Journal of Scientific Computing

, Volume 51, Issue 1, pp 183–212 | Cite as

To CG or to HDG: A Comparative Study

  • Robert M. Kirby
  • Spencer J. Sherwin
  • Bernardo Cockburn
Article

Abstract

Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.

We demonstrate that the HDG approach generates a global trace space system for the unknown that although larger in rank than the traditional static condensation system in CG, has significantly smaller bandwidth at moderate polynomial orders. We show that if one ignores set-up costs, above approximately fourth-degree polynomial expansions on triangles and quadrilaterals the HDG method can be made to be as efficient as the CG approach, making it competitive for time-dependent problems even before taking into consideration other properties of DG schemes such as their superconvergence properties and their ability to handle hp-adaptivity.

Keywords

High-order finite elements Spectral/hp elements Discontinuous Galerkin method Hybridization Domain decomposition 

References

  1. 1.
    Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999) CrossRefGoogle Scholar
  2. 2.
    Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985) MathSciNetMATHGoogle Scholar
  3. 3.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin methods for elliptic problems. In: Discontinuous Galerkin Methods: Theory, Computation and Applications, pp. 89–101. Springer, Berlin (2000) CrossRefGoogle Scholar
  4. 4.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Blackburn, H.M., Sherwin, S.J.: Instability modes and transition of pulsatile stenotic flow: pulse-period dependence. J. Fluid Mech. 573, 57 (2007) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Blackburn, H.M., Barkley, D., Sherwin, S.J.: Convective instability and transient growth in flow over a backwards-facing step. J. Fluid Mech. 603, 271–304 (2008) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Boost c++ libraries Google Scholar
  8. 8.
    Bramble, J.H., Xu, J.: A local post-processing technique for improving the accuracy in mixed finite-element approximations. SIAM J. Numer. Anal. 26(6), 1267–1275 (1989) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Brezzi, F., Douglas, J., Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Buffa, A., Hughes, T.J.R., Sangalli, G.: Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. 44, 1420–1440 (2006). (electronic) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Carmo, B.S., Sherwin, S.J., Bearman, P.W., Willden, R.H.J.: Wake transition in the flow around two circular cylinders in staggered arrangements. J. Fluid Mech. 597, 1–29 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Chen, Z.: Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems. Math. Comput. 4, 1–33 (1996) MATHGoogle Scholar
  14. 14.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) MATHGoogle Scholar
  15. 15.
    Cockburn, B.: Discontinuous Galerkin methods. Z. Angew. Math. Mech. 83, 731–754 (2003) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Cockburn, B.: Discontinuous Galerkin methods for computational fluid dynamics. In: Borst, R., Stein, E., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, pp. 91–123. Wiley, New York (2004) Google Scholar
  17. 17.
    Cockburn, B., Gopalakrishnan, J.: A characterization of hybridized mixed methods for second order elliptic problems. SIAM J. Numer. Anal. 42, 283–301 (2004) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Cockburn, B., Gopalakrishnan, J.: Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput. 74, 1653–1677 (2005) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77, 1887–1916 (2008) MATHCrossRefGoogle Scholar
  20. 20.
    Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Cockburn, B., Dong, B., Guzmán, J., Restelli, M., Sacco, R.: A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput. 31, 3827–3846 (2009) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    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, 1319–1365 (2009) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Cockburn, B., Gopalakrishnan, J., Wang, H.: Locally conservative fluxes for the continuous Galerkin method. SIAM J. Numer. Anal. 45, 1742–1776 (2007) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Cockburn, B., Guzmán, J., Soon, S.-C., Stolarski, H.K.: An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems. SIAM J. Numer. Anal. 47, 2686–2707 (2009) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comput. 78, 1–24 (2009) MATHCrossRefGoogle Scholar
  27. 27.
    Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer, Berlin (2000) MATHCrossRefGoogle Scholar
  28. 28.
    Comodi, M.I.: The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing. Math. Comput. 52, 17–29 (1989) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Darekar, R., Sherwin, S.J.: Flow past a square-section cylinder with a wavy stagnation face. J. Fluid Mech. 426, 263 (2001) MATHCrossRefGoogle Scholar
  30. 30.
    Deville, M.O., Mund, E.H., Fischer, P.F.: High Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambridge (2002) MATHCrossRefGoogle Scholar
  31. 31.
    Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6, 345–390 (1991) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Fraejis de Veubeke, B.M.: Displacement and equilibrium models in the finite element method. In: Zienkiewicz, O.C., Holister, G. (eds.) Stress Analysis, pp. 145–197. Wiley, New York (1977) Google Scholar
  33. 33.
    Gastaldi, L., Nochetto, R.H.: Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. RAIRO Modél. Math. Anal. Numér. 23, 103–128 (1989) MathSciNetMATHGoogle Scholar
  34. 34.
    Guermond, J.L., Shen, J.: Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41, 112–134 (2003) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, New York (1987) MATHGoogle Scholar
  36. 36.
    Hughes, J.T.R., Scovazzi, G., Bochev, P.B., Buffa, A.: A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method. Comput. Methods Appl. Mech. Eng. 195, 2761–2787 (2006) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for CFD, 2nd edn. Oxford University Press, London (2005) CrossRefGoogle Scholar
  38. 38.
    Karniadakis, G.E., Israeli, M., Orszag, S.A.: High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97(2), 414–443 (1991) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Theory and Applications of Special Functions. Academic Press, San Diego (1975) Google Scholar
  40. 40.
    Proriol, J.: Sur une famille de polynomes á deux variables orthogonaux dans un triangle. C.R. Acad. Sci Paris 257, 2459–2461 (1957) MathSciNetGoogle Scholar
  41. 41.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of Finite Element Method. Lecture Notes in Math., vol. 606, pp. 292–315. Springer, New York (1977) CrossRefGoogle Scholar
  42. 42.
    Schwab, Ch.: p- and hp- Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics. Oxford University Press, London (1999) Google Scholar
  43. 43.
    Sherwin, S.J.: Hierarchical hp finite elements in hybrid domains. Finite Elem. Anal. Des. 27, 109 (1997) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Sherwin, S.J., Karniadakis, G.E.: A triangular spectral element method; applications to the incompressible Navier-stokes equations. Comput. Methods Appl. Mech. Eng. 123, 189–229 (1995) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Soon, S.-C., Cockburn, B., Stolarski, H.K.: A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Eng. 80, 1058–1092 (2009) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53, 513–538 (1988) MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Stenberg, R.: Postprocessing schemes for some mixed finite elements. RAIRO Modél. Math. Anal. Numér. 25, 151–167 (1991) MathSciNetMATHGoogle Scholar
  48. 48.
    Szabó, B.A., Babuška, I.: Finite Element Analysis. Wiley, New York (1991) MATHGoogle Scholar
  49. 49.
    Vos, P.E.J., Sherwin, S.J., Kirby, R.M.: From h to p efficiently: Implementing finite and spectral/hp element methods to achieve optimal performance for low and high order discretisations. J. Comput. Phys. 29(13), 5161–5181 (2010) MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: Fluid Mechanics, vol. 3, 5th edn. Butterworth-Heinemann, Oxford (2000) MATHGoogle Scholar
  51. 51.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: Solid Mechanics, vol. 2, 5th edn. Butterworth-Heinemann, Oxford (2000) Google Scholar
  52. 52.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: The Basis, vol. 1, 5th edn. Butterworth-Heinemann, Oxford (2000) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Robert M. Kirby
    • 1
  • Spencer J. Sherwin
    • 2
  • Bernardo Cockburn
    • 3
  1. 1.School of ComputingUniv. of UtahSalt Lake CityUSA
  2. 2.Department of AeronauticsImperial College LondonLondonUK
  3. 3.School of MathematicsUniv. of MinnesotaMinneapolisUSA

Personalised recommendations