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Robust Algebraic Multilevel Preconditioners for Anisotropic Problems

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Book cover Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 45))

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Abstract

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.

Mathematics Subject Classification (2010): 65N20, 65M25

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Acknowledgements

This work has been supported by the Austrian Science Fund (grant P22989-N18), the Bulgarian NSF (grant DCVP 02/1), and FP7-REGPOT-2012-CT2012-316087-AComIn Grant.

Author information

Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, A-4040, Linz, Austria

    J. Kraus

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 25A, 1113, Sofia, Bulgaria

    M. Lymbery & S. Margenov

Authors
  1. J. Kraus
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  2. M. Lymbery
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  3. S. Margenov
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Corresponding author

Correspondence to S. Margenov .

Editor information

Editors and Affiliations

  1. , Dept. of Flow and Material Simulation, Fraunhofer Institute for Industrial Math, Fraunhofer-Platz 1, Kaiserslautern, 67663, Germany

    Oleg P. Iliev

  2. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. 25A, Sofia, 1113, Bulgaria

    Svetozar D. Margenov

  3. , Dept. of Mathematical and Statistical Sc, University of Alberta, 677 Central Academic Building, Edmonton, T6G 2G1, Alberta, Canada

    Peter D Minev

  4. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, East Avenue 7000, Livermore, 94550, California, USA

    Panayot S. Vassilevski

  5. , Department of Mathematics, Pennsylvania State University, 310 McAllister Building, University Park, 16802, Pennsylvania, USA

    Ludmil T Zikatanov

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Kraus, J., Lymbery, M., Margenov, S. (2013). Robust Algebraic Multilevel Preconditioners for Anisotropic Problems. In: Iliev, O., Margenov, S., Minev, P., Vassilevski, P., Zikatanov, L. (eds) Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications. Springer Proceedings in Mathematics & Statistics, vol 45. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7172-1_12

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