Multigrid for an HDG method

B. Cockburn; O. Dubois; J. Gopalakrishnan; S. Tan

doi:10.1093/imanum/drt024

Multigrid for an HDG method

B. Cockburn, O. Dubois, J. Gopalakrishnan, S. Tan

Mathematics

Research output: Contribution to journal › Article › peer-review

53 Scopus citations

Abstract

We analyse the convergence of a multigrid algorithm for the hybridizable discontinuous Galerkin (HDG) method for diffusion problems. We prove that a nonnested multigrid V-cycle, with a single smoothing step per level, converges at a mesh-independent rate. Along the way, we study conditioning of the HDG method, prove new error estimates for it and identify an abstract class of problems for which a non-nested two-level multigrid cycle with one smoothing step converges even when the prolongation norm is greater than 1. Numerical experiments verifying our theoretical results are presented.

Original language	English (US)
Pages (from-to)	1386-1425
Number of pages	40
Journal	IMA Journal of Numerical Analysis
Volume	34
Issue number	4
DOIs	https://doi.org/10.1093/imanum/drt024
State	Published - Apr 16 2014

Bibliographical note

Funding Information:
This research was partially supported by the National Science Foundation (Grant DMS-0712955) and by the Minnesota Supercomputing Institute (to B.C.) and supported in part by the National Science Foundation under Grant DMS-1211635 (to J.G.).

Publisher Copyright:
© 2013 The authors. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Keywords

discontinuous Galerkin methods
hybrid methods
multigrid methods

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title = "Multigrid for an HDG method",

abstract = "We analyse the convergence of a multigrid algorithm for the hybridizable discontinuous Galerkin (HDG) method for diffusion problems. We prove that a nonnested multigrid V-cycle, with a single smoothing step per level, converges at a mesh-independent rate. Along the way, we study conditioning of the HDG method, prove new error estimates for it and identify an abstract class of problems for which a non-nested two-level multigrid cycle with one smoothing step converges even when the prolongation norm is greater than 1. Numerical experiments verifying our theoretical results are presented.",

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N2 - We analyse the convergence of a multigrid algorithm for the hybridizable discontinuous Galerkin (HDG) method for diffusion problems. We prove that a nonnested multigrid V-cycle, with a single smoothing step per level, converges at a mesh-independent rate. Along the way, we study conditioning of the HDG method, prove new error estimates for it and identify an abstract class of problems for which a non-nested two-level multigrid cycle with one smoothing step converges even when the prolongation norm is greater than 1. Numerical experiments verifying our theoretical results are presented.

AB - We analyse the convergence of a multigrid algorithm for the hybridizable discontinuous Galerkin (HDG) method for diffusion problems. We prove that a nonnested multigrid V-cycle, with a single smoothing step per level, converges at a mesh-independent rate. Along the way, we study conditioning of the HDG method, prove new error estimates for it and identify an abstract class of problems for which a non-nested two-level multigrid cycle with one smoothing step converges even when the prolongation norm is greater than 1. Numerical experiments verifying our theoretical results are presented.

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Multigrid for an HDG method

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