Journal of Scientific Computing

, Volume 50, Issue 2, pp 446–461 | Cite as

Well-conditioned Orthonormal Hierarchical \(\mathcal{L}_{2}\) Bases on ℝ n Simplicial Elements



We construct well-conditioned orthonormal hierarchical bases for simplicial \(\mathcal{L}_{2}\) finite elements. The construction is made possible via classical orthogonal polynomials of several variables. The basis functions are orthonormal over the reference simplicial elements in two and three dimensions. The mass matrices M are identity while the conditioning of the stiffness matrices S grows as \(\mathcal{O}(p^{3})\) with respect to the order p. The diagonally normalized stiffness matrices are well conditioned. The diagonally normalized composite matrices ζM+S are also well conditioned for a wide range of ζ. For the mass, stiffness and composite matrices, the bases in this study have much better conditioning than existing high-order hierarchical bases.


Hierarchical bases Simplicial \(\mathcal{L}_{2}\)-conforming elements Matrix conditioning 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adjerid, S., Aiffa, M., Flaherty, J.E.: Hierarchical finite element bases for triangular and tetrahedral elements. Comput. Methods Appl. Mech. Eng. 190, 2925–2941 (2001) CrossRefMATHGoogle Scholar
  2. 2.
    Ainsworth, M., Coyle, J.: Hierarchic hp-edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Eng. 190, 6709–6733 (2001) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ainsworth, M., Coyle, J.: Conditioning of hierarchic p-version Nédélec elements on meshes of curvilinear quadrilaterals and hexahedra. SIAM J. Numer. Anal. 41, 731–750 (2003) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Ainsworth, M., Coyle, J.: Hierarchic finite element bases on unstructured tetrahedral meshes. Int. J. Numer. Methods Eng. 58, 2103–2130 (2003) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Ayuso de Dios, B., Zikatanov, L.: Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput. 40, 4–36 (2009) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Brenner, S.C., Gudi, T., Sung, L.-Y.: A posteriori error control for a weakly over-penalized symmetric interior penalty method. J. Sci. Comput. 40, 37–50 (2009) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Carnevali, P., Morris, R.B., Tsuji, Y., Taylor, G.: New basis functions and computational procedures for p-version finite element analysis. Int. J. Numer. Methods Eng. 36, 3759–3779 (1993) CrossRefMATHGoogle Scholar
  8. 8.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4. North-Holland, New York (1978) CrossRefMATHGoogle Scholar
  9. 9.
    Cockburn, B., Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194, 588–610 (2004) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. J. Comput. Phys. 141, 199–224 (1998) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6, 345–390 (1991) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 81. Cambridge University Press, Cambridge (2001) CrossRefMATHGoogle Scholar
  14. 14.
    Hu, N., Guo, X.-Z., Katz, I.N.: Bounds for eigenvalues and condition numbers in the p-version of the finite element method. Math. Comput. 67, 1423–1450 (1998) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kellogg, O.D.: Foundations of Potential Theory. Die Grundlehren der Mathematischen Wissenschaften, vol. 31. Springer, New York (1967). Reprint from the first edition of 1929 MATHGoogle Scholar
  16. 16.
    Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Askey, R.A. (ed.) Theory and Application of Special Functions, Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975, pp. 435–495. Academic Press, New York (1975) Google Scholar
  17. 17.
    Lasaint, P., Raviart, P.-A.: On a finite element method for solving the neutron transport equation. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–145. Academic Press, New York (1974) Google Scholar
  18. 18.
    Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Die Grundlehren der mathematischen Wissenschaften, vol. 52. Springer, New York (1966). Third enlarged edition MATHGoogle Scholar
  19. 19.
    Reed, W., Hill, T.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973) Google Scholar
  20. 20.
    Remacle, J.-F., Flaherty, J.E., Shephard, M.S.: An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev. 45, 53–72 (2003) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Schöberl, J., Zaglmayr, S.: High order Nédélec elements with local complete sequence properties. Compel 24, 374–384 (2005) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Webb, J.P.: Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements. IEEE Trans. Antennas Propag. 47, 1244–1253 (1999) CrossRefMATHGoogle Scholar
  23. 23.
    Xin, J., Cai, W.: A well-conditioned hierarchical basis for triangular \(\mathcal{H}(\mathbf{curl})\)-conforming elements. Commun. Comput. Phys. 9, 780–806 (2011) MathSciNetGoogle Scholar
  24. 24.
    Xin, J., Pinchedez, K., Flaherty, J.E.: Implementation of hierarchical bases in FEMLAB for simplicial elements. ACM Trans. Math. Softw. 31, 187–200 (2005) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotteUSA

Personalised recommendations