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Advances in Computational Mathematics

, Volume 45, Issue 2, pp 757–785 | Cite as

On the exponent of exponential convergence of p-version FEM spaces

  • Zhaonan DongEmail author
Open Access
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Abstract

We study the exponent of the exponential rate of convergence in terms of the number of degrees of freedom for various non-standard p-version finite element spaces employing reduced cardinality basis. More specifically, we show that serendipity finite element methods and discontinuous Galerkin finite element methods with total degree \(\mathcal {P}_{p}\) basis have a faster exponential convergence with respect to the number of degrees of freedom than their counterparts employing the tensor product \(\mathcal {Q}_{p}\) basis for quadrilateral/hexahedral elements, for piecewise analytic problems under p-refinement. The above results are proven by using a new p-optimal error bound for the L2-orthogonal projection onto the total degree \(\mathcal {P}_{p}\) basis, and for the H1-projection onto the serendipity finite element space over tensor product elements with dimension d ≥ 2. These new p-optimal error bounds lead to a larger exponent of the exponential rate of convergence with respect to the number of degrees of freedom. Moreover, these results show that part of the basis functions in \(\mathcal {Q}_{p}\) basis plays no roles in achieving the hp-optimal error bound in the Sobolev space. The sharpness of theoretical results is also verified by a series of numerical examples.

Keywords

hp-finite element method Discontinuous Galerkin method Serendipity basis \(\mathcal {P}_{p}\) basis Reduced cardinality basis Exponential convergence 

Mathematics Subject Classification (2010)

65N30 65N15 65N50 

Notes

Acknowledgments

The author wishes to express his gratitude to Emmanuil Georgoulis (University of Leicester and National Technical University of Athens) and Andrea Cangiani (University of Leicester) for their helpful comments.

Funding information

Z. D. was supported by the Leverhulme Trust (grant no. RPG-2015-306).

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Volume 55 of National Bureau of Standards Applied Mathematics Series. For Sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)Google Scholar
  2. 2.
    Ainsworth, M., Pinchedez, K.: hp-approximation theory for BDFM and RT finite elements on quadrilaterals. SIAM J. Numer. Anal. 40(6), 2047–2068 (2003). 2002MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnold, D.N., Awanou, G.: The serendipity family of finite elements. Found. Comput. Math. 11(3), 337–344 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babuška, I., Guo, B.Q.: The h-p version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal. 25(4), 837–861 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Babuška, I., Guo, B.Q.: Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted besov spaces. part I: Approximability of functions in the weighted besov spaces. SIAM J. Numer. Anal. 39(5), 1512–1538 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Batir, N: Inequalities for the gamma function. Arch. Math. 91(6), 554–563 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer-Verlag, Berlin-New York Grundlehren der Mathematischen Wissenschaften, No. 223 (1976)Google Scholar
  8. 8.
    Cangiani, A, Dong, Z, Georgoulis, E.H.: hp-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes. SIAM J. Sci. Comput. 39(4), A1251–A1279 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cangiani, A., Dong, Z., Georgoulis, E.H., Houston, P.: hp-version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes. M2AN Math. Model. Numer. Anal. 50(3), 699–725 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cangiani, A., Georgoulis, E.H., Houston, P.: hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24(10), 2009–2041 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38(157), 67–86 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Davis, P.J.: Interpolation and Approximation. Courier Corporation (1975)Google Scholar
  13. 13.
    Dong, Z.: Discontinuous Galerkin Methods on Polytopic Meshes. PhD thesis, University of Leicester (2016)Google Scholar
  14. 14.
    Dong, Z.: On the exponent of exponential convergence of hp-finite element spaces. arXiv:1704.08046 (2017)
  15. 15.
    Georgoulis, E.H.: Discontinuous Galerkin Methods on Shape-regular and Anisotropic Meshes. D.Phil. Thesis, University of Oxford (2003)Google Scholar
  16. 16.
    Gui, W., Babuška, I.: The h, p and h-p versions of the finite element method in 1 dimension. I–III. Numer. Math. 49 (6), 577–683 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guo, B.Q.: The h-p version of the finite element method for elliptic equations of order 2m. Numer. Math. 53(1-2), 199–224 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guo, B.Q.: The h-p version of the finite element method for solving boundary value problems in polyhedral domains. In: Boundary Value Problems and Integral Equations in Nonsmooth Domains (Luminy, 1993), vol. 167 of Lecture Notes in Pure and Appl. Math., pp. 101–120. Dekker, New York (1995)Google Scholar
  19. 19.
    Guo, B.Q., Babuška, I.: The hp version of the finite element method. Part I: the basic approximation results. Comput. Mech. 1(1), 21–41 (1986)CrossRefzbMATHGoogle Scholar
  20. 20.
    Guo, B.Q., Babuška, I.: The hp version of the finite element method. Part II: general results and applications. Comput. Mech. 1(1), 203–220 (1986)CrossRefzbMATHGoogle Scholar
  21. 21.
    Houston, P., Schwab, C., Süli, E.: Stabilized hp-finite element methods for first-order hyperbolic problems. SIAM J. Numer. Anal. 37(5), 1618–1643 (2000). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kretzschmar, F., Moiola, A., Perugia, I., Schnepp, S. M.: A priori, error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems. IMA J. Numer Anal. 36(4), 1599–1635 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Melenk, J.M., Schwab, C.: An hp–finite element method for convection-diffusion problems in one dimension. IMA J. Numer. Anal. 19(3), 425–453 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schötzau, D., Schwab, C.: Exponential convergence for hp-version and spectral finite element methods for elliptic problems in polyhedra. Math. Models Methods Appl. Sci. 25(9), 1617–1661 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Schötzau, D., Schwab, C., Wihler, T.P.: hp-dGFEM for second-order elliptic problems in polyhedra I Stability on geometric meshes. SIAM J. Numer. Anal. 51(3), 1610–1633 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schötzau, D., Schwab, C., Wihler, T.P.: hp-dGFEM for second order elliptic problems in polyhedra II Exponential convergence. SIAM J. Numer. Anal. 51 (4), 2005–2035 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schötzau, D., Schwab, C., Wihler, T.P.: hp-dGFEM, for second-order mixed elliptic problems in polyhedra. Math. Comput. 85(299), 1051–1083 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schwab, C.: p– and hp–Finite element methods: Theory and applications in solid and fluid mechanics Oxford University Press: Numerical mathematics and scientific computation (1998)Google Scholar
  30. 30.
    Szabó, B., Babuška, I.: Finite Element Analysis. A Wiley-Interscience Publication. Wiley, New York (1991)Google Scholar
  31. 31.
    Wihler, T.P., Frauenfelder, P., Schwab, C.: Exponential convergence of the hp-DGFEM, for diffusion problems. Comput. Math. Appl. 46, 183–205 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LeicesterLeicesterUK

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