A robust finite element solver for a multiharmonic parabolic optimal control problem

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.