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

Journal Article · · SIAM Journal on Numerical Analysis
DOI:https://doi.org/10.1137/100805248· OSTI ID:1226946
 [1];  [2];  [3]
  1. Univ. of Bath (United Kingdom). Dept. of Mathematical Sciences
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  3. Pennsylvania State Univ., University Park, PA (United States). Dept. of Mathematics
+ Show Author Affiliations

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.

Research Organization:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE
DOE Contract Number:
AC52-07NA27344
OSTI ID:
1226946
Report Number(s):
LLNL-JRNL-451252
Journal Information:
SIAM Journal on Numerical Analysis, Vol. 50, Issue 3; ISSN 0036-1429
Publisher:
Society for Industrial and Applied Mathematics
Country of Publication:
United States
Language:
English

References (15)

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
  • Falgout, Robert D.; Vassilevski, Panayot S.; Zikatanov, Ludmil T.
  • Numerical Linear Algebra with Applications, Vol. 12, Issue 5-6 https://doi.org/10.1002/nla.437
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

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Related Subjects

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