A substructuring preconditioner with vertex-related interface solvers for elliptic-type equations in three dimensions

  • Qiya HuEmail author
  • Shaoliang HuEmail author


In this paper, we propose a variant of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the proposed preconditioner, we use the simplest coarse solver associated with the finite element space induced by the coarse partition and construct inexact interface solvers based on overlapping domain decomposition with small overlaps. This new preconditioner has an important merit: its construction and efficiency do not depend on the concrete form of the considered elliptic-type equations. We apply the proposed preconditioner to solve the linear elasticity problems and Maxwell’s equations in three dimensions. Numerical results show that the convergence rate of PCG method with the preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations.


Domain decomposition Substructuring preconditioner Linear elasticity problems Maxwell’s equations PCG iteration Convergence rate 

Mathematics Subject Classification (2010)

65N30 65N55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Bramble, J., Pasciak, J., Schatz, A.: The construction of preconditioners for elliptic problems by substructuring, IV. Math. Comp. 53, 1–24 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Brenner, S., Sung, L.: BDDC And FETI-DP without matrices or vectors. Comput. Methods Appl. Mech. Engrg. 196, 1429–1435 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cai, M., Pavarino, L. F., Widlund, O. B.: Overlapping Schwarz methods with a standard coarse space for almost incompressible elasticity. SIAM J. Sci. Comput. 37(2), 811–830 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cai, X.: An additive Schwarz algorithms for parabolic convection-diffusion equation. Numer. Math. 601991(1), 41–61 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cai, X.: The use of pointwise interpolation in domain decomposition methods with nonnested meshes. SIAM J. Sci. Comput. 16, 250–256 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cai, X., Sarkis, M.: A restricted additive Schwarz preconditioner for general sparse linear system. SIAM J. Sci. Comput., 21(2), 792–797 (1999). Springer-Verlag, Berlin, Heidelberg, New York, 2008. Third editionMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cessenat, M.: Mathematical Methods in Electromagnetism. World Scientific, River Edge (1998)zbMATHGoogle Scholar
  8. 8.
    Chan, T., Zou, J.: Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes. Numer. Algorithm. 8, 329–346 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chan, T., Smith, B., Zou, J.: Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids. Numer. Math. 73(2), 149–167 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, X., Hu, Q.: Inexact solvers for saddle-point system arising from domain decomposition of linear elatcity problems in three dimensions. Int. J. Numer. Anal. Model. 8(1), p156–173 (2011)Google Scholar
  11. 11.
    Chung, E., Kim, H., Widlund, O.: Two-level overlapping Schwarz algorithms for a staggered discontinuous Galerkin Method. SIAM J. Numer. Anal. 51(1), 47–67 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dohrmann, C.: A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25(1), 246–258 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dohrmann, C., Widlund, O.: An iterative substructuring algorithm for two-dimensional problems in H(curl). SIAM J. Numer. Anal. 50(3), 1004–1028 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dohrmann, C., Widlund, O.: A BDDC Algorithm with Deluxe Scaling for Three-Dimensional H(curl) Problems, Comm. Pure Appl Math. (2015)
  15. 15.
    Dryja, M., Galvis, J., Sarkis, M.: BDDC Methods for discontinuous Galerkin discretization of elliptic problems. J. Complexity 23, 715–739 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dryja, M., Smith, F., Widlund, O.: Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal. 31(6), 1662–1694 (1994)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dryja, M., Widlund, O. B.: Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15, 604–620 (1994)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dryja, M., Widlund, O.: Schwarz methods of Neumann-Neumann type for three- dimensional elliptic finite element problems. Comm. Pure Appl. Math. 48, 121–155 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dubois, O., Gander, M.: Optimized Schwarz methods for a diffusion problem with discontinuous coefficient, to appear in Numerical AlgorithmsGoogle Scholar
  20. 20.
    Farhat, C., Roux, F.: A method of finite element tearing and interconnecting and its parallel solution algorithm, Internat. J. Numer. Methods Eng. 32, 1205–1227 (1991)CrossRefGoogle Scholar
  21. 21.
    Farhat, C., Lesoinne, M., Pierson, K.: A scalable dual-primal domain decomposition method. Numer. Linear Algebra Appl. 7, 687–714 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Farhat, C., Mandel, J., Roux, F.: Optimal convergence properties of the FETI domain decomposition method. Comput. Methods. Appl. Mech. Eng. 115, 365–388 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Frommer, A., Szyld, D.: An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms. SIAM J. Numer. Anal. 39, 463–479 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gander, M.: Optimized Schwarz Methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Gander, M., Kwok, F.: Best Robin parameters for optimized Schwarz methods at cross points. SIAM J. Sci. Comput. 34, 1849–1879 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Geuzaine, C., Remacle, J. -F.: Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities[J]. Int. J. Numer. Methods Eng. 79 (11), 1309–1331 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hu, Q., Shi, Z., Yu, D.: Efficient solvers for saddle-point problems arising from domain decompositions with Lagrange multipliers. SIAM J. Numer. Anal. 42 (3), 905–933 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hu, Q.: A regularized domain decomposition method with lagrange multiplier. Adv. Comput. Math. 26(4), 367–401 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Hu, Q., Shu, S., Wang, J.: Nonoverlapping domain decomposition methods with a simple coarse space for elliptic problems. Math. Comput. 79(272), 2059–2078 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hu, Q., Shu, S., Zou, J.: A substructuring preconditioner of three-dimensional Maxwell’s equations. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering XX (No. 91 in Lecture Notes in Computational Science and Engineering). Proceedings of the Twentieth International Conference on Domain Decomposition Methods, held at the University of California at San Diego, CA, February 9-13, 2011, pp. 73–84. Springer, Heidelberg-Berlin (2013)Google Scholar
  32. 32.
    Hu, Q., Zou, J.: A nonoverlapping domain decomposition method for Maxwells equations in three dimensions. SIAM J. Numer. Anal. 41(5), 1682–1708 (2003)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Hu, Q., Zou, J.: Substructuring preconditioners for saddle-point problems arising from Maxwells equations in three dimensions. Math. Comp. 73(245), 35–61 (2004). (electronic)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev space. Acta Math. 147(1–2), 71–88 (1981)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Karypis, G.: METIS a Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices Version 5.1.0. University of Minnesota, Department of Computer Science and Engineering, Minneapolis, MN (2013)Google Scholar
  36. 36.
    Klawonn, A., Widlund, O. B.: A domain decomposition method with Lagrange multipliers and A inexact solvers for linear elasticity. SIAM J. Sci Comput. 22, 1199–1219 (2000)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Klawonn, A., Widlund, O., Dryja, M.: Dual-primal FETI methods for three-dimensional elliptic problems with Heterogeneous coefficients. SIAM J. Numer. Anal. 40, 159–179 (2002)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Klawonn, A., Rheinbach, O., Widlund, O. B.: An analysis of a FETI-DP algorithm on irregular subdomains in the plane. SIAM J. Numer. Anal. 46, 2484–2504 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kim, H., Tu, X.: A three-level BDDC algorithm for mortar discretizations. SIAM J. Numer. Anal. 47, 1576–1600 (2009)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Li, J., Widlund, O.: FETI-DP,BDDC,andblockCholeskymethods. Internat. J. Numer. Methods Eng. 66(2), 250C271 (2006). CrossRefGoogle Scholar
  41. 41.
    Li, J., Widlund, O.: On the use of inexact subdomain solvers for BDDC algorithms. Comput. Methods Appl. Mech. Eng. 196, 1415–1428 (2007)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Mandel, J., Brezina, M.: Balancing domain decomposition for problems with large jumps in coefficients. Math. Comput. 65, 1387–1401 (1996)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Mandel, J., Dohrmann, C.: Convergence of a balancing domain decomposition by constraints and energy minimization, Numer Linear Algebra Appl. (2003)Google Scholar
  44. 44.
    Mandel, J., Dohrmann, C., Tezaur, R.: An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer Math. 54, 167–193 (2005)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Monk, P.: Finite element methods for maxwells equations. Oxford University Press, Oxford (2003)CrossRefGoogle Scholar
  46. 46.
    Nécas, J.: Les méthodes directes en théOrie des équations elliptiques. Academia, Prague (1967)zbMATHGoogle Scholar
  47. 47.
    Si, H.: TetGen, A Quality Tetrahedral Mesh Generator and 3D Delaunay Triangulator, Version 1.5.Google Scholar
  48. 48.
    Smith, B.: An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems. SIAM J. Sci. Stat. Comput. 13(1), 364–378 (1992)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Toselli, A.: Overlapping Schwarz methods for Maxwells equations in three dimensions. Numer. Math. 86, 733–752 (2000)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Toselli, A.: Dual-primal FETI algorithms for edge finite element approximations in 3D. IMA J. Numer. Anal. 26, 96–130 (2006)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Toselli, A., Widlund, O.: Domain Decomposition Methods: Algorithms and Theory. Springer, Berlin (2005)CrossRefGoogle Scholar
  52. 52.
    Veiga, L., Cho, D., Pavarino, L., Scacchi, S.: Overlapping Schwarz methods for Isogeometric analysis. SIAM J. Numer. Anal. 50, 1394–1416 (2012)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Veiga, L., Pavarino, L., Scacchi, S., Widlund, O., Zampini, S.: Isogeometric BDDC preconditioners with deluxe scaling. SIAM J. Sci. Comput. 36 (3), 1118–1139 (2014)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Xu, J., Zhu, Y.: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. M3AS 18, 77–105 (2008)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Xu, J., Zou, J.: Some non-overlapping domain decomposition methods, SIAM Review, 24 (1998)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LSEC, ICMSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

Personalised recommendations