Multilevel methods for elliptic problems with highly varying coefficients on nonaligned coarse grids

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

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

In this paper 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. An 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. Numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.

Original languageEnglish (US)
Pages (from-to)1675-1694
Number of pages20
JournalSIAM Journal on Numerical Analysis
Volume50
Issue number3
DOIs
StatePublished - Sep 5 2012

Fingerprint

Multilevel Methods
Varying Coefficients
Elliptic Problems
Boundary value problems
Grid
Coefficient
Discontinuity
Experiments
Quasi-interpolants
Schwarz Methods
Sharpness
Elliptic Boundary Value Problems
Condition number
Piecewise Linear
Diffusion Coefficient
Overlapping
Convergence Rate
Numerical Experiment
Partition
Mesh

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

@article{5c544a46c56f443684bdc3a001465e5d,
title = "Multilevel methods for elliptic problems with highly varying coefficients on nonaligned coarse grids",
abstract = "In this paper 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. An 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 Poincar{\'e}-type inequalities. Numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.",
author = "Robert Scheichl and Vassilevski, {Panayot S.} and Zikatanov, {Ludmil T.}",
year = "2012",
month = "9",
day = "5",
doi = "10.1137/100805248",
language = "English (US)",
volume = "50",
pages = "1675--1694",
journal = "SIAM Journal on Numerical Analysis",
issn = "0036-1429",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",

}

Multilevel methods for elliptic problems with highly varying coefficients on nonaligned coarse grids. / Scheichl, Robert; Vassilevski, Panayot S.; Zikatanov, Ludmil T.

In: SIAM Journal on Numerical Analysis, Vol. 50, No. 3, 05.09.2012, p. 1675-1694.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Multilevel methods for elliptic problems with highly varying coefficients on nonaligned coarse grids

AU - Scheichl, Robert

AU - Vassilevski, Panayot S.

AU - Zikatanov, Ludmil T.

PY - 2012/9/5

Y1 - 2012/9/5

N2 - In this paper 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. An 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. Numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.

AB - In this paper 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. An 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. Numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.

UR - http://www.scopus.com/inward/record.url?scp=84865592671&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865592671&partnerID=8YFLogxK

U2 - 10.1137/100805248

DO - 10.1137/100805248

M3 - Article

AN - SCOPUS:84865592671

VL - 50

SP - 1675

EP - 1694

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 3

ER -