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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    Anderson, E., Bai, Z., Bischof, C.: LAPACK Users’ guide, vol. 9. Society for Industrial Mathematics (1999)

  2. 2.

    Brezina, M., Heberton, C., Mandel, J., Vaněk, P.: An iterative method with convergence rate chosen a priori. Technical Report 140, University of Colorado Denver, CCM, University of Colorado Denver : Presented at the 1998 Copper Mountain Conference on Iterative Methods, April (1999, 1998)

  3. 3.

    Chartier, T., Falgout, R.D., Henson, V.E., Jones, J., Manteuffel, T., McCormick, S., Ruge, J., Vassilevski, P.S.: Spectral AMGe (\(\rho \)AMGe). SIAM J. Sci. Comput. 25(1), 1–26 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Chevalier, C., Pellegrini, F.: PT-SCOTCH: a tool for efficient parallel graph ordering. Parallel Comput. 6–8(34), 318–331 (2008)

    Article  MathSciNet  Google Scholar 

  5. 5.

    Dohrmann, C.R., Widlund, O.B.: An overlapping Schwarz algorithm for almost incompressible elasticity. SIAM J. Numer. Anal. 47(4), 2897–2923 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Dohrmann, C.R., Widlund, O.B.: Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity. Int. J. Numer. Method Eng. 82(2), 157–183 (2010)

    MATH  MathSciNet  Google Scholar 

  7. 7.

    Dolean, V., Nataf, F., Scheichl, R., Spillane, N.: Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps. Comput. Methods Appl. Math. 12(4), 391–414 (2012)

    Article  MathSciNet  Google Scholar 

  8. 8.

    Dolean, V., Nataf, F., Spillane, N., Xiang, H.: A coarse space construction based on local Dirichlet to Neumann maps. SIAM J. Sci. Comput. 33, 1623–1642 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Dryja, M., Sarkis, M.V., Widlund, O.B.: Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72(3), 313–348 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Dryja, M., Smith, B.F., Widlund, O.B.: Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal. 31(6), 1662–1694 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Dryja, Maksymilian, Widlund, Olof B.: Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15(3), 604–620 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Efendiev, Y., Galvis, J., Lazarov, R., Willems, J.: Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46(05), 1175–1199 (2012)

    Google Scholar 

  13. 13.

    Efendiev, Y., Galvis, J., Vassilevski, P.S.: Spectral element agglomerate algebraic multigrid methods for elliptic problems with high contrast coefficients. In: Huang, Y., Kornhuber, R., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering XIX, Lecture Notes in Computational Science and Engineering, vol. 78, pp. 407–414. Springer, Berlin (2011)

  14. 14.

    Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8(4), 1461–1483 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model Simul 8(5), 1621–1644 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Graham, I.G., Lechner, P.O., Scheichl, R.: Domain decomposition for multiscale PDEs. Numer. Math. 106(4), 589–626 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Frédéric Hecht. FreeFem++. Numerical Mathematics and Scientific Computation. Laboratoire J.L. Lions, Université Pierre et Marie Curie, 3.7 edn. http://www.freefem.org/ff++/ (2010)

  18. 18.

    Karypis, G., Kumar, V.: METIS: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices. Technical report, Department of Computer Science, University of Minnesota. http://glaros.dtc.umn.edu/gkhome/views/metis (1998)

  19. 19.

    Knyazev, A.V.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23(2), 517–541 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Lehoucq, R. B., Sorensen, D. C., Yang, C.: ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Society for Industrial Mathematics (1998)

  21. 21.

    Mandel, J., Brezina, M.: Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp. 65(216), 1387–1401 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Nataf, F., Xiang, H., Dolean, V.: A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps. C. R. Mathématique 348(21–22), 1163–1167 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Parlett, B.N.: The symmetric eigenvalue problem, vol. 20. Society for Industrial Mathematics (1998)

  24. 24.

    Pechstein, C., Scheichl, R.: Scaling up through domain decomposition. Appl. Anal. 88(10–11), 1589–1608 (2009)

    Google Scholar 

  25. 25.

    Pechstein, C., Scheichl, R.: Weighted Poincaré inequalities. IMA J. Numer. Anal. 33(3), 652–686 (2013)

    Google Scholar 

  26. 26.

    Saad, Y.: Iterative methods for sparse linear systems. Society for Industrial Mathematics (2003)

  27. 27.

    Scheichl, R., Vainikko, E.: Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients. Computing 80(4), 319–343 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Scheichl, R., Vassilevski, P.S., Zikatanov, L.T.: Mutilevel methods for elliptic problems with highly varying coefficients on non-aligned coarse grids. SIAM J. Numer. Anal. 50(3), 1675–1694 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. 29.

    Smith, B. F., Bjørstad, P. E., Gropp, W.: Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (2004)

  30. 30.

    Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: A robust two level domain decomposition preconditioner for systems of PDEs. Comptes Rendus Mathématique 349(23–24), 1255–1259 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. 31.

    Toselli, A., Widlund, O.B.: Domain decomposition methods—algorithms and theory. In: Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

  32. 32.

    Vassilevski, P.S.: Multilevel block factorization preconditioners. Springer, New York (2008)

    Google Scholar 

  33. 33.

    Willems, J.: Spectral coarse spaces in robust two-level methods. Technical Report 2012–20, RICAM, Linz (2012)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to N. Spillane.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Spillane, N., Dolean, V., Hauret, P. et al. Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126, 741–770 (2014). https://doi.org/10.1007/s00211-013-0576-y

Download citation

Keywords

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

Mathematics Subject Classification (2000)

  • 65F10
  • 65N22
  • 65N30
  • 65N55