Eine einfache Darstellung der Grundgleichungen für 'Große Deformationen' und Herleitung eines geeigneten Fehlerschätzers für die adaptive FEM.

A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using \emph{anisotropic meshes} which can improve the accuracy of the discretization considerably. The main focus is on \emph{a posteriori} error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient \emph{a posteriori} error estimation is achieved for the finite volume method on anisotropic meshes.

Some problems and results in connection with error estimators for modern elements of the Mindlin Reissner equation for plates are discussed.

Es werden die vorgestellten Fehlerindikatoren in Bezug auf die Netzsteuerung anhand von drei Beispielen analysiert. Im weiteren werden auch die einzelen MITC-Elemente und ihre Besonderheiten bei dieser Analyse der Netzsteuerung mit berücksichtigt. Als Abschluss werden einige spezielle Fehlerindikatoren vorgestellt, die für die weitere Entwicklung einige interessante Eigenschaften aufzeigen. Im zweiten Teil geht es um die Auswertung mit dem speziellen Ziel der Findung einer optimalen Netzsteuerung. Dabei wird auf die Besonderheiten der Elemente eingegangen sowie auf die Plattendicke und auf ihre Wirkung bei den Fehlerindikatoren. Mit diesen Erkenntnissen wird ein spezieller Fehlerindikator vorgestellt, der die Vorteile aller Fehlerindikatoren aus Teil I vereint.

A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using \emph{anisotropic meshes} which can improve the accuracy of the discretization considerably. The main focus is on \emph{a posteriori} error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient \emph{a posteriori} error estimation is achieved for the finite volume method on anisotropic meshes.

It is well-known that the solution of a so-called singularly perturbed differential equation exhibits layers. These are small regions in the domain where the solution changes drastically. These layers deteriorate the convergence of standard numerical algorithms, such as the finite element method on a uniform mesh. In the past many approaches were developed to overcome this difficulty. In this context it was very helpful to understand the structure of the solution - especially to know where the layers can occur. Therefore, we have a lot of analysis in the literature concerning the properties of solutions of such problems. Nevertheless, this field is far from being understood conclusively. More recently, there is an increasing interest in the numerics of optimal control problems subject to a singularly perturbed convection-diffusion equation and box constraints for the control. However, it is not much known about the solutions of such optimal control problems. The proposed solution methods are based on the experience one has from scalar singularly perturbed differential equations, but so far, the analysis presented does not use the structure of the solution and in fact, the provided bounds are rather meaningless for solutions which exhibit boundary layers, since these bounds scale like epsilon^(-1.5) as epsilon converges to 0. In this thesis we strive to prove bounds for the solution and its derivatives of the optimal control problem. These bounds show that there is an additional layer that is weaker than the layers one expects knowing the results for scalar differential equation problems, but that weak layer deteriorates the convergence of the proposed methods. In Chapter 1 and 2 we discuss the optimal control problem for the one-dimensional case. We consider the case without control constraints and the case with control constraints separately. For the case without control constraints we develop a method to prove bounds for arbitrary derivatives of the solution, given the data is smooth enough. For the latter case we prove bounds for the derivatives up to the second order. Subsequently, we discuss several discretization methods. In this context we use special Shishkin meshes. These meshes are piecewise equidistant, but have a very fine subdivision in the region of the layers. Additionally, we consider different ways of discretizing the control constraints. The first one enforces the compliance of the constraints everywhere and the other one enforces it only in the mesh nodes. For each proposed algorithm we prove convergence estimates that are independent of the parameter epsilon. Hence, they are meaningful even for small values of epsilon. As a next step we turn to the two-dimensional case. To be able to adapt the proofs of Chapter 2 to this case we require bounds for the solution of the scalar differential equation problem for a right hand side f only in W^(1,infty). Although, a lot of results for this problem can be found in the literature but we can not apply any of them, because they require a smooth right hand side f in C^(2,alpha) for some alpha in (0,1). Therefore, we dedicate Chapter 3 to the analysis of the scalar differential equations problem only using a right hand side f that is not very smooth. In Chapter 4 we strive to prove bounds for the solution of the optimal control problem in the two dimensional case. The analysis for this problem is not complete. Especially, the characteristic layers induce subproblems that are not understood completely. Hence, we can not prove sharp bounds for all terms in the solution decomposition we construct. Nevertheless, we propose a solution method. Numerical results indicate an epsilon-independent convergence for the considered examples - although we are not able to prove this.

Eine einfache Darstellung der Grundgleichungen für 'Große Deformationen' und Herleitung eines geeigneten Fehlerschätzers für die adaptive FEM.

Some problems and results in connection with error estimators for modern elements of the Mindlin Reissner equation for plates are discussed.

Es werden die vorgestellten Fehlerindikatoren in Bezug auf die Netzsteuerung anhand von drei Beispielen analysiert. Im weiteren werden auch die einzelen MITC-Elemente und ihre Besonderheiten bei dieser Analyse der Netzsteuerung mit berücksichtigt. Als Abschluss werden einige spezielle Fehlerindikatoren vorgestellt, die für die weitere Entwicklung einige interessante Eigenschaften aufzeigen. Im zweiten Teil geht es um die Auswertung mit dem speziellen Ziel der Findung einer optimalen Netzsteuerung. Dabei wird auf die Besonderheiten der Elemente eingegangen sowie auf die Plattendicke und auf ihre Wirkung bei den Fehlerindikatoren. Mit diesen Erkenntnissen wird ein spezieller Fehlerindikator vorgestellt, der die Vorteile aller Fehlerindikatoren aus Teil I vereint.

Diese Arbeit befasst sich mit Untersuchung der Hölder Calmness, eines Stabilitätskonzeptes das man als Verallgemeinerung des Begriffs der Calmness erhält. Ausgehend von Charakterisierungen dieser Eigenschaft für Niveaumengen von Funktionen, werden, unter der Voraussetzung der Hölder Calmness, Prozeduren zur Bestimmung von Elementen dieser Mengen analysiert. Ebenso werden hinreichende Bedingungen für Hölder Calmness studiert. Da Hölder Calmness (nichtleerer) Lösungsmengen endlicher Ungleichungssysteme mittels (lokaler) Fehlerabschätzungen beschrieben werden kann, werden auch Erweiterungen der lokalen zu globalen Ergebnissen diskutiert. Als Anwendung betrachten wir speziell den Fall von Niveaumengen von Polynomen bzw. allgemeine Lösungsmengen polynomialer Gleichungen und Ungleichungen. Eine konkrete Frage, die wir beantworten wollen, ist die nach dem Zusammenhang zwischen dem größten Grad der beteiligten Polynome sowie dem Typ, d.h. dem auftretenden Exponenten, der Hölder Calmness des entsprechenden Systems. / This thesis is concerned with an analysis of Hölder calmness, a stability property derived from the concept of calmness. On the basis of its characterization for (sub)level sets, we will cogitate about procedures to determine points in such sets under a Hölder calmness assumption. Also sufficient conditions for Hölder calmness of (sub)level sets and of inequality systems will be given and examined. Further, since Hölder calmness of (nonempty) solution sets of finite inequality systems may be described in terms of (local) error bounds, we will as well amplify the local propositions to global ones. As an application we investigate the case of (sub)level sets of polynomials and of general solution sets of polynomial equations and inequalities. A concrete question we want to answer here is, in which way the maximal degree of the involved polynomials is connected to the exponent of Hölder calmness or of the error bound for the system in question.