Schwarz Preconditioning for High Order Simplicial Finite Elements

  • Joachim Schöberl
  • Jens M. Melenk
  • Clemens G. A. Pechstein
  • Sabine C. Zaglmayr
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 55)


This paper analyzes two-level Schwarz methods for matrices arising from the p-version finite element method on triangular and tetrahedral meshes. The coarse level consists of the lowest order finite element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a model problem.


Domain Decomposition Domain Decomposition Method Interpolation Operator Tetrahedral Mesh Coarse Space 
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  1. 1.
    M. Ainsworth, A hierarchical domain decomposition preconditioner for h-p finite element approximation on locally refined meshes, SIAM J. Sci. Comput., 17 (1996), pp. 1395–1413.zbMATHCrossRefGoogle Scholar
  2. 2.
    M. Ainsworth, A preconditioner based on domain decomposition for h-p finite element approximation on quasi-uniform meshes, SIAM J. Numer. Anal., 33 (1996), pp. 1358–1376.zbMATHCrossRefGoogle Scholar
  3. 3.
    I. Babuška, A. Craig, J. Mandel, and J. Pitkäranta, Effcient preconditioning for the p-version finite element method in two dimensions, SIAM J. Numer. Anal., 28 (1991), pp. 624–661.CrossRefGoogle Scholar
  4. 4.
    I. Babuška and M. Suri, The p and hp versions of the finite element method: basic principles and properties, SIAM Review, 36 (1994), pp. 578–632.CrossRefGoogle Scholar
  5. 5.
    A. Bećirović, P. Paule, V. Pillwein, A. Riese, C. Schneider, and J. Schöberl, Hypergeometric summation algorithms for high order finite elements, Tech. Rep. 2006–8, SFB F013, Johannes Kepler University, Numerical and Symbolic Scientific Computing, Linz, Austria, 2006.Google Scholar
  6. 6.
    S. Beuchler, R. Schneider, and C. Schwab, Multiresolution weighted norm equivalences and applications, Numer. Math., 98 (2004), pp. 67–97.zbMATHCrossRefGoogle Scholar
  7. 7.
    S. Beuchler and J. Schöberl, Optimal extensions on tensor-product meshes, Appl. Numer. Math., 54 (2005), pp. 391–405.zbMATHCrossRefGoogle Scholar
  8. 8.
    I. Bica, Iterative substructuring methods for the p-version finite element method for elliptic problems, PhD thesis, Courant Institute of Mathematical Sciences, New York University, New York, September 1997.Google Scholar
  9. 9.
    M. A. Casarin, Jr., Quasi-optimal Schwarz methods for the conforming spectral element discretization, SIAM J. Numer. Anal., 34 (1997), pp. 2482–2502.zbMATHCrossRefGoogle Scholar
  10. 10.
    P. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numer., (1975), pp. 77–84.Google Scholar
  11. 11.
    M. Dryja and O. B. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., SIAM, Philadelphia, PA, 1990, pp. 3–21.Google Scholar
  12. 12.
    T. Eibner and J. M. Melenk, A local error analysis of the boundary concentrated FEM, IMA J. Numer. Anal., (2006). To appear.Google Scholar
  13. 13.
    M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative schwarz algorithms, Numerische Mathematik, 70 (1995), pp. 163–180.zbMATHCrossRefGoogle Scholar
  14. 14.
    B. Guo and W. Cao, An additive Schwarz method for the h-p version of the finite element method in three dimensions, SIAM J. Numer. Anal., 35 (1998), pp. 632–654.zbMATHCrossRefGoogle Scholar
  15. 15.
    G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, 1999.Google Scholar
  16. 16.
    V. G. Korneev and S. Jensen, Domain decomposition preconditioning in the hierarchical p-version of the finite element method, Appl. Numer. Math., 29 (1999), pp. 479–518.zbMATHCrossRefGoogle Scholar
  17. 17.
    J. Mandel, Iterative solvers by substructuring for the p-version finite element method, Comput. Methods Appl. Mech. Eng., 80 (1990), pp. 117–128.zbMATHCrossRefGoogle Scholar
  18. 18.
    J. M. Melenk, On condition numbers in hp-FEM with Gauss-Lobatto based shape functions, J. Comp. Appl. Math., 139 (2002), pp. 21–48.zbMATHCrossRefGoogle Scholar
  19. 19.
    R. M. noz Sola, Polynomial liftings on a tetrahedron and applications to the hp-version of the finite element method in three dimensions, SIAM J. Numer. Anal., 34 (1997), pp. 282–314.CrossRefGoogle Scholar
  20. 20.
    L. F. Pavarino, Additive Schwarz methods for the p-version finite element method, Numer. Math., 66 (1994), pp. 493–515.zbMATHCrossRefGoogle Scholar
  21. 21.
    A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999.Google Scholar
  22. 22.
    J. Schöberl, J. M. Melenk, C. G. A. Pechstein, and S. C. Zaglmayr, Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements, Tech. Rep. 2005–11, RICAM, Johann Radon Institute for Computational and Applied Mathematics, Austria Academy of Sciences, Linz, Austria, 2005.Google Scholar
  23. 23.
    C. Schwab, p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, Oxford Science Publications, 1998.Google Scholar
  24. 24.
    S. J. Sherwin and M. A. Casarin, Low-energy basis preconditioning for elliptic substructured solvers based on unstructured spectral/hp element discretization, J. Comput. Phys., 171 (2001), pp. 394–417.zbMATHCrossRefGoogle Scholar
  25. 25.
    B. F. Smith, P. E. Bjørstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 1996.Google Scholar
  26. 26.
    B. Szabó and I. Babuška, Finite Element Analysis, John Wiley & Sons, New York, 1991.zbMATHGoogle Scholar
  27. 27.
    B. Szabó, A. Düster, and E. Rank, The p-version of the finite element method, in Encyclopedia of Computational Mechanics, E. Stein, R. de Borst, and T. J. R. Hughes, eds., vol. 1, John Wiley & Sons, 2004, ch. 5.Google Scholar
  28. 28.
    A. Toselli and O. B. Widlund, Domain Decomposition Methods - Algorithms and Theory, vol. 34 of Series in Computational Mathematics, Springer, 2005.Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Joachim Schöberl
    • 1
  • Jens M. Melenk
    • 2
  • Clemens G. A. Pechstein
    • 3
  • Sabine C. Zaglmayr
    • 1
  1. 1.Radon Institute for Computational and Applied Mathematics (RICAM)Austria
  2. 2.Department of MathematicsThe University of ReadingUK
  3. 3.Institute for Computational MathematicsJohannes Kepler UniversityLinzAustria

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