Inexact Additive Schwarz Solvers for hp-FEM Discretizations in Three Dimensions

  • Sven Beuchler
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 66)


In this paper, a boundary value problem of second order in three space dimensions is discretized by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned conjugate gradient method with an overlapping domain decomposition preconditioner with in-exact subproblem solvers. In addition to a global solver for the low order functions, the ingredients of this preconditioner are local solvers for the patches. Here, a solver is used which utilizes the tensor product structure of the patches. The efficiency in time and iteration numbers of the presented solver is shown in several numerical examples for diffusion like problems as well as for problems in linear elasticity.


Bilinear Form Domain Decomposition Coarse Mesh Multigrid Method Spectral Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ainsworth, M.: A preconditioner based on domain decomposition for h-p finite element approximation on quasi-uniform meshes. SIAM J. Numer. Anal. 33(4), 1358–1376 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Antonietti, P.F., Houston, P.: A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods. J. Sci. Comput. 46(1), 124–149 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Babuška, I., Craig, A., Mandel, J., Pitkäranta, J.: Efficent preconditioning for the p-version finite element method in two dimensions. SIAM J. Numer. Anal. 28(3), 624–661 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beuchler, S.: Multi-grid solver for the inner problem in domain decomposition methods for p-FEM. SIAM J. Numer. Anal. 40(3), 928–944 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Beuchler, S.: A domain decomposition preconditioner for p-FEM discretizations of two-dimensional elliptic problems. Computing 74(4), 299–317 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Beuchler, S.: Wavelet solvers for hp-FEM discretizations in 3D using hexahedral elements. Comput. Methods Appl. Mech. Engrg. 198(13-14), 1138–1148 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Beuchler, S., Meyer, A., Pester, M.: SPC-PM3AdH v1.0-programmers manual. Technical Report SFB393 01-08, Technische Universität Chemnitz (2001)Google Scholar
  8. 8.
    Beuchler, S., Schneider, R., Schwab, C.: Multiresolution weighted norm equivalences and applications. Numer. Math. 98(1), 67–97 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Braess, D.: Finite elements. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  10. 10.
    Bramble, J., Pasciak, J., Xu, J.: Parallel multilevel preconditioners. Math. Comp. 55(191), 1–22 (1991)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Canuto, C., Gervasio, P., Quarteroni, A.: Finite-element preconditioning of G-NI spectral methods. SIAM J. Sci. Comput. 31(6), 4422–4451 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Costabel, M., Dauge, M., Demkowicz, L.: Polynomial extension operators for H 1, H(curl) and H(div)-spaces on a cube. Math. Comp. 77(264), 1967–1999 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Demkowicz, L.: Computing with hp Finite Elements. CRC Press, Taylor and Francis (2006)Google Scholar
  14. 14.
    Deville, M.O., Mund, E.H.: Finite element preconditioning for pseudospectral solutions of elliptic problems. SIAM J. Sci. Stat. Comp. 18(2), 311–342 (1990)MathSciNetGoogle Scholar
  15. 15.
    Guo, B., Gao, W.: Domain decomposition method for the hp-version finite element method. Comp. Methods Appl. Mech. Eng. 157, 524–440 (1998)Google Scholar
  16. 16.
    Guo, B., Zhang, J.: Stable and compatible polynomial extensions in three dimensions and applications to the p and h-p finite element method. SIAM J. Numer. Anal. 47(2), 1195–1225 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Haase, G., Langer, U., Meyer, A.: Domain decomposition preconditioners with inexact subdomain solvers. Technical Report 192, TU Chemnitz (1991)Google Scholar
  18. 18.
    Hackbusch, W.: Multigrid Methods and Applications. Springer, Heidelberg (1985)Google Scholar
  19. 19.
    Jensen, S., Korneev, V.G.: On domain decomposition preconditioning in the hierarchical p −version of the finite element method. Comput. Methods Appl. Mech. Eng. 150(1–4), 215–238 (1997)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Karniadakis, G.M., Sherwin, S.J.: Spectral/HP Element Methods for CFD. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  21. 21.
    Korneev, V.G., Langer, U., Xanthis, L.: On fast domain decomposition methods solving procedures for hp-discretizations of 3d elliptic problems. Comp. Methods Appl. Math. 3(4), 536–559 (2003)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Korneev, V.G., Rytov, A.: Fast domain decomposition algorithm discretizations of 3-d elliptic equations by spectral elements. Comput. Methods Appl. Mech. Engrg. 197(17-18), 1443–1446 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Korneev, V.G.: An almost optimal method for Dirichlet problems on decomposition subdomains of the hierarchical hp-version. Differ. Equ. 37(7), 1008–1018 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Mandel, J.: Iterative solvers by substructuring for the p-version finite element method. Comput. Methods Appl. Mech. Eng. 80(1-3), 117–128 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Melenk, J.M., Gerdes, K., Schwab, C.: Fully discrete hp-finite elements: Fast quadrature. Comput. Methods Appl. Mech. Eng. 190, 4339–4364 (1999)CrossRefGoogle Scholar
  26. 26.
    Munoz-Sola, R.: Polynomial liftings on a tetrahedron and applications to the h-p version of the finite element method in three dimensions. SIAM J. Numer. Anal. 34(1), 282–314 (1996)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pavarino, L.F.: Additive Schwarz methods for the p-version finite element method. Numer. Math. 66(4), 493–515 (1994)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Pavarino, L.F., Widlund, O.B.: Iterative substructuring methods for spectral elements in three dimensions. In: Krizek, M., et al. (eds.) Finite Element Methods. 50 Years of the Courant Element, University of Conference held at the Jyväskylä, Finland. Inc. Lect. Notes Pure Appl. Math., vol. 164, pp. 345–355. Marcel Dekker, New York (1994)Google Scholar
  29. 29.
    Pavarino, L.F., Widlund, O.B.: A polylogarithimc bound for an iterative substructuring method for spectral elements in three dimensions. SIAM J. Numer. Anal. 33(4), 1303–1335 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Ruge, J.W., Stüben, K.: Algebraic Multigrid. Multigrid methods, ch. 4, pp. 73–130. SIAM, Philadelphia (1987)Google Scholar
  31. 31.
    Schöberl, J., Melenk, J.M., Pechstein, C., Zaglmayr, S.: Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements. IMA J. Numer. Anal. 28(1), 1–24 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Schwab, C.: p − and hp −finite element methods. Theory and applications in solid and fluid mechanics. Clarendon Press, Oxford (1998)Google Scholar
  33. 33.
    Solin, P., Segeth, K., Dolezel, I.: Higher-Order Finite Element Methods. Chapman and Hall, CRC Press (2003)Google Scholar
  34. 34.
    Toselli, A., Widlund, O.B.: Domain Decomposition Methods - Algorithms and Theory. Springer (2005)Google Scholar
  35. 35.
    Zhang, X.: Multilevel Schwarz methods. Numer. Math. 63, 521–539 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut für Numerische SimulationRheinische Friedrich–Wilhelms–Universität BonnBonnGermany

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