2. Mesh Refinement in General

Numerical solutions of continuous mathematical problems join a wide field of different disciplines of scientific interests and applications such as exploration of physical phenomena and processes. One common sine qua non for all numerical schemes is the necessity of discretization by means of sampling. This work is mostly focused on three-dimensional spatial discretization in general, and the optimization via mesh refinement in particular to improve numerical solutions. Across-the-board the term ``mesh refinement'' is used ambiguously, it summarizes a wide class of mesh adaptation and modification techniques which come more and more into the light in the field of numerical calculation.
The construction of spatial sampling in terms of partitioning is in general called mesh generation which is a well-established scientific discipline since the mid 70's [2]. As already noticed in Section 1.2.1, several commercial and non-commercial software products for two and three-dimensional cases have been developed during the years with different scopes and tasks. However, based on already existing three-dimensional tessellations, numerical calculations can largely benefit from mesh refinement techniques, since they improve accuracy and therefore also reliability in a tremendous way.

There are a lot of collections of tools and goodies available for mesh data structures with different adaptation techniques and computational geometry efforts. For example the GrAL library [26] which is a generic library for grid (or mesh) data structures and algorithms operating on them. Another very interesting project is the so-called CGAL library, which is a collaborative effort of several sites in Europe and Israel. The goal is to make the most important of the solutions and methods developed in computational geometry available to users in industry and academia in a C++ library [27]. One very general library according to the demands of graph computations is the so called Boost Graph Library [28] which is part of the Boost C++ Libraries project [29]. And there are many more.

At the Institute for Microlectronics a great deal of investigations regarding three-dimensional mesh generation and adaptation was carried out [17,30,31,32] over the last decade. According to all previous work which dealt almost exclusively with three-dimensional, unstructured, tetrahedron based meshes, it was a logical step to spend effort on mesh adaptation techniques suitable for process and device simulation.