On the robustness and optimality of algebraic multilevel methods for reaction–diffusion type problems

Abstract

This paper is on preconditioners for reaction---diffusion 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 zero-order 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 $$\gamma $$? in the strengthened Cauchy---Bunyakowski---Schwarz inequality are computed for both mass and stiffness matrices in case of a general $$m$$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 $$m=3$$m=3. Together with the estimates for $$\gamma $$? this shows that the construction of a uniformly convergent AMLI method with optimal complexity is possible (for $$m \ge 3$$m?3). Finally, we discuss the practical application of this preconditioning technique in the context of time-periodic parabolic optimal control problems.