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

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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.

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Notes

  1. For \({{\mathbf {x}}}\in \mathbb {R}^2\) and \(A,B\subset \mathbb {R}^2\), we denote by \({{\mathrm{dist}}}({{\mathbf {x}}},A)\) the set–point distance \(\inf _{{{\mathbf {y}}}\in A}\left| {{\mathbf {x}}}-{{\mathbf {y}}}\right| \) and by \({{\mathrm{dist}}}(A,B)\) the set–set distance \(\inf _{{{\mathbf {x}}}\in A,{{\mathbf {y}}}\in B}\left| {{\mathbf {x}}}-{{\mathbf {y}}}\right| \).

  2. We set \(B_r({{\mathbf {x}}}_0):=\{{{\mathbf {x}}}\in \mathbb {R}^2:\ \left| {{\mathbf {x}}}-{{\mathbf {x}}}_0\right| <r\}\), and \(B_r:=B_r({\mathbf {0}})\).

References

  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.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Babuška and J. M. Melenk, The partition of unity method, Internat. J. Numer. Methods Engrg., 40 (1997), 727–758.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. C. Brenner and L. R. Scott, Mathematical theory of finite element methods, 3rd ed., Texts Appl. Math., Springer-Verlag, New York, 2008.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems, J. Comput. Phys., 225 (2007), 1961–1984.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  16. U. Hetmaniuk, Stability estimates for a class of Helmholtz problems, Commun. Math. Sci., 5 (2007), 665–678.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  22. T. Huttunen, P. Monk, and J. P. Kaipio, Computational aspects of the ultra-weak variational formulation, J. Comput. Phys., 182 (2002), 27–46.

    Article  MathSciNet  MATH  Google Scholar 

  23. T. Luostari, T. Huttunen, and P. Monk, Improvements for the ultra weak variational formulation, Internat. J. Numer. Methods Engrg., 94 (2013), 598–624.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.

    MATH  Google Scholar 

  25. J. M. Melenk, On Generalized Finite Element Methods, PhD thesis, University of Maryland, 1995.

  26. J. M. Melenk, hp-finite element methods for singular perturbations, vol. 1796 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.

  27. J. M. Melenk, On approximation in meshless methods, in Frontiers of numerical analysis, Universitext, Springer, Berlin, 2005, pp. 65–141.

    Google Scholar 

  28. J. M. Melenk, A. Parsania, and S. Sauter, General DG-methods for highly indefinite Helmholtz problems, Journal of Scientific Computing, (2013), 1–46.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. A. Moiola, R. Hiptmair, and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions, Z. Angew. Math. Phys., 62 (2011), 809–837.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Moiola, R. Hiptmair, and I. Perugia, Vekua theory for the Helmholtz operator, Z. Angew. Math. Phys., 62 (2011), 779–807.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Moiola and E. A. Spence, Is the Helmholtz equation really sign-indefinite?, SIAM Rev., 56 (2014), 274–312.

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, 2003.

  35. P. Monk and D. Wang, A least squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 175 (1999), 121–136.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  39. E. A. Spence, Wavenumber-explicit bounds in time-harmonic acoustic scattering, SIAM J. Math. Anal., 46 (2014), 2987–3024.

    Article  MathSciNet  MATH  Google Scholar 

  40. I. N. Vekua, New methods for solving elliptic equations, North Holland, 1967.

  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.

    Article  MathSciNet  MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. Seminar of Applied Mathematics, ETH Zürich, 8092, Zurich, Switzerland

    R. Hiptmair

  2. Department of Mathematics and Statistics, University of Reading, Whiteknights, Reading, Berkshire, RG6 6AX, UK

    A. Moiola

  3. Faculty of Mathematics, University of Vienna, 1090, Vienna, Austria

    I. Perugia

  4. Department of Mathematics, University of Pavia, 27100, Pavia, Italy

    I. Perugia

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  1. R. Hiptmair
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  2. A. Moiola
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Correspondence to A. Moiola.

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Communicated by Douglas Arnold.

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Hiptmair, R., Moiola, A. & Perugia, I. Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the \(hp\)-Version. Found Comput Math 16, 637–675 (2016). https://doi.org/10.1007/s10208-015-9260-1

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