This paper is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds.
This paper is on preconditioners for reactiondiffusion problems that are both, uniform with respect to the reaction-diffusion coefficients, and optimal in terms of computational complexity. The considered preconditioners belong to the class of so-called algebraic multilevel iteration (AMLI) methods, which are based on a multilevel block factorization and polynomial stabilization. The main focus of this work is on the construction and on the analysis of a hierarchical splitting of the conforming finite element space of piecewise linear functions that allows to meet the optimality conditions for the related AMLI preconditioner in case of second-order elliptic problems with non-vanishing zeroorder term. The finite element method (FEM) then leads to a system of linear equations with a system matrix that is a weighted sum of stiffness and mass matrices. Bounds for the constant γ in the strengthened Cauchy-BunyakowskiSchwarz inequality are computed for both mass and stiffness matrices in case of a general m-refinement. Moreover, an additive preconditioner is presented for the pivot blocks that arise in the course of the multilevel block factorization. Its optimality is proven for the case m = 3. Together with the Communicated by Gabriel Wittum. estimates for γ this shows that the construction of a uniformly convergent AMLI method with optimal complexity is possible (for m ≥ 3). Finally, we discuss the practical application of this preconditioning technique in the context of time-periodic parabolic optimal control problems.
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