Multiharmonic Finite Element Analysis of a Time-Periodic Parabolic Optimal Control Problem

Dipl.-Ing.in Monika Wolfmayr

Jan. 22, 2013, 3:30 p.m. S2 059

In this talk, we present the multiharmonic analysis of a distributed
parabolic optimal control problem in a time-periodic setting including
existence and uniqueness results of the solution of some weak space-time
variational formulation for the parabolic time-periodic boundary value
problem appearing in the constraints for the optimal control problem.
Since the cost functional is quadratic, the optimal control problem is
uniquely solvable as well. 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 which decouples into smaller systems. We construct
preconditioners for these systems which yield robust and fast
convergence rates for the preconditioned minimal residual method. All
systems can be solved totally in parallel. Moreover, we present an AMLI
preconditioner which leads to a robust and fast solver of optimal
complexity. Finally, a complete analysis for the error introduced by the
multiharmonic finite element discretization is presented as well as some
numerical results confirming our theoretical findings including examples
with different desired states. In addition, we consider jumping material
coefficients in one of the examples.