Adaptive Finite Element Methods for Contact Problems embedded in a Fictitious Domain - Simulation and Optimal Control

The purpose of the thesis at hand is the simulation and the optimal control of contact problems using the finite cell method (FCM). The FCM combines the fictitious domain concept with the finite element method (FEM). This enables us to use different domains in the original continuous setting and in the discrete one. In particular, we start off with a smoothly penalized formulation of the contact problems. They are then embedded into a larger fictitious domain. At this point the FEM is applied to discretize the problem. Thereby, the transition from the original to the purely fictitious area is resolved by a possibly inexact composite quadrature. Finally, the semi-linear discrete problem is solved by a globalized Newton method introducing a linearized discrete problem. The overall a-priori theory provides the convergence of the considered approach. We note especially the contribution to the a-priori error analysis of the FCM for semi-linear equations. In order to get computable approximations of the arising error quantities, we perform a thorough a-posteriori error analysis. For this purpose, we work with a goal oriented approach. In particular, the dual weighted residual method is applied and error identities are proven in the case of the simulation as well as in the case of the optimal control. The error identities consists in five additive terms and a higher order remainder. Each term is assigned to one error source: regularizing the contact conditions, applying the fictitious domain concept, discretizing by means of the FEM, using an inexact quadrature, and solving iteratively the non-linear discrete system. Based on the acquired insights a-posteriori error estimators are derived for each error source and they are controlled in an adaptive algorithm. Finally, numerical experiments substantiate the theoretical findings.