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

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## 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 \(m\)-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 estimates for \(\gamma \) this shows that the construction of a uniformly convergent AMLI method with optimal complexity is possible (for \(m \ge 3\)). Finally, we discuss the practical application of this preconditioning technique in the context of time-periodic parabolic optimal control problems.

## Keywords

Algebraic multilevel iteration Reaction–diffusion problems Uniform convergence Optimal complexity## Notes

### Acknowledgments

We want to thank Ulrich Langer for encouraging us to look deeper into this very interesting topic. Moreover, we want to thank Veronika Pillwein for introducing us to the symbolic computational algorithm Cylindrical Algebraic Decomposition (CAD). The research has been supported by the Doctoral Program “Computational Mathematics: Numerical Analysis and Symbolic Computation”. The authors gratefully acknowledge the financial support by the Austrian Science Fund (FWF) under the grant W1214, project DK4, the Johannes Kepler University of Linz and the Federal State of Upper Austria.

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