Subject:  The Multiharmonic Finite Element and Boundary Element Method for Simulation and Control of Eddy Current Problems 
Author:  Dipl.Ing. Michael Kolmbauer 
Supervision:  O.Univ.Prof. Dr. Ulrich Langer 
External referee:  Prof. Dr. Fredi Tröltzsch (Technische Universität Berlin) 
This thesis deals with the simulation and control of timedependent, but timeperiodic eddy current problems in unbounded domains in R^{3}. In order to discretize such problems in the full spacetime cylinder, we use a nonstandard spacetime discretization method, namely, the multiharmonic finite element and boundary element method. This discretization technique yields large systems of linear algebraic equations, whereas the fast solution of these systems determines the efficiency of this method. Here, suitable preconditioners are needed in order to ensure efficient and parameterrobust convergence rates of the applied iterative method. Therefore, the main focus of this thesis lies on the construction and analysis of robust and efficient preconditioning strategies for the resulting systems of linear equations. The basic idea of the multiharmonic approach is to use a Fourier series approximation as a discretization technique in time. This allows to switch from the time domain to the frequency domain, and therefore, to replace the solution of a timedependent problem by the solution of a system of timeindependent problems for the Fourier coefficients. Due to the infinite exterior domain on the one hand, and inhomogeneities and the possible presence of sources in the interior domains on the other hand, the symmetric finite element – boundary element coupling method is used to discretize the Fourier coefficients in space. The main challenge in the construction of efficient and parameterrobust preconditioners for the resulting frequency domain equations is indicated by the full range of crucial model, regularization and discretization parameters, that imping on the convergence rate of any iterative method. We use matrix and operator interpolation techniques to construct parameterrobust, blockdiagonal preconditioners in a straightforward fashion. Furthermore, we prove rigorous bounds for the condition numbers of the preconditioned frequency domain equations, that are independent of all involved model, regularization, and discretization parameters. Numerical examples illustrate the robustness of these blockdiagonal preconditioners. In order to obtain efficient preconditioners, the diagonal blocks have to be replaced by efficient (and robust) preconditioners. The individual diagonal blocks rely on the solution of standard H^{1} or H(curl) problems, for which efficient and robust preconditioners are already available. First, we consider and analyze the eddy current problem. We start by investigating the pure multiharmonic finite element approach on a bounded domain, before we tackle the case of unbounded domains in terms of the multiharmonic finite element – boundary element coupling method. For both, we construct blockdiagonal preconditioners for various types of variational formulations. Second, we consider the eddy current optimal control problem with distributed control. Again we apply the multiharmonic finite element – boundary element coupling method to discretize in time and space. After deriving the optimality system, efficient and parameterrobust preconditioners are constructed for the resulting system of linear equations. A challenging topic in optimal control problems is the incorporation of various constraints imposed on the state or control variables in the optimization procedure. These constraints render the resulting system of equations nonlinear. There, the semismooth Newton linearization leads to problems, where our constructed blockdiagonal preconditioners can also be useful. 
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