Robust preconditioning techniques for multiharmonic finite element method with application to time-periodic parabolic optimal control problems

We are concerned with efficient solutions of the time-periodic parabolic optimal control problems. By using the multiharmonic FEM, the linear algebraic equations characterizing the first-order optimality conditions can be decoupled into a series of parallel solvable block 4 × 4 linear systems with respect to the cosine and sine Fourier coefficients of the state and scaled control variables for different frequencies. Parameter robust preconditioners are proposed for solving these linear systems along with information on practical algorithm implementation and detailed spectral analysis. Problem independent eigenvalue bounds and upper bound approximations of the condition numbers of the eigenvector matrices are obtained for the preconditioned matrices. Such results ensure efficient Krylov subspace acceleration methods and a parameter-free Chebyshev acceleration method, which are both robust in view of all discretization and model parameters. Numerical experiments are presented to demonstrate the robustness and effectiveness of the proposed preconditioners within both Krylov subspace and Chebyshev accelerations compared with some already available preconditioned Krylov subspace methods.