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Numerische Mathematik

, Volume 122, Issue 2, pp 197–225 | Cite as

Sparsity optimized high order finite element functions for H(div) on simplices

  • Sven Beuchler
  • Veronika Pillwein
  • Sabine Zaglmayr
Article

Abstract

This paper deals with conforming high-order finite element discretizations of the vector-valued function space H(Div) in 2 and 3 dimensions. A new set of hierarchic basis functions on simplices with the following two main properties is introduced. When working with an affine, simplicial triangulation, first, the divergence of the basis functions is L 2-orthogonal, and secondly, the L 2-inner product of the interior basis functions is sparse with respect to the polynomial degree p. The construction relies on a tensor-product based construction with properly weighted Jacobi polynomials as well as an explicit splitting of the higher-order basis functions into solenoidal and non-solenoidal ones. The basis is suited for fast assembling strategies. The general proof of the sparsity result is done by the assistance of computer algebra software tools. Several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter-dependent Div - Div problem. Even on curved elements a tremendous improvement in condition numbers is observed. The precomputed mass and stiffness matrix entries in general form are available online.

Mathematics Subject Classification

65N30 65N22 33C45 

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References

  1. 1.
    Abramowitz, M., Stegun, I. (eds): Handbook of Mathematical Functions. Dover, NY (1965)Google Scholar
  2. 2.
    Ainsworth M., Coyle J.: Hierarchical finite element bases on unstructured tetrahedral meshes. Int. J. Numer. Methods Eng. 58(14), 2103–2130 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. In: Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  4. 4.
    Arnold, D.N., Falk, R.S., Winther, R.: Differential complexes and stability of finite element methods I: the de Rham complex. In: Arnold, D., Bochev, P., Lehoucq, R., Nicolaides, R., Shaskov, M. (eds.) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications, vol. 142, pp. 23–46. Springer, Berlin (2006)Google Scholar
  5. 5.
    Babuška, I., Suri, M.: The p- and h-p versions of the finite element method, an overview. Comput. Methods Appl. Mech. Eng. 80(1–3), 5–26 (1990, spectral and high order methods for partial differential equations (Como, 1989))Google Scholar
  6. 6.
    Bećirović A., Paule P., Pillwein V., Riese A., Schneider C., Schöberl J.: Hypergeometric summation algorithms for high-order finite elements. Computing 78(3), 235–249 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Beuchler S., Pillwein V.: Shape functions for tetrahedral p-fem using integrated Jacobi polynomials. Computing 80, 345–375 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Beuchler, S., Pillwein, V.: Completions to sparse shape functions for triangular and tetrahedral p-fem. In: Langer, U., Discacciati, M., Keyes, D.E., Widlund, O.B., Zulehner, W., (eds.) Proceedings of the 17th International Conference on Domain Decomposition Methods in Science and Engineering XVII held at St. Wolfgang/Strobl, Austria, (July 3–7, 2006). Lecture Notes in Computational Science and Engineering, vol. 60, pp. 435–442, Springer, Heidelberg (2008)Google Scholar
  9. 9.
    Beuchler, S., Pillwein, V., Zaglmayr S.: Sparsity optimized high order finite element functions for H(div) on simplices. Technical Report 2010-04, DK Computational Mathematics, JKU Linz (2010)Google Scholar
  10. 10.
    Beuchler S., Schöberl J.: New shape functions for triangular p-fem using integrated Jacobi polynomials. Numer. Math. 103, 339–366 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bossavit, A.: Computational electromagnetism: variational formulation, complementary, edge elements. In: Electromagnetism. Academic Press, London (1989)Google Scholar
  12. 12.
    Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)zbMATHCrossRefGoogle Scholar
  13. 13.
    Demkowicz L.: Computing with hp Finite Elements. CRC Press/Taylor & Francis, Boca Raton/ London (2006)CrossRefGoogle Scholar
  14. 14.
    Demkowicz L., Buffa A.: H 1, H(curl) and H(div)-conforming projection-based interpolation in three dimensions. quasi-optimal p-interpolation estimates. Comput. Methods Appl. Mech. Eng. 194, 267–296 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Demkowicz, L., Kurtz, J., Pardo, D., Paszyński, M., Rachowicz, W., Zdunek, A.: Computing with hp-adaptive finite elements. In: Frontiers: three dimensional elliptic and Maxwell problems with applications. Applied Mathematics and Nonlinear Science Series, vol. 2. Chapman & Hall/CRC Press, Boca Raton (2008)Google Scholar
  16. 16.
    Demkowicz L., Monk P., Vardapetyan L., Rachowicz W.: De Rham diagram for hp finite element spaces. Comput. Math. Appl. 39(7–8), 29–38 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Dubiner M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6, 345 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations. In: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)Google Scholar
  19. 19.
    Karniadakis G.M., Sherwin S.J.: Spectral/HP Element Methods for CFD. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  20. 20.
    Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975). Math. Res. Center, Univ. Wisconsin, Publ. No. 35, pp. 435–495. Academic Press, New York (1975)Google Scholar
  21. 21.
    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
  22. 22.
    Nédélec J.C.: Mixed finite elements in \({\mathbb{R}^3}\) . Numerische Mathematik 35(35), 315–341 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Nédélec J.C.: A new family of mixed finite elements in \({\mathbb{R}^3}\) . Numerische Mathematik 50(35), 57–81 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Orszag, S.A.: Spectral methods for problems in complex geometries. J. Comp. Phys. 37–80 (1980)Google Scholar
  25. 25.
    Paule, P., Pillwein, V., Schneider, C., Schöberl, J.: Hypergeometric Summation Techniques for High Order Finite Elements. In: PAMM, vol. 6, pp. 689–690. Wiley InterScience, Weinheim (2006). doi: 10.1002/pamm.200610325
  26. 26.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical aspects of finite element methods. Lecture Notes in Mathematics, vol. 606. Berlin (1977)Google Scholar
  27. 27.
    Schöberl, J., Zaglmayr, S.: High order Nédélec elements with local complete sequence properties. COMPEL 24(2) (2005)Google Scholar
  28. 28.
    Schwab C.: p- and hp-finite element methods. Theory and applications in solid and fluid mechanics. Clarendon Press, Oxford (1998)Google Scholar
  29. 29.
    Sherwin S.J.: Hierarchical hp finite elements in hybrid domains. Finite Elem. Anal. Des. 27, 109–119 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Sherwin S.J., Karniadakis G.E.: A new triangular and tetrahedral basis for high-order finite element methods. Int. J. Numer. Methods Eng. 38, 3775–3802 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Solin P., Segeth K., Dolezel I.: Higher-Order Finite Element Methods. CRC Press/Chapman & Hall, Boca Raton/London (2003)Google Scholar
  32. 32.
    Szabo B., Duester A., Rank E.: The p-version of the finite element method. In: Stein, E., Borst, R., Hughes, T.J. (eds) Encyclopedia of Computational Mechanics, Wiley, London (2004)Google Scholar
  33. 33.
    Tricomi F.G., Vorlesungen über F.G.: Orthogonalreihen. Springer, Berlin (1955)zbMATHGoogle Scholar
  34. 34.
    Zaglmayr, S.: High Order Finite Elements for Electromagnetic Field Computation. PhD thesis, Johannes Kepler University, Linz, Austria (2006)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Sven Beuchler
    • 1
  • Veronika Pillwein
    • 2
  • Sabine Zaglmayr
    • 3
  1. 1.Institute for Numerical SimulationUniversity of BonnBonnGermany
  2. 2.Research Institute for Symbolic ComputationLinzAustria
  3. 3.Computer Simulation TechnologyDarmstadtGermany

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