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Multigrid Methods for the Stokes Equations using Distributive Gauss–Seidel Relaxations based on the Least Squares Commutator

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Abstract

A distributive Gauss–Seidel relaxation based on the least squares commutator is devised for the saddle-point systems arising from the discretized Stokes equations. Based on that, an efficient multigrid method is developed for finite element discretizations of the Stokes equations on both structured grids and unstructured grids. On rectangular grids, an auxiliary space multigrid method using one multigrid cycle for the Marker and Cell scheme as auxiliary space correction and least squares commutator distributive Gauss–Seidel relaxation as a smoother is shown to be very efficient and outperforms the popular block preconditioned Krylov subspace methods.

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Acknowledgments

The work of the first author was supported by 2010-2012 China Scholarship Council (CSC). The second author was supported by NSF Grant DMS-1115961, and in part by Department of Energy prime award # DE-SC0006903. The authors are grateful to the discussions with Professors A. Brandt, J. Hu, J. Xu, I. Yavneh, and L. Zikatanov. The second author was introduced to DGS by Prof. Yavenh during the IMA workshop ‘Numerical Solutions of Partial Differential Equations: Fast Solution Techniques’, Dec 2010, and then had further discussion with these experts in the workshop ‘Algebraic Multigrid Methods with Applications to Fluids and Structure Interactions and Other Multi-Physical Systems’ in Kunming Aug 2011. We would also like to thank IMA and Kunming University of Science and Technology for their support and hospitality, as well as for their exciting research atmosphere. The authors also would like to thank two referees for their thorough review. The quality of the paper is improved by addressing their constructive comments and suggestions.

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  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Ming Wang

  2. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Long Chen

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  1. Ming Wang
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Correspondence to Long Chen.

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Wang, M., Chen, L. Multigrid Methods for the Stokes Equations using Distributive Gauss–Seidel Relaxations based on the Least Squares Commutator. J Sci Comput 56, 409–431 (2013). https://doi.org/10.1007/s10915-013-9684-1

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