ubc Adaptivity in anisotropic finite element calculations 2006-05-09 [Electronic ed.] prv Universitätsbibliothek Chemnitz Universitätsbibliothek Chemnitz, Chemnitz Fakultät für Mathematik male Rostow am Don male male male male When the finite element method is used to solve boundary value problems, the corresponding finite element mesh is appropriate if it is reflects the behavior of the true solution. A posteriori error estimators are suited to construct adequate meshes. They are useful to measure the quality of an approximate solution and to design adaptive solution algorithms. Singularly perturbed problems yield in general solutions with anisotropic features, e.g. strong boundary or interior layers. For such problems it is useful to use anisotropic meshes in order to reach maximal order of convergence. Moreover, the quality of the numerical solution rests on the robustness of the a posteriori error estimation with respect to both the anisotropy of the mesh and the perturbation parameters. There exist different possibilities to measure the a posteriori error in the energy norm for the singularly perturbed reaction-diffusion equation. One of them is the equilibrated residual method which is known to be robust as long as one solves auxiliary local Neumann problems exactly on each element. We provide a basis for an approximate solution of the aforementioned auxiliary problem and show that this approximation does not affect the quality of the error estimation. Another approach that we develope for the a posteriori error estimation is the hierarchical error estimator. The robustness proof for this estimator involves some stages including the strengthened Cauchy-Schwarz inequality and the error reduction property for the chosen space enrichment. In the rest of the work we deal with adaptive algorithms. We provide an overview of the existing methods for the isotropic meshes and then generalize the ideas for the anisotropic case. For the resulting algorithm the error reduction estimates are proven for the Poisson equation and for the singularly perturbed reaction-difussion equation. The convergence for the Poisson equation is also shown. Numerical experiments for the equilibrated residual method, for the hierarchical error estimator and for the adaptive algorithm confirm the theory. The adaptive algorithm shows its potential by creating the anisotropic mesh for the problem with the boundary layer starting with a very coarse isotropic mesh. 510 Anisotropes Gitter Finite-Elemente-Methode Konvergenz adaptive algorithm anisotropic a posteriori error estimation anisotropic mesh equilibrated residual method finite elements hierarchical error estimator singularly perturbed reaction-diffusion equation triangular mesh urn:nbn:de:swb:ch1-200600815 Technische Universität Chemnitz dgg Technische Universität Chemnitz, Chemnitz Sergey Grosman M. Sc. 1978-08-15 aut Thomas Apel Prof. Dr. dgs rev Bernd Heinrich Prof. Dr. rev Gert Lube Prof. Dr. rev eng 2006-02-02 2006-04-21 born digital doctoral_thesis