Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids

Scheichl, Robert; Vassilevski, Panayot S.; Zikatanov, Ludmil T.

doi:10.1137/100805248

Title: Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids

Abstract

We generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the Schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters. One assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small. We do not need to align them with possible discontinuities in the coefficients. The proofs make use of novel stable splittings based on weighted quasi-interpolants and weighted Poincaré-type inequalities. Finally, numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.

Authors:

Scheichl, Robert ^[1]; Vassilevski, Panayot S. ^[2]; Zikatanov, Ludmil T. ^[3]

Univ. of Bath (United Kingdom). Dept. of Mathematical Sciences
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Pennsylvania State Univ., University Park, PA (United States). Dept. of Mathematics

Publication Date:: Thu Jun 21 00:00:00 EDT 2012

Research Org.:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1226946

Report Number(s):: LLNL-JRNL-451252
Journal ID: ISSN 0036-1429

DOE Contract Number:: AC52-07NA27344

Resource Type:: Journal Article

Journal Name:: SIAM Journal on Numerical Analysis

Additional Journal Information:: Journal Volume: 50; Journal Issue: 3; Journal ID: ISSN 0036-1429

Publisher:: Society for Industrial and Applied Mathematics

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; coarse spaces; multigrid; overlapping Schwarz method; large coefficient jumpts

Citation Formats


                    Scheichl, Robert, Vassilevski, Panayot S., and Zikatanov, Ludmil T. Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids.  United States: N. p., 2012. 
        Web.  doi:10.1137/100805248.

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                    Scheichl, Robert, Vassilevski, Panayot S., & Zikatanov, Ludmil T. Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids.  United States.  https://doi.org/10.1137/100805248

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                    Scheichl, Robert, Vassilevski, Panayot S., and Zikatanov, Ludmil T. 2012.  
        "Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids".  United States.  https://doi.org/10.1137/100805248.  https://www.osti.gov/servlets/purl/1226946.

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@article{osti_1226946,

  title        = {Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids},

  author       = {Scheichl, Robert and Vassilevski, Panayot S. and Zikatanov, Ludmil T.},

  abstractNote = {We generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the Schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters. One assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small. We do not need to align them with possible discontinuities in the coefficients. The proofs make use of novel stable splittings based on weighted quasi-interpolants and weighted Poincaré-type inequalities. Finally, numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.},

  doi          = {10.1137/100805248},

  url          = {https://www.osti.gov/biblio/1226946},
  journal      = {SIAM Journal on Numerical Analysis},

  issn         = {0036-1429},

  number       = 3,

  volume       = 50,

  place        = {United States},

  year         = {Thu Jun 21 00:00:00 EDT 2012},

  month        = {Thu Jun 21 00:00:00 EDT 2012}

}

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Works referenced in this record:

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Similar Records

Title: Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids

Abstract

Citation Formats

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part I journal, January 2005

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part II journal, January 2005

Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids journal, April 1996

Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions journal, January 1996

On two-grid convergence estimates journal, January 2005

Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media journal, January 2010

Unstructured Additive Schwarz--Conjugate Gradient Method for Elliptic Problems with Highly Discontinuous Coefficients journal, January 1999

Domain decomposition for multiscale PDEs journal, April 2007

On the abstract theory of additive and multiplicative Schwarz algorithms journal, March 1995

Analysis of FETI methods for multiscale PDEs. Part II: interface variation journal, February 2011

Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements journal, September 1997

Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients journal, July 2007

Uniform Convergent Multigrid Methods for Elliptic Problems with Strongly Discontinuous Coefficients journal, January 2008

Iterative Methods by Space Decomposition and Subspace Correction journal, December 1992

Domain decomposition preconditioners for elliptic equations with jump coefficients journal, January 2008

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part I
journal, January 2005

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part II
journal, January 2005

Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids
journal, April 1996

Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions
journal, January 1996

On two-grid convergence estimates
journal, January 2005

Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media
journal, January 2010

Unstructured Additive Schwarz--Conjugate Gradient Method for Elliptic Problems with Highly Discontinuous Coefficients
journal, January 1999

Domain decomposition for multiscale PDEs
journal, April 2007

On the abstract theory of additive and multiplicative Schwarz algorithms
journal, March 1995

Analysis of FETI methods for multiscale PDEs. Part II: interface variation
journal, February 2011

Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements
journal, September 1997

Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients
journal, July 2007

Uniform Convergent Multigrid Methods for Elliptic Problems with Strongly Discontinuous Coefficients
journal, January 2008

Iterative Methods by Space Decomposition and Subspace Correction
journal, December 1992

Domain decomposition preconditioners for elliptic equations with jump coefficients
journal, January 2008