Condition number estimates for the nonoverlapping optimized Schwarz method and the 2-Lagrange multiplier method for general domains and cross points

Research output: Contribution to journalArticle

Abstract

The optimized Schwarz method and the closely related 2-Lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations. Although these methods are known to work well in special cases (e. g., when the domain is a square and the two subdomains are rectangles), the problem has never been systematically stated nor analyzed for general domains with general subdomains. The problem of cross points (when three or more subdomains meet at a single vertex) has been particularly vexing. We introduce a 2-Lagrange multiplier method for domain decompositions with cross points. We estimate the condition number of the iteration and provide an optimized Robin parameter for general domains. We hope that this new systematic theory will allow broader utilization of optimized Schwarz and 2-Lagrange multiplier preconditioners.

Original languageEnglish
Pages (from-to)3062-3083
Number of pages22
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number6
DOIs
StatePublished - 2013

Keywords

  • domain decomposition
  • Schwarz method
  • partial differential equation
  • parallel preconditioner
  • Krylov space
  • 2-Lagrange multiplier
  • WAVE-FORM RELAXATION
  • ABSORBING BOUNDARY-CONDITIONS
  • SHALLOW-WATER EQUATIONS
  • DECOMPOSITION METHODS
  • DIFFUSION PROBLEMS

Cite this

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title = "Condition number estimates for the nonoverlapping optimized Schwarz method and the 2-Lagrange multiplier method for general domains and cross points",
abstract = "The optimized Schwarz method and the closely related 2-Lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations. Although these methods are known to work well in special cases (e. g., when the domain is a square and the two subdomains are rectangles), the problem has never been systematically stated nor analyzed for general domains with general subdomains. The problem of cross points (when three or more subdomains meet at a single vertex) has been particularly vexing. We introduce a 2-Lagrange multiplier method for domain decompositions with cross points. We estimate the condition number of the iteration and provide an optimized Robin parameter for general domains. We hope that this new systematic theory will allow broader utilization of optimized Schwarz and 2-Lagrange multiplier preconditioners.",
keywords = "domain decomposition, Schwarz method, partial differential equation, parallel preconditioner, Krylov space, 2-Lagrange multiplier, WAVE-FORM RELAXATION, ABSORBING BOUNDARY-CONDITIONS, SHALLOW-WATER EQUATIONS, DECOMPOSITION METHODS, DIFFUSION PROBLEMS",
author = "Sebastien Loisel",
year = "2013",
doi = "10.1137/100803316",
volume = "51",
pages = "3062--3083",
journal = "SIAM Journal on Numerical Analysis",
issn = "0036-1429",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "6",

}

TY - JOUR

T1 - Condition number estimates for the nonoverlapping optimized Schwarz method and the 2-Lagrange multiplier method for general domains and cross points

AU - Loisel,Sebastien

PY - 2013

Y1 - 2013

N2 - The optimized Schwarz method and the closely related 2-Lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations. Although these methods are known to work well in special cases (e. g., when the domain is a square and the two subdomains are rectangles), the problem has never been systematically stated nor analyzed for general domains with general subdomains. The problem of cross points (when three or more subdomains meet at a single vertex) has been particularly vexing. We introduce a 2-Lagrange multiplier method for domain decompositions with cross points. We estimate the condition number of the iteration and provide an optimized Robin parameter for general domains. We hope that this new systematic theory will allow broader utilization of optimized Schwarz and 2-Lagrange multiplier preconditioners.

AB - The optimized Schwarz method and the closely related 2-Lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations. Although these methods are known to work well in special cases (e. g., when the domain is a square and the two subdomains are rectangles), the problem has never been systematically stated nor analyzed for general domains with general subdomains. The problem of cross points (when three or more subdomains meet at a single vertex) has been particularly vexing. We introduce a 2-Lagrange multiplier method for domain decompositions with cross points. We estimate the condition number of the iteration and provide an optimized Robin parameter for general domains. We hope that this new systematic theory will allow broader utilization of optimized Schwarz and 2-Lagrange multiplier preconditioners.

KW - domain decomposition

KW - Schwarz method

KW - partial differential equation

KW - parallel preconditioner

KW - Krylov space

KW - 2-Lagrange multiplier

KW - WAVE-FORM RELAXATION

KW - ABSORBING BOUNDARY-CONDITIONS

KW - SHALLOW-WATER EQUATIONS

KW - DECOMPOSITION METHODS

KW - DIFFUSION PROBLEMS

U2 - 10.1137/100803316

DO - 10.1137/100803316

M3 - Article

VL - 51

SP - 3062

EP - 3083

JO - SIAM Journal on Numerical Analysis

T2 - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 6

ER -