Hyperbolic Maxwell Variational Inequalities in Type-II Superconductivity
This thesis aims at getting better insights in the theory and numerics of hyperbolic Maxwell variational inequalities of the second kind. Motivated by Bean's critical state model for type-II (high-temperature) superconductivity replacing the classical Ohm's law in Maxwell's equations, we establish a novel well-posedness result for the concerned problem by means of a fully discrete scheme and a rigorous convergence analysis thereof. One major advantage of this approach is the natural derivation of a numerical algorithm to compute the corresponding solution based on the semismooth Newton method and a Moreau--Yosida penalization.
Moreover, from a physical point of view, the problem features unknown interfaces between the superconducting and normal regions of the domain. Therefore, we propose an adaptive mesh refinement algorithm based on a posteriori error estimators in careful combination with the mentioned penalization. This results not only in an increased numerical accuracy, but it also provides a way to identify these unknown interfaces without any additional a priori assumption. The main results consist of the equivalence of the estimators to the actual error between the analytical solution and its penalized finite element approximation, as well as rigorous verification of the convergence of our novel algorithm.
The final chapter is dedicated to a shape optimization problem subject to the variational inequality of the second kind. We compute the shape derivative of a penalized problem to cope with the low regularity of the dual variable mapping due to the underlying variational inequality structure. Thereafter, a limiting analysis with respect to the penalization parameter yields the existence of a minimizer to the original shape optimization problem. Finally, we establish an efficient level set method where the shape derivative provides a descent direction. All our analytical results in this thesis are complemented by various numerical experiments.