Chemnitz, Techn. Univ., Diss., [2006].

On a large class of two-dimensional anisotropic meshes, the inf-sup condition (stability) is proved for the triangular and quadrilateral finite element pairs suggested by Bernardi/Raugel and Fortin. As a consequence the pairs ${\cal P}_2-{\cal P}_0$, ${\cal Q}_2-{\cal P}_0$, and ${\cal Q}_2^\prime-{\cal P}_0$ turn out to be stable independent of the aspect ratio of the elements.

Adaptive anisotropic refinement of finite element meshes allows to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem. We present a new approach to adaptive refinement and demonstrate that this allows to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.

On a large class of two-dimensional anisotropic meshes, the inf-sup condition (stability) is proved for the triangular and quadrilateral finite element pairs suggested by Bernardi/Raugel and Fortin. As a consequence the pairs ${\cal P}_2-{\cal P}_0$, ${\cal Q}_2-{\cal P}_0$, and ${\cal Q}_2^\prime-{\cal P}_0$ turn out to be stable independent of the aspect ratio of the elements.

Viele technische Prozesse führen auf Randwertprobleme mit partiellen Differentialgleichungen, die mit Finite-Elemente-Methoden näherungsweise gelöst werden können. Spezielle Varianten dieser Methoden sind Finite-Elemente-Mortar-Methoden. Sie erlauben das Arbeiten mit an Teilgebietsschnitträndern nichtzusammenpassenden Netzen, was für Probleme mit komplizierten Geometrien, Randschichten, springenden Koeffizienten sowie für zeitabhängige Probleme von Vorteil sein kann. Ebenso können unterschiedliche Diskretisierungsmethoden in den einzelnen Teilgebieten miteinander gekoppelt werden. In dieser Arbeit wird das Finite-Elemente-Mortaring nach einer Methode von Nitsche für elliptische Randwertprobleme auf zweidimensionalen polygonalen Gebieten untersucht. Von besonderem Interesse sind dabei nichtreguläre Lösungen (u \in H^{1+\delta}(\Omega), \delta>0) mit Eckensingularitäten für die Poissongleichung sowie die Lamé-Gleichung mit gemischten Randbedingungen. Weiterhin werden singulär gestörte Reaktions-Diffusions-Probleme betrachtet, deren Lösungen zusätzlich zu Eckensingularitäten noch anisotropes Verhalten in Randschichten aufweisen. Für jede dieser drei Problemklassen wird das Nitsche-Mortaring dargelegt. Es werden einige Eigenschaften der Mortar-Diskretisierung angegeben und a-priori-Fehlerabschätzungen in einer H^1-artigen sowie der L_2-Norm durchgeführt. Auf lokal verfeinerten Dreiecksnetzen können auch für Lösungen mit Eckensingularitäten optimale Konvergenzordnungen nach gewiesen werden. Bei den Lösungen mit anisotropen Verhalten werden zusätzlich anisotrope Dreiecksnetze verwendet. Es werden auch hier Konvergenzordnungen wie bei klassischen Finite-Elemente-Methoden ohne Mortaring erreicht. Numerische Experimente illustrieren die Methode und die Aussagen zur Konvergenz.

The work selects a specific issue from the numerical analysis of optimal control problems. We investigate a linear-quadratic optimal control problem based on a partial differential equation on 3-dimensional non-convex domains. Based on efficient solution methods for the partial differential equation an algorithm known from control theory is applied. Now the main objectives are to prove that there is no degradation in efficiency and to verify the result by numerical experiments. We describe a solution method which has second order convergence, although the intermediate control approximations are piecewise constant functions. This superconvergence property is gained from a special projection operator which generates a piecewise constant approximation that has a supercloseness property, from a sufficiently graded mesh which compensates the singularities introduced by the non-convex domain, and from a discretization condition which eliminates some pathological cases. Both isotropic and anisotropic discretizations are investigated and similar superconvergence properties are proven. A model problem is presented and important results from the regularity theory of solutions to partial differential equation in non-convex domains have been collected in the first chapters. Then a collection of statements from the finite element analysis and corresponding numerical solution strategies is given. Here we show newly developed tools regarding error estimates and projections into finite element spaces. These tools are necessary to achieve the main results. Known fundamental statements from control theory are applied to the given model problems and certain conditions on the discretization are defined. Then we describe the implementation used to solve the model problems and present all computed results.

The work selects a specific issue from the numerical analysis of optimal control problems. We investigate a linear-quadratic optimal control problem based on a partial differential equation on 3-dimensional non-convex domains. Based on efficient solution methods for the partial differential equation an algorithm known from control theory is applied. Now the main objectives are to prove that there is no degradation in efficiency and to verify the result by numerical experiments. We describe a solution method which has second order convergence, although the intermediate control approximations are piecewise constant functions. This superconvergence property is gained from a special projection operator which generates a piecewise constant approximation that has a supercloseness property, from a sufficiently graded mesh which compensates the singularities introduced by the non-convex domain, and from a discretization condition which eliminates some pathological cases. Both isotropic and anisotropic discretizations are investigated and similar superconvergence properties are proven. A model problem is presented and important results from the regularity theory of solutions to partial differential equation in non-convex domains have been collected in the first chapters. Then a collection of statements from the finite element analysis and corresponding numerical solution strategies is given. Here we show newly developed tools regarding error estimates and projections into finite element spaces. These tools are necessary to achieve the main results. Known fundamental statements from control theory are applied to the given model problems and certain conditions on the discretization are defined. Then we describe the implementation used to solve the model problems and present all computed results.

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.

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.

Viele technische Prozesse führen auf Randwertprobleme mit partiellen Differentialgleichungen, die mit Finite-Elemente-Methoden näherungsweise gelöst werden können. Spezielle Varianten dieser Methoden sind Finite-Elemente-Mortar-Methoden. Sie erlauben das Arbeiten mit an Teilgebietsschnitträndern nichtzusammenpassenden Netzen, was für Probleme mit komplizierten Geometrien, Randschichten, springenden Koeffizienten sowie für zeitabhängige Probleme von Vorteil sein kann. Ebenso können unterschiedliche Diskretisierungsmethoden in den einzelnen Teilgebieten miteinander gekoppelt werden. In dieser Arbeit wird das Finite-Elemente-Mortaring nach einer Methode von Nitsche für elliptische Randwertprobleme auf zweidimensionalen polygonalen Gebieten untersucht. Von besonderem Interesse sind dabei nichtreguläre Lösungen (u \in H^{1+\delta}(\Omega), \delta>0) mit Eckensingularitäten für die Poissongleichung sowie die Lamé-Gleichung mit gemischten Randbedingungen. Weiterhin werden singulär gestörte Reaktions-Diffusions-Probleme betrachtet, deren Lösungen zusätzlich zu Eckensingularitäten noch anisotropes Verhalten in Randschichten aufweisen. Für jede dieser drei Problemklassen wird das Nitsche-Mortaring dargelegt. Es werden einige Eigenschaften der Mortar-Diskretisierung angegeben und a-priori-Fehlerabschätzungen in einer H^1-artigen sowie der L_2-Norm durchgeführt. Auf lokal verfeinerten Dreiecksnetzen können auch für Lösungen mit Eckensingularitäten optimale Konvergenzordnungen nach gewiesen werden. Bei den Lösungen mit anisotropen Verhalten werden zusätzlich anisotrope Dreiecksnetze verwendet. Es werden auch hier Konvergenzordnungen wie bei klassischen Finite-Elemente-Methoden ohne Mortaring erreicht. Numerische Experimente illustrieren die Methode und die Aussagen zur Konvergenz.