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Numerical Algorithms

, Volume 64, Issue 3, pp 519–547 | Cite as

A new triangular spectral element method I: implementation and analysis on a triangle

  • Michael Daniel Samson
  • Huiyuan Li
  • Li-Lian Wang
Original Paper

Abstract

This paper serves as our first effort to develop a new triangular spectral element method (TSEM) on unstructured meshes, using the rectangle–triangle mapping proposed in the conference note (Li et al. 2011). Here, we provide some new insights into the originality and distinctive features of the mapping, and show that this transform only induces a logarithmic singularity, which allows us to devise a fast, stable and accurate numerical algorithm for its removal. Consequently, any triangular element can be treated as efficiently as a quadrilateral element, which affords a great flexibility in handling complex computational domains. Benefited from the fact that the image of the mapping includes the polynomial space as a subset, we are able to obtain optimal L 2- and H 1-estimates of approximation by the proposed basis functions on triangle. The implementation details and some numerical examples are provided to validate the efficiency and accuracy of the proposed method. All these will pave the way for developing an unstructured TSEM based on, e.g., the hybridizable discontinuous Galerkin formulation.

Keywords

Rectangle–triangle mapping Consistency condition Triangular spectral elements Spectral accuracy 

Mathematics Subject Classifications (2010)

65N35 65N22 65F05 35J05 

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References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Acadmic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Boyd, J.P., Yu, F.: Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions. J. Comput. Phys. 230(4), 1408–1438 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Scientific Computation. Springer-Verlag, Berlin (2006)Google Scholar
  4. 4.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Scientific Computation. Springer, Berlin (2007)Google Scholar
  5. 5.
    Chen, L., Shen, J., Xu, C.: A triangular spectral method for the Stokes equations. Numer. Math.: Theory Methods Appl. 4, 158–179 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, Q., Babuška, I.M.: Approximate optimal points for polynomial interpolation of real functions 529 in an interval and in a triangle. Comput. Methods Appl. Math. Eng. 128(2), 405–417 (1995)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chernov, A.: Optimal convergence estimates for the trace of the polynomial L 2-projection operator on a simplex. Math. Comput. 81(278), 765–787 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, The Netherlands (1978)zbMATHGoogle Scholar
  9. 9.
    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Deville, M.O., Fischer, P.F., Mund, E.H.: High-Order Methods for Incompressible Fluid Flow. Cambridge Monographs on Applied and Computational Mathematics, vol. 9. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6(4), 345–390 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Duffy, M.G.: Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J. Numer. Anal. 19(6), 1260–1262 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gautschi, W.: Gauss quadrature routines for two classes of logarithmic weight functions. Numer. Algorithms 55(2–3), 265–277 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gordon, W.J., Hall, C.A.: Construction of curvilinear co-ordinate systems and applications to mesh generation. Int. J. Numer. Methods Eng. 7, 461–477 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 7th Edn. Academic Press, New York (2007)zbMATHGoogle Scholar
  16. 16.
    Guo, B.Y., Shen, J., Wang, L.: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput 27(1–3), 305–322 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guo, B.Y., Wang, L.: Error analysis of spectral method on a triangle. Adv. Comput. Math. 26(4), 473–496 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Heinrichs, W.: Spectral collocation schemes on the unit disc. J. Comput. Phys. 199, 55–86 (2004)MathSciNetGoogle Scholar
  19. 19.
    Helenbrook, B.T.: On the existence of explicit hp-finite element methods using Gauss–Lobatto integration on the triangle. SIAM J. Numer. Anal. 47(2), 1304–1318 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hesthaven, J.S.: From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35(2), 655–676 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hylleraas, E.A.: Linearization of products of Jacobi polynomials. Math. Scand. 10, 189–200 (1962)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics. Numerical Mathematics and Scientific Computation, 2nd Edn. Oxford University Press, New York (2005)Google Scholar
  23. 23.
    Kirby, R.M., Sherwin, S.J., Cockburn, B.: To CG or to HDG: a comparative study. J. Sci. Comput. 51(1), 183–212 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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) pp. 435–495. Math. Res. Center, Univ. Wisconsin, Publ. No, p. 35. Academic Press, New York (1975)Google Scholar
  25. 25.
    Li, H., Shen, J.: Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle. Math. Comput. 79(271), 1621–1646 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Li, H., Wang, L.: A spectral method on tetrahedra using rational basis functions. Int. J. Numer. Anal. Model. 7(2), 330–355 (2010)MathSciNetGoogle Scholar
  27. 27.
    Li, Y., Wang, L., Li, H., Ma, H.: A New Spectral Method on Triangles. In: Spectral and High Order Methods for Partial Differential Equations: Selected papers from the ICOSAHOM ’09 conference, June 22–26, Trondheim, Norway. Lecture Notes in Computational Sciences and Engineering, Vol. 76, pp. 237–246. Springer, New York (2011)CrossRefGoogle Scholar
  28. 28.
    Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable Discontinuous Galerkin Methods. In: Spectral and High Order Methods for Partial Differential Equations: Selected Papers from the ICOSAHOM ’09 Conference, June 22–26, Trondheim, Norway. Lecture Notes in Computational Sciences and Engineering, Vol. 76, pp. 63–84. Springer, New York (2011)CrossRefGoogle Scholar
  29. 29.
    Pasquetti, R., Rapetti, F.: Spectral element methods on unstructured meshes: comparisons and recent advances. J. Sci. Comput. 27(1–3), 377–387 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pasquetti, R., Rapetti, F.: Spectral element methods on unstructured meshes: which interpolation points? Numer. Algorithms 55(2–3), 349–366 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Patera, A.T.: A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54(3), 468–488 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Schwab, C.: p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation. Oxford Science, Oxford, UK (1998) 587Google Scholar
  33. 33.
    Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics, Vol. 41. Springer-Verlag, Berlin Heidelberg (2011)CrossRefGoogle Scholar
  34. 34.
    Shen, J., Wang, L., Li, H.: A triangular spectral element method using fully tensorial rational basis functions. SIAM J. Numer. Anal. 47(3), 1619–1650 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Szegö, G.: Orthogonal Polynomials, 4th Edn, Vol. 23. AMS Coll. Publ., Providence, RI (1975)Google Scholar
  36. 36.
    Taylor, M.A., Wingate, B.A., Vincent, R.E.: An algorithm for computing Fekete points in the triangle. SIAM J. Numer. Anal. 38(5), 1707–1720 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Weber, H.: Lehrbuch der Algebra. Erster Band, Braunschweig (1912)zbMATHGoogle Scholar
  38. 38.
    Xie, Z., Wang, L., Zhao, X.: On exponential convergence of Gegenbauer interpolation and spectral differentiation. Math. Comput., electronically published on 21 August 2012Google Scholar
  39. 39.
    Xu, Y.: Common Zeros of Polynomials in Several Variables and Higher Dimensional Quadrature. Chapman & Hall/CRC, London, UK (1994)zbMATHGoogle Scholar
  40. 40.
    Xu, Y.: On Gauss–Lobatto integration on the triangle. SIAM J. Numer. Anal. 49(2), 541–548 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Michael Daniel Samson
    • 1
  • Huiyuan Li
    • 2
  • Li-Lian Wang
    • 1
  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  2. 2.Institute of SoftwareChinese Academy of SciencesBeijingChina

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