Footnote:
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Description:
This thesis deals with adaptive meshes in two dimensions, which are used in, e.g., the numerical computation of solutions to partial differential equations (PDEs) and in computer graphics. This thesis consists of four research articles. The first article deals with different refinement strategies for two-dimensional meshes built from triangles or quadrilaterals. The focus is on the efficient realization of the strategies in Matlab and pursues the overall goal to provide a suitable toolbox (ameshref) for research and teaching. The second article concerns the coarsening of triangular meshes that have been refined with the classical red-green-blue refinement method. Here, coarsening is to be understood as a reversal of the refinement. We make use of information about the refinement history implicitly contained in the mesh data. It will be discussed which challenges arise in this context and how coarsening can be realized efficiently and robustly by easy-to-verify criteria. The third article can be read as a thematic follow-up of the second one. Coarsening algorithms for the refinement strategies presented in the first article are discussed and implemented in Matlab. Thus, the toolbox is extended by a coarsening option (ameshcoars). Once again, the challenge lies in extracting relevant information quickly and efficiently from implicit data. The last article deals with the H1-stability of the L2-projection onto finite element spaces on adaptively refined meshes. We discuss local criteria for adaptively refined meshes and apply them to the refinement strategies for quadrilateral meshes implemented in the first article. Thus, H1-stability is shown for a selection of adaptive refinement strategies for finite element spaces with polynomial degree between two and nine. ; In dieser Arbeit beschäftigen wir uns mit adaptiven Netzen in zwei Dimensionen, wie sie zum Beispiel bei der numerischen Berechnung von Lösungen partieller Differentialgleichungen sowie in der Computergraphik Anwendung finden. Diese Arbeit beinhaltet vier ...