Numerische Simulation Auf Massiv Parallelen Rechnern Anisotropic Finite Elements: Local Estimates and Applications Preprint-reihe Des Chemnitzer Sfb 393

Habilitationsschrift zur Erlangung des akademischen Grades doctor rerum naturalium habilitatus Preface The solution of elliptic boundary value problems may have anisotropic behaviour in parts of the domain. That means that the solution varies signiicantly only in certain directions. Examples include diiusion problems in domains with edges and singularly perturbed convection-diiusion-reaction problems where boundary or interior layers appear. In such cases it is an obvious idea to reeect this anisotropy in the discretization by using anisotropic meshes with a small mesh size in the direction of the rapid variation of the solution and a larger mesh size in the perpendicular direction. Anisotropic meshes can also be advantageous if surfaces with strongly anisotropic curvature (the front side of a wing of an airplane) or thin layers of diierent material are to be discretized. In order to describe anisotropic elements mathematically we introduce the term aspect ratio. The aspect ratio is the ratio of the diameter of the element e and the supremum of the diameters of all balls contained in e. A nite element is called anisotropic if the aspect ratio tends to innnity when the mesh size or some (small perturbation) parameter tends to zero. Contrary, elements are called isotropic if the aspect ratio is bounded by a moderate constant. Triangular elements are isotropic if they satisfy Zllmal's minimal angle condition. Already in the fties and seventies it was shown that certain local interpolation error estimates can be proved for some classes of anisotropic nite elements. The minimalangle condition is replaced by the weaker maximal angle condition. Nevertheless, the majority of papers and books on the nite element method excludes anisotropic nite elements. Since the end of the eighties anisotropic elements are considered in the international literature more intensively. Examples of using anisotropic elements include interpolation tasks, singular perturbation and ow problems, the treatment of edge singularities, and adaptive procedures. The corresponding papers lead to two conclusions. First, anisotropic mesh reenement ooers a great potential for the construction of eecient numerical procedures (interpolation, nite element method, boundary element method, nite volume method), more eecient than it is possible with the restriction to a bounded aspect ratio. So one can expect a broad utilization of such meshes. Second, there are still challenges to set all the ingredients of such methods on a solid mathematical basis. These ingredients include a-priori and a-posteriori error estimates and the solution of the arising system of algebraic …