is inefficient and also inaccurate. For instance, it is not possible to determine, if a point is on an edge or within a triangle, by methods of floating point comparison. It is inefficient to search all triangles which are within the neighborhood of a point or an edge, because a list or a tree has to be searched and it has to be checked, if the respective elements are incident.

The rule for the collocation of functions and their local support is briefly reviewed in order to show the requirements on the underlying topological structure: Functions which are collocated with the interior of triangle have their local support only on the respective triangle, functions which are collocated with an edge point of a triangle have their local support on both triangles which are supersets of the common edge, and functions which are collocated with the corner points of the triangle have their local support on all triangles which cover the respective point.

The properties which are required in all of these rules are of pure topological nature. This means, that only properties of sets such as unions, intersections, subsets, and supersets are relevant, whereas the geometrical properties such as coordinates and distances are irrelevant for the execution of this rule.

In the following a structure is introduced which covers all the topological properties of the initially described geometrical structure without describing its geometrical properties comprising coordinates, distances, and angles. A method is briefly introduced which provides proper means for handling the topological operations, e.g. unions, intersections and operations for finding the local neighborhood of a given element. Furthermore an association of basis functions basis on the topological space is given.