Numerische Mathematik

, Volume 126, Issue 4, pp 741–770 | Cite as

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

  • N. SpillaneEmail author
  • V. Dolean
  • P. Hauret
  • F. Nataf
  • C. Pechstein
  • R. Scheichl


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.


Coarse spaces Overlapping Schwarz method Two-level methods Generalized eigenvectors Problems with large coefficient variation 

Mathematics Subject Classification (2000)

65F10 65N22 65N30 65N55 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • N. Spillane
    • 1
    Email author
  • V. Dolean
    • 2
  • P. Hauret
    • 3
  • F. Nataf
    • 1
  • C. Pechstein
    • 4
  • R. Scheichl
    • 5
  1. 1.Laboratoire Jacques-Louis Lions, CNRS UMR 7598Université Pierre et Marie CurieParisFrance
  2. 2.Laboratoire J.-A.. Dieudonné, CNRS UMR 7351Université de Nice-Sophia AntipolisNice Cedex 02France
  3. 3.Centre de Technologie de LadouxManufacture des Pneumatiques MichelinClermont-Ferrand Cedex 09France
  4. 4.Institute of Computational MathematicsJohannes Kepler UniversityLinzAustria
  5. 5.Department of Mathematical SciencesUniversity of BathBathUK

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