Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

N. Spillane, Victorita Dolean Maini, P. Hauret, F. Nataf, C. Pechstein, R. Scheichl

Research output: Contribution to journalArticle

58 Citations (Scopus)

Abstract

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.
LanguageEnglish
Pagesn/a
Number of pages30
JournalNumerische Mathematik
Volumen/a
Issue numbern/a
Early online date15 Aug 2013
DOIs
Publication statusPublished - 2013

Fingerprint

Generalized Eigenproblem
Systems of Partial Differential Equations
Partial differential equations
Overlap
Partial differential equation
Generalized Eigenvalue Problem
Domain Decomposition Method
Domain decomposition methods
Differential System
Scalability
Robustness
Numerical Examples
Three-dimensional
Coefficient
Term

Keywords

  • coarse spaces
  • overlapping Schwarz method
  • two-level methods
  • generalized eigenvectors
  • large coefficient variation

Cite this

Spillane, N. ; Dolean Maini, Victorita ; Hauret, P. ; Nataf, F. ; Pechstein, C. ; Scheichl, R. / Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. In: Numerische Mathematik. 2013 ; Vol. n/a, No. n/a. pp. n/a.
@article{d4c603ced8604e4fa9be229518e70b22,
title = "Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps",
abstract = "Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.",
keywords = "coarse spaces, overlapping Schwarz method, two-level methods, generalized eigenvectors, large coefficient variation",
author = "N. Spillane and {Dolean Maini}, Victorita and P. Hauret and F. Nataf and C. Pechstein and R. Scheichl",
year = "2013",
doi = "10.1007/s00211-013-0576-y",
language = "English",
volume = "n/a",
pages = "n/a",
journal = "Numerische Mathematik",
issn = "0029-599X",
number = "n/a",

}

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. / Spillane, N.; Dolean Maini, Victorita; Hauret, P.; Nataf, F.; Pechstein, C.; Scheichl, R.

In: Numerische Mathematik, Vol. n/a, No. n/a, 2013, p. n/a.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

AU - Spillane, N.

AU - Dolean Maini, Victorita

AU - Hauret, P.

AU - Nataf, F.

AU - Pechstein, C.

AU - Scheichl, R.

PY - 2013

Y1 - 2013

N2 - Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.

AB - Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.

KW - coarse spaces

KW - overlapping Schwarz method

KW - two-level methods

KW - generalized eigenvectors

KW - large coefficient variation

UR - http://link.springer.com/journal/211

U2 - 10.1007/s00211-013-0576-y

DO - 10.1007/s00211-013-0576-y

M3 - Article

VL - n/a

SP - n/a

JO - Numerische Mathematik

T2 - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - n/a

ER -