Subject: Multiharmonic Finite Element Analysis of Parabolic Time-Periodic Simulation and Optimal Control Problems
Author: Dipl.-Ing.in Monika Wolfmayr
Supervision: O.Univ.-Prof. Dr. Ulrich Langer
Priv.-Doz. Dr. Johannes Kraus
External referee: Prof. Dr. Alfio Borzì (Universität Würzburg)

This thesis is devoted to the construction, analysis and implementation of efficient and robust numerical methods for linear parabolic time-periodic simulation and optimal control problems. The discretization of these problems is based on the multiharmonic finite element method whereas new algebraic multilevel preconditioned minimal residual methods are developed for solving the discrete problems, which have saddle point structure.

The mathematical and numerical analysis include existence and uniqueness results in a new variational framework and full a priori and a posteriori error estimates in space and time. Since we consider time-periodic problems, the multiharmonic finite element method is a very natural approach to discretize this type of parabolic problems. More precisely, we expand all -- given and unknown -- functions into Fourier series in time, truncate them, and then approximate the Fourier coefficients by the finite element method. This method reduces a large linear time-dependent problem to a sequence of smaller time-independent ones.

The multiharmonic finite element discretization of linear parabolic time-periodic simulation and optimal control problems leads to large systems of symmetric but indefinite linear algebraic equations, which fortunately decouple into smaller linear systems each of them defining the cosine and sine Fourier coefficients with respect to a single frequency. The resulting smaller systems have saddle point structure and can be solved by the preconditioned minimal residual method totally in parallel. Hence, we construct block-diagonal preconditioners resulting in fast converging minimal residual solvers with parameter-independent convergence rates. The diagonal blocks of these preconditioners are sums of stiffness and mass matrices, which can be seen as finite element discretization of reaction-diffusion type problems with heterogeneous reaction and diffusion coefficients.

Moreover, we present efficient preconditioners for reaction-diffusion type problems that are optimal in terms of the computational complexity and robust with respect to the reaction and diffusion coefficients. The considered preconditioners belong to the class of so-called algebraic multilevel iteration methods, which are based on multilevel block factorization and polynomial stabilization. One of the main achievements of this thesis is not only the construction of preconditioners via the algebraic multilevel iteration method but also the presentation of a rigorous proof of the robustness and optimal complexity of these preconditioners. This analysis benefits from the use of symbolic techniques.

Although the main focus of this thesis is the numerical analysis of linear parabolic time-periodic simulation and optimal control problems, we finally implement the algorithms in C++, perform many numerical experiments and discuss numerical results which impressively confirm our theoretical findings.

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