SIAM Journal on Scientific and Statistical Computing
On dynamic iteration methods for solving time-periodic differential equations
SIAM Journal on Numerical Analysis
Iterative methods for solving linear systems
Iterative methods for solving linear systems
A Note on Preconditioning for Indefinite Linear Systems
SIAM Journal on Scientific Computing
Nonlinear Analysis: Theory, Methods & Applications
Numerical Methods for Time-Periodic Solutions of Nonlinear Parabolic Boundary Value Problems
SIAM Journal on Numerical Analysis
Numerical analysis of nonlinear multiharmonic eddy current problems
Numerische Mathematik
Efficient solvers for nonlinear time-periodic eddy current problems
Computing and Visualization in Science
Computers & Mathematics with Applications
SIAM Journal on Matrix Analysis and Applications
Nested multigrid methods for time-periodic, parabolic optimal control problems
Computing and Visualization in Science
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This paper presents the analysis of a distributed parabolic optimal control problem in a multiharmonic setting. In particular, the desired state is assumed to be multiharmonic. After eliminating the control from the optimality system, we arrive at the reduced optimality system for the state and the co-state that is nothing but a coupled system of a forward and a backward parabolic partial differential equation. Due to the linearity, the state and the co-state are multiharmonic as well. We discretize the Fourier coefficients by the finite element method. This leads to a large system of algebraic equations, which fortunately decouples into smaller systems each of them defining the cosine and sine Fourier coefficients for the state and co-state with respect to a single frequency. For these smaller systems, we construct preconditioners resulting in a fast converging minimal residual solver with a parameter-independent convergence rate. All these systems can be solved totally in parallel.