Robust Algebraic Multilevel Preconditioners for Anisotropic Problems

We present an overview on the state of the art of robust AMLI preconditioners for anisotropic elliptic problems. The included theoretical results summarize the convergence analysis of both linear and nonlinear AMLI methods for finite element discretizations by conforming and nonconforming linear elements and by conforming quadratic elements. The initially proposed hierarchical basis approach leads to robust multilevel algorithms for linear but not for quadratic elements for which an alternative AMLI method based on additive Schur complement approximation (ASCA) has been developed by the authors just recently. The presented new numerical results are focused on cases beyond the limitations of the rigorous AMLI theory. They reveal the potential and prospects of the ASCA approach to enhance the robustness of the resulting AMLI methods especially in situations when the matrix-valued coefficient function is not resolved on the coarsest mesh in the multilevel hierarchy.

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