Fundamentals of Natural Indexing for Simplex Finite Elements in Two and Three Dimensions

In this report the foundation of a new methodology for denoting the components of nite element meshes is presented. The proposed methodology is an indexing technique that utilizes properties often met with in dealing with nite element meshes, namely that there is a coarse triangulation to which local re nement is applied (hierarchical meshes). Correspondingly, our index scheme consists of a coarse and a local index scheme. The coarse index scheme enumerates the components of the coarse triangulation. The local index scheme utilize coordinates that are natural with respect to the shape of the coarse elements. For example, we are using barycentric coordinates for the case of simplex elements. Therefore we refer to our methodology as Natural Indexing. The index scheme provides \useful names" for the mesh components in the following sense: 1.) it expresses the rich spatial structures of nite element meshes in a compact manner, 2.) it enables a convenient description of fast solution methods, and 3.) the scheme is suitable for high-performance computing platforms (in particular parallel ones). We explain in detail how the index scheme is constructed for simplex nite element methods in two and three dimensions. Hereby we rely on commonly applied re nement techniques. In two dimensions this leads two a very compact local index scheme where fundamental mesh relations can be expressed by simple index arithmetic. In three dimensions the index formulas are more complicated. This is related to quality deterioration of tetrahedra created by applying standard renement procedures|an issue that cannot be resolved by an indexing technique.