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A nearly optimal multigrid method for general unstructured grids

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Abstract

In this paper, we develop a multigrid method on unstructured shape-regular grids. For a general shape-regular unstructured grid of \({\mathcal O}(N)\) elements, we present a construction of an auxiliary coarse grid hierarchy on which a geometric multigrid method can be applied together with a smoothing on the original grid by using the auxiliary space preconditioning technique. Such a construction is realized by a cluster tree which can be obtained in \({\mathcal O}(N\log N)\) operations for a grid of N elements. This tree structure in turn is used for the definition of the grid hierarchy from coarse to fine. For the constructed grid hierarchy we prove that the convergence rate of the multigrid preconditioned CG for an elliptic PDE is \(1 - {\mathcal O}({1}/{\log N})\). Numerical experiments confirm the theoretical bounds and show that the total complexity is in \({\mathcal O}(N\log N)\).

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Authors and Affiliations

  1. Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056, Aachen, Germany

    Lars Grasedyck

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

    Lu Wang

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Jinchao Xu

Authors
  1. Lars Grasedyck
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  2. Lu Wang
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  3. Jinchao Xu
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Correspondence to Lu Wang.

Additional information

This work was supported by NSF DOE DE-SC0006903, DMS-1217142 and Center for Computational Mathematics and Applications at Penn State.

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Grasedyck, L., Wang, L. & Xu, J. A nearly optimal multigrid method for general unstructured grids. Numer. Math. 134, 637–666 (2016). https://doi.org/10.1007/s00211-015-0785-7

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  • DOI: https://doi.org/10.1007/s00211-015-0785-7

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