Adaptive optimal control of the signorini's problem

In this article, we present a-posteriori error estimations in context of optimal control of contact problems; in particular of Signorini’s problem. Due to the contact side-condition, the solution operator of the underlying variational inequality is not differentiable, yet we want to apply Newton’s method. Therefore, the non-smooth problem is regularized by penalization and afterwards discretized by finite elements. We derive optimality systems for the regularized formulation in the continuous as well as in the discrete case. This is done explicitly for Signorini’s contact problem, which covers linear elasticity and linearized surface contact conditions. The latter creates the need for treating trace-operations carefully, especially in contrast to obstacle contact conditions, which exert in the domain. Based on the dual weighted residual method and these optimality systems, we deduce error representations for the regularization, discretization and numerical errors. Those representations are further developed into error estimators. The resulting error estimator for regularization error is defined only in the contact area. Therefore its computational cost is especially low for Signorini’s contact problem. Finally, we utilize the estimators in an adaptive refinement strategy balancing regularization and discretization errors. Numerical results substantiate the theoretical findings. We present different examples concerning Signorini’s problem in two and three dimensions

Multigrid methods for obstacle problems

In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set

Convergence analysis of a conforming adaptive finite element method for an obstacle problem

The adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps 'SOLVE', 'ESTIMATE', 'MARK', and 'REFINE'. The techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE---in fact, there is no modification near the discrete free boundary necessary for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle "chi" and homogeneous Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in the unconstrained case. Numerical results are given to illustrate the performance of the error estimator

Schätzerreduktion und Konvergenz adaptiver FEM für Hindernisprobleme

In dieser Arbeit zeigen wir die Konvergenz einer adaptiven P1 Finite-Elemente-Methode für elliptische Hindernisprobleme. Zur Steuerung der adaptiven Verfeinerung verwenden wir - abhängig von der Art des Hindernisses - verschiedene residualbasierte a posteriori Fehlerschätzer. Wir erweitern das aus dem linearen Fall bekannte Prinzip der Schätzerreduktion auf das (nichtlineare) Hindernisproblem und umgehen somit die Notwendigkeit der diskreten lokalen Effizienz des Fehlerschätzers für unseren Konvergenzbeweis. Als direkte Folge erhalten wir Konvergenzresultate, die von der Art der lokalen Netzverfeinerung weitestgehend unabhängig sind. Zudem gelingt es uns, eine Kontraktionseigenschaft der Datenoszillationen nachzuweisen, so dass diese nicht mehr, wie in der Literatur üblich, explizit kontrolliert werden müssen. Im Falle nicht affiner Hindernisse erkennen wir außerdem zusätzliche Probleme, die auf die Grenzen des Prinzips der Schätzerreduktion hinweisen und somit zum Gesamtverständnis dieser Methode beitragen.In this thesis, we show the convergence of some adaptive P1 finite element method for elliptic obstacle problems. For adaptive mesh refinement we use different residual-based a posteriori error estimators, depending on the shape of the obstacle. We extend the principle of estimator reduction, which is known from the linear case, to the (nonlinear) obstacle problem. In this way, we circumvent the need for the discrete local efficiency of the underlying error estimator for our convergence analysis. A direct consequence are thus convergence results which are basically independent of the local mesh refinement strategy. We further show that the data oscillation terms satisfy some contraction property, such that it becomes unnecessary to control their decay explicitly, as it is usually done in the literature. In the case of non-affine obstacles, we also glimpse additional problems that could not be foreseen within the linear setting and thus contribute to a broader understanding of the principle of estimator reduction