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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

An Obstacle Problem for Mean Curvature Flow

Logaritsch, Philippe 25 October 2016 (has links) (PDF)
We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.
2

A posteriori error estimators based on duality techniques from the calculus of variations

Buß, Hinderk. Unknown Date (has links) (PDF)
University, Diss., 2003--Heidelberg.
3

Optimization problems with complementarity constraints in infinite-dimensional spaces

Wachsmuth, Gerd 10 August 2017 (has links) (PDF)
In this thesis we consider optimization problems with complementarity constraints in infinite-dimensional spaces. On the one hand, we deal with the general situation, in which the complementarity constraint is governed by a closed convex cone. We use the local decomposition approach, which is known from finite dimensions, to derive first-order necessary optimality conditions of strongly stationary type. In the non-polyhedric case, stronger conditions are obtained by an additional linearization argument. On the other hand, we consider the optimal control of the obstacle problem. This is a classical example for a problem with complementarity constraints in infinite dimensions. We are concerned with the control-constrained case. Due to the lack of surjectivity, a system of strong stationarity is not necessarily satisfied for all local minimizers. We identify assumptions on the data of the optimal control problem under which strong stationarity of local minimizers can be verified. Moreover, without any additional assumptions on the data, we show that a system of M-stationarity is satisfied provided that some sequence of multipliers converges in capacity. Finally, we also discuss the notion of polyhedric sets. These sets have many applications in infinite-dimensional optimization theory. Since the results concerning polyhedricity are scattered in the literature, we provide a review of the known results. Furthermore, we give some new results concerning polyhedricity of intersections and provide counterexamples which demonstrate that intersections of polyhedric sets may fail to be polyhedric. We also prove a new polyhedricity result for sets in vector-valued Sobolev spaces.
4

An Obstacle Problem for Mean Curvature Flow

Logaritsch, Philippe 19 October 2016 (has links)
We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.
5

Optimization problems with complementarity constraints in infinite-dimensional spaces

Wachsmuth, Gerd 19 June 2017 (has links)
In this thesis we consider optimization problems with complementarity constraints in infinite-dimensional spaces. On the one hand, we deal with the general situation, in which the complementarity constraint is governed by a closed convex cone. We use the local decomposition approach, which is known from finite dimensions, to derive first-order necessary optimality conditions of strongly stationary type. In the non-polyhedric case, stronger conditions are obtained by an additional linearization argument. On the other hand, we consider the optimal control of the obstacle problem. This is a classical example for a problem with complementarity constraints in infinite dimensions. We are concerned with the control-constrained case. Due to the lack of surjectivity, a system of strong stationarity is not necessarily satisfied for all local minimizers. We identify assumptions on the data of the optimal control problem under which strong stationarity of local minimizers can be verified. Moreover, without any additional assumptions on the data, we show that a system of M-stationarity is satisfied provided that some sequence of multipliers converges in capacity. Finally, we also discuss the notion of polyhedric sets. These sets have many applications in infinite-dimensional optimization theory. Since the results concerning polyhedricity are scattered in the literature, we provide a review of the known results. Furthermore, we give some new results concerning polyhedricity of intersections and provide counterexamples which demonstrate that intersections of polyhedric sets may fail to be polyhedric. We also prove a new polyhedricity result for sets in vector-valued Sobolev spaces.
6

On efficient a posteriori error analysis for variational inequalities

Köhler, Karoline Sophie 14 November 2016 (has links)
Effiziente und zuverlässige a posteriori Fehlerabschätzungen sind eine Hauptzutat für die effiziente numerische Berechnung von Lösungen zu Variationsungleichungen durch die Finite-Elemente-Methode. Die vorliegende Arbeit untersucht zuverlässige und effiziente Fehlerabschätzungen für beliebige Finite-Elemente-Methoden und drei Variationsungleichungen, nämlich dem Hindernisproblem, dem Signorini Problem und dem Bingham Problem in zwei Raumdimensionen. Die Fehlerabschätzungen hängen vom zum Problem gehörenden Lagrange Multiplikator ab, der eine Verbindung zwischen der Variationsungleichung und dem zugehörigen linearen Problem darstellt. Effizienz und Zuverlässigkeit werden bezüglich eines totalen Fehlers gezeigt. Die Fehleranschätzungen fordern minimale Regularität. Die Approximation der exakten Lösung erfüllt die Dirichlet Randbedingungen und die Approximation des Lagrange Multiplikators ist nicht-positiv im Falle des Hindernis- und Signoriniproblems, und hat Betrag kleiner gleich 1 für das Bingham Problem. Dieses allgemeine Vorgehen ermöglicht das Einbinden nicht-exakter diskreter Lösungen, welche im Kontext dieser Ungleichungen auftreten. Aus dem Blickwinkel der Anwendungen ist Effizienz und Zuverlässigkeit im Bezug auf den Fehler der primalen Variablen in der Energienorm von großem Interesse. Solche Abschätzungen hängen von der Wahl eines effizienten diskreten Lagrange Multiplikators ab. Im Falle des Hindernis- und Signorini Problems werden postive Beispiele für drei Finite-Elemente Methoden, der konformen Courant Methode, der nicht-konformen Crouzeix-Raviart Methode und der gemischten Raviart-Thomas Methode niedrigster Ordnung hergeleitet. Partielle Resultate liegen im Fall des Bingham Problems vor. Numerischer Experimente heben die theoretischen Ergebnisse hervor und zeigen Effizienz und Zuverlässigkeit. Die numerischen Tests legen nahe, dass der aus den Abschätzungen resultierende adaptive Algorithmus mit optimaler Konvergenzrate konvergiert. / Efficient and reliable a posteriori error estimates are a key ingredient for the efficient numerical computation of solutions for variational inequalities by the finite element method. This thesis studies such reliable and efficient error estimates for arbitrary finite element methods and three representative variational inequalities, namely the obstacle problem, the Signorini problem, and the Bingham problem in two space dimensions. The error estimates rely on a problem connected Lagrange multiplier, which presents a connection between the variational inequality and the corresponding linear problem. Reliability and efficiency are shown with respect to some total error. Reliability and efficiency are shown under minimal regularity assumptions. The approximation to the exact solution satisfies the Dirichlet boundary conditions, and an approximation of the Lagrange multiplier is non-positive in the case of the obstacle and Signorini problem and has an absolute value smaller than 1 for the Bingham flow problem. These general assumptions allow for reliable and efficient a posteriori error analysis even in the presence of inexact solve, which naturally occurs in the context of variational inequalities. From the point of view of the applications, reliability and efficiency with respect to the error of the primal variable in the energy norm is of great interest. Such estimates depend on the efficient design of a discrete Lagrange multiplier. Affirmative examples of discrete Lagrange multipliers are presented for the obstacle and Signorini problem and three different first-order finite element methods, namely the conforming Courant, the non-conforming Crouzeix-Raviart, and the mixed Raviart-Thomas FEM. Partial results exist for the Bingham flow problem. Numerical experiments highlight the theoretical results, and show efficiency and reliability. The numerical tests suggest that the resulting adaptive algorithms converge with optimal convergence rates.

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