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Dispersion Analysis of HDG Methods

  • Jay Gopalakrishnan
  • Manuel Solano
  • Felipe Vargas
Article
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Abstract

This work presents a dispersion analysis of the Hybrid Discontinuous Galerkin (HDG) method. Considering the Helmholtz system, we quantify the discrepancies between the exact and discrete wavenumbers. In particular, we obtain an analytic expansion for the wavenumber error for the lowest order Single Face HDG (SFH) method. The expansion shows that the SFH method exhibits convergence rates of the wavenumber errors comparable to that of the mixed hybrid Raviart–Thomas method. In addition, we observe the same behavior for the higher order cases in numerical experiments.

Keywords

Dispersion analysis Hybridizable discontinuous Galerkin Helmholtz equation 

Notes

Acknowledgements

This paper is the outgrowth of a suggestion from Prof. Bernardo Cockburn to go beyond the standard HDG method while comparing dispersion relations. On this occasion for celebrating Prof. Cockburn’s contributions, the authors would like to place on record their deep appreciation for his tireless efforts to nurture the mathematical community of researchers in discontinuous Galerkin methods over the years. This work was initiated while the student author F. Vargas was visiting Portland State University, thanks to the support from CONICYT, Chile. M. Solano was partially supported by Conicyt-Chile through Fondecyt project No. 1160320 and project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal.

References

  1. 1.
    Ainsworth, M.: Discrete dispersion relation for \(hp\)-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553–575 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ainsworth, M., Monk, P., Muniz, W.: Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second order wave equation. J. Sci. Comput. 27(1–3), 5–40 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Babuška, I.M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34(6), 2392–2423 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chung, E., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. SIAM J. Numer. Anal. 52(2), 915–932 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chung, E., Engquist, B.: Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47, 3820–3848 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77(264), 1887–1916 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cockburn, B., Gopalakrishnan, J., Lazaron, 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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cui, J., Zhang, W.: An analysis of HDG methods for the Helmholtz equation. IMA J. Numer. Anal. 34(1), 279–295 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    De Basabe, J.D., Sen, M.K., Wheeler, M.F.: The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion. Geophys. J. Int. 175(1), 83–93 (2014)CrossRefGoogle Scholar
  12. 12.
    Deraemaeker, A., Babuška, I.M., Bouillard, P.: Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int. J. Numer. Meth. Eng. 46(4), 471–499 (1999)CrossRefMATHGoogle Scholar
  13. 13.
    Giorgiani, G., Fernández-Méndez, S., Huerta, A.: Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems. Int. J. Numer. Meth. Fluids 72(12), 1244–1262 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gittelson, C.J., Hiptmair, R.: Dispersion analysis of plane wave discontinuous Galerkin methods. Int. J. Numer. Meth. Eng. 98(5), 313–323 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gopalakrishnan, J., Lanteri, S., Olivares, N., Perrusel, R.: Stabilization in relation to wavenumber in HDG methods. Adv. Model. Simul. Eng. Sci. 2(1), 13 (2015)CrossRefGoogle Scholar
  16. 16.
    Gopalakrishnan, J., Muga, I., Olivares, N.: Dispersive and dissipative errors in the DPG method with scaled norms for the Helmholtz equation. SIAM J. Sci. Comput. 36(1), A20–A39 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Griesmaier, R., Monk, P.: Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation. J. Sci. Comput. 49(2), 291–310 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hu, F.Q., Hussaini, M., Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151(2), 921–946 (1999)CrossRefMATHGoogle Scholar
  19. 19.
    Kirby, R.C.: Singularity-free evaluation of collapsed-coordinate orthogonal polynomials. ACM Trans. Math. Softw. 37, 5 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Sherwin, S.: Dispersion Analysis of the Continuous and Discontinuous Galerkin Formulations, pp. 426–431. Springer Berlin Heidelberg, Berlin, Heidelberg (2000)MATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsPortland State UniversityPortlandUSA
  2. 2.Departamento de Ingeniería Matemática and Centro de Investigación en Ingeniería Matemática (CI2MA)Universidad de ConcepciónConcepciónChile

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