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Foundations of Computational Mathematics

, Volume 16, Issue 3, pp 637–675 | Cite as

Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the \(hp\)-Version

  • R. Hiptmair
  • A. MoiolaEmail author
  • I. Perugia
Article

Abstract

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.

Keywords

Helmholtz equation Approximation by plane waves Trefftz-discontinuous Galerkin method \(hp\)-version A priori convergence analysis Exponential convergence 

Mathematical Subject Classification

65N30 65N15 35J05 

Notes

Acknowledgments

The authors are grateful to Markus Melenk for advice on how to establish the analytic regularity result reported in Theorem 2.3. They also wish to thank Monique Dauge and Euan A. Spence for their help in strengthening some of the results. They also appreciate the valuable suggestions of the reviewers, which led to substantial enhancements compared to the first version of the manuscript: the wavenumber dependence in Proposition 2.1 and Lemma 4.5 is improved, the bounds in Sect. 5 are sharper, and the proof of Theorem 6.5 is simplified. Ilaria Perugia acknowledges support of the Italian Ministry of Education, University and Research (MIUR) through the project PRIN-2012HBLYE4.

References

  1. 1.
    M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol. 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.
    I. Babuška and B. Q. Guo, The h-p version of the finite element method for domains with curved boundaries, SIAM J. Numer. Anal., 25 (1988), 837–861.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    I. Babuška and J. M. Melenk, The partition of unity method, Internat. J. Numer. Methods Engrg., 40 (1997), 727–758.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    T. Betcke, S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and M. Lindner, Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation., Numer. Methods Partial Differ. Equations, 27 (2011), 31–69.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. C. Brenner and L. R. Scott, Mathematical theory of finite element methods, 3rd ed., Texts Appl. Math., Springer-Verlag, New York, 2008.Google Scholar
  6. 6.
    A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation, M2AN, Math. Model. Numer. Anal., 42 (2008), 925–940.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), 255–299.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Dauge, Elliptic boundary value problems on corner domains, vol. 1341 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions.Google Scholar
  9. 9.
    S. Esterhazy and J. Melenk, On stability of discretizations of the Helmholtz equation, in Numerical Analysis of Multiscale Problems, I. Graham, T. Hou, O. Lakkis, and R. Scheichl, eds., vol. 83 of Lecture Notes in Computational Science and Engineering, Springer Verlag, 2011, pp. 285–324.Google Scholar
  10. 10.
    X.-B. Feng and H.-J. Wu, hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 80 (2011), 1997–2024.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems, J. Comput. Phys., 225 (2007), 1961–1984.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Gander, I. Graham, and E. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed? Numerische Mathematik (2015). doi: 10.1007/s00211-015-0700-2.
  13. 13.
    C. J. Gittelson, R. Hiptmair, and I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version, M2AN Math. Model. Numer. Anal., 43 (2009), 297–332.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. Grisvard, Singularities in boundary value problems, vol. 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris; Springer-Verlag, Berlin, 1992.Google Scholar
  15. 15.
    W. Gui and I. Babuska, The h, p and h-p versions of the finite element method in one dimension. II. The error analysis of the h- and h-p versions, Numer. Math., 49 (1986), 613–657.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    U. Hetmaniuk, Stability estimates for a class of Helmholtz problems, Commun. Math. Sci., 5 (2007), 665–678.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    R. Hiptmair, A. Moiola, and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: Analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264–284.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    R. Hiptmair, A. Moiola, and I. Perugia, Stability results for the time-harmonic Maxwell equations with impedance boundary conditions, Math. Models Methods Appl. Sci., 21 (2011), 2263–2287.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    R. Hiptmair, A. Moiola, and I. Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes, Appl. Num. Math., 79 (2014), 79–91.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R. Hiptmair, A. Moiola, I. Perugia, and C. Schwab, Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-DGFEM, Math. Modelling Numer. Analysis, 48 (2014), 727–752.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    P. Houston, C. Schwab, and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133–2163.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    T. Huttunen, P. Monk, and J. P. Kaipio, Computational aspects of the ultra-weak variational formulation, J. Comput. Phys., 182 (2002), 27–46.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    T. Luostari, T. Huttunen, and P. Monk, Improvements for the ultra weak variational formulation, Internat. J. Numer. Methods Engrg., 94 (2013), 598–624.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.zbMATHGoogle Scholar
  25. 25.
    J. M. Melenk, On Generalized Finite Element Methods, PhD thesis, University of Maryland, 1995.Google Scholar
  26. 26.
    J. M. Melenk, hp-finite element methods for singular perturbations, vol. 1796 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.Google Scholar
  27. 27.
    J. M. Melenk, On approximation in meshless methods, in Frontiers of numerical analysis, Universitext, Springer, Berlin, 2005, pp. 65–141.Google Scholar
  28. 28.
    J. M. Melenk, A. Parsania, and S. Sauter, General DG-methods for highly indefinite Helmholtz problems, Journal of Scientific Computing, (2013), 1–46.Google Scholar
  29. 29.
    J. M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation, SIAM J. Numer. Anal., 49 (2011), 1210–1243.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A. Moiola, Trefftz-discontinuous Galerkin methods for time-harmonic wave problems, PhD thesis, Seminar for applied mathematics, ETH Zürich, 2011. doi: 10.3929/ethz-a-006698757.
  31. 31.
    A. Moiola, R. Hiptmair, and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions, Z. Angew. Math. Phys., 62 (2011), 809–837.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    A. Moiola, R. Hiptmair, and I. Perugia, Vekua theory for the Helmholtz operator, Z. Angew. Math. Phys., 62 (2011), 779–807.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    A. Moiola and E. A. Spence, Is the Helmholtz equation really sign-indefinite?, SIAM Rev., 56 (2014), 274–312.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    P. Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, 2003.Google Scholar
  35. 35.
    P. Monk and D. Wang, A least squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 175 (1999), 121–136.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    D. Schötzau, C. Schwab, and T. P. Wihler, hp-dGFEM for Second-Order Elliptic Problems in Polyhedra I: Stability on Geometric Meshes, SIAM J. Numer. Anal., 51 (2013), 1610–1633.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    D. Schötzau, C. Schwab, and T. P. Wihler, hp-DGFEM for Second Order Elliptic Problems in Polyhedra II: Exponential Convergence, SIAM J. Numer. Anal., 51 (2013), 2005–2035.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    C. Schwab, p- and hp-Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford, 1998.zbMATHGoogle Scholar
  39. 39.
    E. A. Spence, Wavenumber-explicit bounds in time-harmonic acoustic scattering, SIAM J. Math. Anal., 46 (2014), 2987–3024.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    I. N. Vekua, New methods for solving elliptic equations, North Holland, 1967.Google Scholar
  41. 41.
    T. P. Wihler, P. Frauenfelder, and C. Schwab, Exponential convergence of the hp-DGFEM for diffusion problems, Comput. Math. Appl., 46 (2003), 183–205.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Seminar of Applied MathematicsETH ZürichZurichSwitzerland
  2. 2.Department of Mathematics and StatisticsUniversity of ReadingReading, BerkshireUK
  3. 3.Faculty of MathematicsUniversity of ViennaViennaAustria
  4. 4.Department of MathematicsUniversity of PaviaPaviaItaly

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