Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
Wolfmayr, Monika
In this talk, we will consider an optimal control problem with a parabolic time-periodic partial differential equation appearing in its PDE constraints. In electromagnetics, such kind of problems arise from the optimal control of two-dimensional eddy current problems. In order to solve the optimal control problem, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations that decouples into smaller systems, which can be solved totally in parallel. In citeKollmannKolmbauerLangerWolfmayrZulehner:ref:1, we construct preconditioners for these systems which yield robust and fast convergence rates for the preconditioned minimal residual method with respect to all parameters. These block diagonal preconditioners are practically implemented by the algebraic multilevel iteration method presented in citeKraus:ref:2. The diagonal blocks of the preconditioners consist of a weighted sum of stiffness and mass matrices. In citeKrausWolfmayr:ref:3, we discuss and prove the robustness and optimality of the AMLI method for solving these reaction-diffusion type problems discretized by the finite element method. Moreover, the multiharmonic finite element analysis of time-periodic parabolic optimal control problems can also be found in citeLangerWolfmayr:ref:4, where different variational setting are investigated and estimates of the complete discretization error are derived. beginthebibliography4 bibitemKollmannKolmbauerLangerWolfmayrZulehner:ref:1 M. Kollmann, M. Kolmbauer, U. Langer, M. Wolfmayr, W. Zulehner: em A Finite Element Solver for a Multiharmonic Parabolic Optimal Control Problem/, Comput. Math. Appl. 65(3), 2013, 469-486. bibitemKraus:ref:2 J. Kraus: em Additive Schur Complement Approximation and Application to Multilevel Preconditioning/, SIAM J. Sci. Comput., 34(6):A2872-A2895, 2012. bibitemKrausWolfmayr:ref:3 J. Kraus, M. Wolfmayr: em On the Robustness and Optimality of the Algebraic Multilevel Method for Reaction-Diffusion Type Problems/, submitted. bibitemLangerWolfmayr:ref:4 U. Langer, M. Wolfmayr: em Multiharmonic Finite Element Analysis of a Time-Periodic Parabolic Optimal Control Problem/, submitted. endthebibliography