A Survey of Trefftz Methods for the Helmholtz Equation

Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 114)

Abstract

Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.

References

  1. 1.
    C.J. Alves, S.S. Valtchev, Numerical comparison of two meshfree methods for acoustic wave scattering. Eng. Anal. Boundary Elem. 29 (4), 371–382 (2005)MATHCrossRefGoogle Scholar
  2. 2.
    M. Amara, R. Djellouli, C. Farhat, Convergence analysis of a discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of Helmholtz problems. SIAM J. Numer. Anal. 47 (2), 1038–1066 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    M. Amara, H. Calandra, R. Dejllouli, M. Grigoroscuta-Strugaru, A stable discontinuous Galerkin-type method for solving efficiently Helmholtz problems. Comput. Struct. 106–107, 258–272 (2012)CrossRefGoogle Scholar
  4. 4.
    M. Amara, S. Chaudhry, J. Diaz, R. Djellouli, S.L. Fiedler, A local wave tracking strategy for efficiently solving mid- and high-frequency Helmholtz problems. Comput. Methods Appl. Mech. Eng. 276, 473–508 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    P.F. Antonietti, I. Perugia, D. Zaliani, Schwarz domain decomposition preconditioners for plane wave discontinuous Galerkin methods, in Numerical Mathematics and Advanced Applications - ENUMATH 2013, ed. by A. Abdulle, S. Deparis, D. Kressner, F. Nobile, M. Picasso. Lecture Notes in Computational Science and Engineering, vol. 103 (Springer, Berlin, 2015), pp. 557–572Google Scholar
  6. 6.
    D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (5), 1749–1779 (2002)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    R.J. Astley, P. Gamallo, Special short wave elements for flow acoustics. Comput. Methods Appl. Mech. Eng. 194 (2–5), 341–353 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A.K. Aziz, M.R. Dorr, R.B. Kellogg, A new approximation method for the Helmholtz equation in an exterior domain. SIAM J. Numer. Anal. 19 (5), 899–908 (1982)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    A.H. Barnett, T. Betcke, Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comput. Phys. 227 (14), 7003–7026 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    A.H. Barnett, T. Betcke, An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons. SIAM J. Sci. Comput. 32 (3), 1417–1441 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (01), 199–214 (2013)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    S. Bergman, Integral Operators in the Theory of Linear Partial Differential Equations. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 23 (Springer, New York, 1969); Second revised printingGoogle Scholar
  13. 13.
    T. Betcke, J. Phillips, Adaptive plane wave discontinuous Galerkin method for Helmholtz problems, in Proceedings of the 10th International Conference on the Mathematical and Numerical Aspects of Waves, Vancouver, 2011, pp. 261–264Google Scholar
  14. 14.
    T. Betcke, J. Phillips, Approximation by dominant wave directions in plane wave methods. Technical Report, UCL (2012). Available at http://discovery.ucl.ac.uk/1342769/ Google Scholar
  15. 15.
    T. Betcke, L.N. Trefethen, Reviving the method of particular solutions. SIAM Rev. 47 (3), 469–491 (2005)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    T. Betcke, M. Gander, J. Phillips, Block Jacobi relaxation for plane wave discontinuous Galerkin methods, in Domain Decomposition Methods in Science and Engineering XXI, ed. by J. Erhel, M.J. Gander, L. Halpern, G. Pichot, T. Sassi, O. Widlund. Lecture Notes in Computational Science and Engineering, vol. 98 (Springer, Berlin, 2014), pp. 577–585Google Scholar
  17. 17.
    A. Buffa, P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation. M2AN, Math. Model. Numer. Anal. 42 (6), 925–940 (2008)Google Scholar
  18. 18.
    O. Cessenat, Application d’une nouvelle formulation variationnelle aux équations d’ondes harmoniques. Problèmes de Helmholtz 2D et de Maxwell 3D. Ph.D. thesis, Université Paris IX Dauphine, 1996Google Scholar
  19. 19.
    O. Cessenat, B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal. 35 (1), 255–299 (1998)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    S.N. Chandler-Wilde, S. Langdon, Acoustic scattering: high-frequency boundary element methods and unified transform methods, in Unified Transform Method for Boundary Value Problems: Applications and Advances, ed. by A. Fokas, B. Pelloni (SIAM, Philadelphia, 2015), pp. 181–226Google Scholar
  21. 21.
    S.N. Chandler-Wilde, I.G. Graham, S. Langdon, E. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer. 21, 89–305 (2012)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Y.K. Cheung, W.G. Jin, O.C. Zienkiewicz, Solution of Helmholtz equation by Trefftz method. Int. J. Numer. Methods Eng. 32 (1), 63–78 (1991)MATHCrossRefGoogle Scholar
  23. 23.
    C.I.R. Davis, B. Fornberg, A spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDEs. Complex Var. Elliptic Equ. 59, 564–577 (2014)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    E. Deckers, B. Bergen, B. Van Genechten, D. Vandepitte, W. Desmet, An efficient wave based method for 2D acoustic problems containing corner singularities. Comput. Methods Appl. Mech. Eng. 241–244, 286–301 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    E. Deckers et al., The wave based method: an overview of 15 years of research. Wave Motion 51 (4), 550–565 (2014); Innovations in Wave ModellingGoogle Scholar
  26. 26.
    W. Desmet, A wave based prediction technique for coupled vibro-acoustic analysis. Ph.D. thesis, KU Leuven, Belgium, 1998Google Scholar
  27. 27.
    W. Desmet et al., The wave based method, in “Mid-Frequency” CAE Methodologies for Mid-Frequency Analysis in Vibration and Acoustics (KU Leuven, Belgium, 2012), pp. 1–60Google Scholar
  28. 28.
    S.C. Eisenstat, On the rate of convergence of the Bergman-Vekua method for the numerical solution of elliptic boundary value problems. SIAM J. Numer. Anal. 11, 654–680 (1974)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    A. El Kacimi, O. Laghrouche, Improvement of PUFEM for the numerical solution of high-frequency elastic wave scattering on unstructured triangular mesh grids. Int. J. Numer. Methods Eng. 84 (3), 330–350 (2010)MathSciNetMATHGoogle Scholar
  30. 30.
    S. Esterhazy, J. Melenk, On stability of discretizations of the Helmholtz equation, in Numerical Analysis of Multiscale Problems, ed. by I. Graham, T. Hou, O. Lakkis, R. Scheichl. Lecture Notes in Computational Science and Engineering, vol. 83 (Springer, Berlin, 2011), pp. 285–324Google Scholar
  31. 31.
    G. Fairweather, A. Karageorghis, P.A. Martin, The method of fundamental solutions for scattering and radiation problems. Eng. Anal. Boundary Elem. 27 (7), 759–769 (2003)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    C. Farhat, I. Harari, L. Franca, The discontinuous enrichment method. Comput. Methods Appl. Mech. Eng. 190 (48), 6455–6479 (2001)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    C. Farhat, I. Harari, U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192 (11), 1389–1419 (2003)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    C. Farhat, R. Tezaur, J. Toivanen, A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers. Int. J. Numer. Methods Eng. 78 (13), 1513–1531 (2009)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    X.B. Feng, H.J. Wu, h p-Discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 80 (4), 1997–2024 (2011)Google Scholar
  36. 36.
    L. Fox, P. Henrici, C. Moler, Approximations and bounds for eigenvalues of elliptic operators. SIAM J. Numer. Anal. 4, 89–102 (1967)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225, 1961–1984 (2007)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    G. Gabard, Exact integration of polynomial-exponential products with application to wave-based numerical methods. Commun. Numer. Methods Eng. 25 (3), 237–246 (2009)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    G. Gabard, P. Gamallo, T. Huttunen, A comparison of wave-based discontinuous Galerkin, ultra-weak and least-square methods for wave problems. Int. J. Numer. Methods Eng. 85 (3), 380–402 (2011)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    P. Gamallo, R.J. Astley, A comparison of two Trefftz-type methods: the ultra-weak variational formulation and the least squares method for solving shortwave 2D Helmholtz problems. Int. J. Numer. Methods Eng. 71, 406–432 (2007)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    M. Gander, I. Graham, E. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed? Numer. Math. 131 (3), 567–614 (2015). doi:10.1007/s00211-015-0700-2. http://dx.doi.org/10.1007/s00211-015-0700-2 MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    E. Giladi, J.B. Keller, A hybrid numerical asymptotic method for scattering problems. J. Comput. Phys. 174 (1), 226–247 (2001)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    A. Gillman, R. Djellouli, M. Amara, A mixed hybrid formulation based on oscillated finite element polynomials for solving Helmholtz problems. J. Comput. Appl. Math. 204 (2), 515–525 (2007)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    C.J. Gittelson, Plane wave discontinuous Galerkin methods. Master’s thesis, SAM, ETH Zürich, Switzerland, 2008. Available at http://www.sam.math.ethz.ch/~hiptmair/StudentProjects/Gittelson/thesis.pdf Google Scholar
  45. 45.
    C.J. Gittelson, R. Hiptmair, Dispersion analysis of plane wave discontinuous Galerkin methods. Int. J. Numer. Methods Eng. 98 (5), 313–323 (2014)MathSciNetCrossRefGoogle Scholar
  46. 46.
    C.J. Gittelson, R. Hiptmair, I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version. M2AN, Math. Model. Numer. Anal. 43 (2), 297–332 (2009)Google Scholar
  47. 47.
    C.I. Goldstein, The weak element method applied to Helmholtz type equations. Appl. Numer. Math. 2 (3–5), 409–426 (1986)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    M. Grigoroscuta-Strugaru, M. Amara, H. Calandra, R. Djellouli, A modified discontinuous Galerkin method for solving efficiently Helmholtz problems. Commun. Comput. Phys. 11 (2), 335–350 (2012)MathSciNetCrossRefGoogle Scholar
  49. 49.
    I. Harari, P. Barai, P.E. Barbone, Numerical and spectral investigations of Trefftz infinite elements. Int. J. Numer. Methods Eng. 46 (4), 553–577 (1999)MATHCrossRefGoogle Scholar
  50. 50.
    P. Henrici, A survey of I. N. Vekua’s theory of elliptic partial differential equations with analytic coefficients. Z. Angew. Math. Phys. 8, 169–202 (1957)MathSciNetMATHGoogle Scholar
  51. 51.
    B. Heubeck, C. Pflaum, G. Steinle, New finite elements for large-scale simulation of optical waves. SIAM J. Sci. Comput. 31 (2), 1063–1081 (2008/09)Google Scholar
  52. 52.
    R. Hiptmair, I. Perugia, Mixed plane wave DG methods, in Domain Decomposition Methods in Science and Engineering XVIII, ed. by M. Bercovier, M.J. Gander, R. Kornhuber, O. Widlund. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2008), pp. 51–62Google Scholar
  53. 53.
    R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49, 264–284 (2011)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    R. Hiptmair, A. Moiola, I. Perugia, C. Schwab, Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz h p-dGFEM. Math. Model. Numer. Anal. 48, 727–752 (2014)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    R. Hiptmair, A. Moiola, I. Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes. Appl. Numer. Math. 79, 79–91 (2014)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version. Found. Comput. Math. (2015). doi:10.1007/s10208-015-9260-1MATHGoogle Scholar
  57. 57.
    C.J. Howarth, New generation finite element methods for forward seismic modelling. Ph.D. thesis, University of Reading, UK, 2014. Available at http://www.reading.ac.uk/maths-and-stats/research/theses/maths-phdtheses.aspx
  58. 58.
    C. Howarth, P. Childs, A. Moiola, Implementation of an interior point source in the ultra weak variational formulation through source extraction. J. Comput. Appl. Math. 271, 295–306 (2014)MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Q. Hu, L. Yuan, A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations. Int. J. Numer. Anal. Model. 11 (3), 587–607 (2014)MathSciNetGoogle Scholar
  60. 60.
    T. Huttunen, P. Monk, J.P. Kaipio, Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182 (1), 27–46 (2002)MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    T. Huttunen, P. Gamallo, R. Astley, A comparison of two wave element methods for the Helmholtz problem. Commun. Numer. Methods Eng. 25 (1), 35–52 (2009)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    D. Huybrechs, S. Olver, Highly oscillatory quadrature, in Highly Oscillatory Problems. London Mathematical Society Lecture Note Series, vol. 366 (Cambridge University Press, Cambridge, 2009), pp. 25–50Google Scholar
  63. 63.
    F. Ihlenburg, I. Babuška, Solution of Helmholtz problems by knowledge-based FEM. Comput. Assist. Mech. Eng. Sci. 4, 397–416 (1997)MATHGoogle Scholar
  64. 64.
    L.M. Imbert-Gérard, Interpolation properties of generalized plane waves. Numer. Math. (2015). doi:10.1007/s00211-015-0704-yMATHGoogle Scholar
  65. 65.
    L.M. Imbert-Gérard, B. Després, A generalized plane-wave numerical method for smooth nonconstant coefficients. IMA J. Numer. Anal. 34 (3), 1072–1103 (2014)MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    S. Kapita, P. Monk, T. Warburton, Residual based adaptivity and PWDG methods for the Helmholtz equation. arXiv:1405.1957v1 (2014)Google Scholar
  67. 67.
    E. Kita, N. Kamiya, Trefftz method: an overview. Adv. Eng. Softw. 24 (1–3), 3–12 (1995)MATHCrossRefGoogle Scholar
  68. 68.
    L. Kovalevsky, P. Ladevéze, H. Riou, The Fourier version of the variational theory of complex rays for medium-frequency acoustics. Comput. Methods Appl. Mech. Eng. 225/228, 142–153 (2012)Google Scholar
  69. 69.
    F. Kretzschmar, A. Moiola, I. Perugia, S.M. Schnepp, A priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems. arXiv:1501.05253v2 (2015)Google Scholar
  70. 70.
    P. Ladevéze, H. Riou, On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics. Comput. Methods Appl. Mech. Eng. 278, 729–743 (2014)MathSciNetCrossRefGoogle Scholar
  71. 71.
    P. Ladevéze, A. Barbarulo, H. Riou, L. Kovalevsky, The variational theory of complex rays, in “Mid-Frequency” CAE Methodologies for Mid-Frequency Analysis in Vibration and Acoustics (KU Leuven, Belgium, 2012), pp. 155–217Google Scholar
  72. 72.
    O. Laghrouche, P. Bettes, R.J. Astley, Modelling of short wave diffraction problems using approximating systems of plane waves. Int. J. Numer. Methods Eng. 54, 1501–1533 (2002)MATHCrossRefGoogle Scholar
  73. 73.
    O. Laghrouche, P. Bettess, E. Perrey-Debain, J. Trevelyan, Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed. Comput. Methods Appl. Mech. Eng. 194 (2–5), 367–381 (2005)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    F. Li, C.W. Shu, A local-structure-preserving local discontinuous Galerkin method for the Laplace equation. Methods Appl. Anal. 13 (2), 215–233 (2006)MathSciNetMATHGoogle Scholar
  75. 75.
    Z.C. Li, T.T. Lu, H.Y. Hu, A.H.D. Cheng, Trefftz and Collocation Methods (WIT Press, Southampton, 2008)MATHGoogle Scholar
  76. 76.
    T. Luostari, Non-polynomial approximation methods in acoustics and elasticity. Ph.D. thesis, University of Eastern Finland, 2013. Available at http://venda.uef.fi/inverse/Frontpage/Publications/Theses
  77. 77.
    T. Luostari, T. Huttunen, P. Monk, Improvements for the ultra weak variational formulation. Int. J. Numer. Methods Eng. 94 (6), 598–624 (2013)MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    P.A. Martin, Multiple scattering, Encyclopedia of Mathematics and Its Applications, vol. 107 (Cambridge University Press, Cambridge, 2006); Interaction of time-harmonic waves with N obstaclesGoogle Scholar
  79. 79.
    P. Mayer, J. Mandel, The finite ray element method for the Helmholtz equation of scattering: first numerical experiments. Technical Report 111, Center for Computational Mathematics, UC Denver, 1997. Available at http://ccm.ucdenver.edu/reports/
  80. 80.
    J.M. Melenk, On generalized finite element methods. Ph.D. thesis, University of Maryland, 1995Google Scholar
  81. 81.
    J.M. Melenk, I. Babuška, The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139 (1–4), 289–314 (1996)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    J.M. Melenk, S. Sauter, Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49 (3), 1210–1243 (2011)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    J.M. Melenk, A. Parsania, S. Sauter, General DG-methods for highly indefinite Helmholtz problems. J. Sci. Comput. 57 (3), 536–581 (2013)MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    A. Moiola, Approximation properties of plane wave spaces and application to the analysis of the plane wave discontinuous Galerkin method. Report 2009-06, SAM, ETH Zürich, 2009Google Scholar
  85. 85.
    A. Moiola, Trefftz-discontinuous Galerkin methods for time-harmonic wave problems. Ph.D. thesis, Seminar for Applied Mathematics, ETH Zürich, 2011. Available at http://e-collection.library.ethz.ch/view/eth:4515
  86. 86.
    A. Moiola, R. Hiptmair, I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62, 809–837 (2011)MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    A. Moiola, R. Hiptmair, I. Perugia, Vekua theory for the Helmholtz operator. Z. Angew. Math. Phys. 62, 779–807 (2011)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    P. Monk, D. Wang, A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 175 (1/2), 121–136 (1999)MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    P. Monk, J. Schöberl, A. Sinwel, Hybridizing Raviart-Thomas elements for the Helmholtz equation. Electromagnetics 30, 149–176 (2010)CrossRefGoogle Scholar
  90. 90.
    E. Moreno, D. Erni, C. Hafner, R. Vahldieck, Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures. J. Opt. Soc. Am. A 19 (1), 101–111 (2002)CrossRefGoogle Scholar
  91. 91.
    N. Nguyen, J. Peraire, F. Reitich, B. Cockburn, A phase-based hybridizable discontinuous Galerkin method for the numerical solution of the Helmholtz equation. J. Comput. Phys. 290, 318–335 (2015)MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    M. Ochmann, The source simulation technique for acoustic radiation problems. Acta Acustica united with Acustica 81 (6), 512–527 (1995)MATHGoogle Scholar
  93. 93.
    M.J. Peake, J. Trevelyan, G. Coates, Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems. Comput. Methods Appl. Mech. Eng. 259, 93–102 (2013)MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    M.J. Peake, J. Trevelyan, G. Coates, The equal spacing of N points on a sphere with application to partition-of-unity wave diffraction problems. Eng. Anal. Boundary Elem. 40, 114–122 (2014)MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    E. Perrey-Debain, Plane wave decomposition in the unit disc: convergence estimates and computational aspects. J. Comput. Appl. Math. 193 (1), 140–156 (2006)MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    E. Perrey-Debain, O. Laghrouche, P. Bettess, Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362 (1816), 561–577 (2004)MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    I. Perugia, P. Pietra, A. Russo, A plane wave virtual element method for the Helmholtz problem. arXiv:1505.04965v1 (2015)Google Scholar
  98. 98.
    B. Pluymers, B. van Hal, D. Vandepitte, W. Desmet, Trefftz-based methods for time-harmonic acoustics. Arch. Comput. Methods Eng. 14 (4), 343–381 (2007)MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    Q.H. Qin, Trefftz finite element method and its applications. Appl. Mech. Rev. 58 (5), 316–337 (2005)CrossRefGoogle Scholar
  100. 100.
    H. Riou, P. Ladevéze, B. Sourcis, The multiscale VTCR approach applied to acoustics problems. J. Comput. Acoust. 16 (4), 487–505 (2008)MATHCrossRefGoogle Scholar
  101. 101.
    H. Riou, P. Ladevéze, B. Sourcis, B. Faverjon, L. Kovalevsky, An adaptive numerical strategy for the medium-frequency analysis of Helmholtz’s problem. J. Comput. Acoust. 20 (01), 1250001 (2012)Google Scholar
  102. 102.
    H. Riou, P. Ladevéze, L. Kovalevsky, The variational theory of complex rays: an answer to the resolution of mid-frequency 3d engineering problems. J. Sound Vib. 332 (8), 1947–1960 (2013)CrossRefGoogle Scholar
  103. 103.
    I.H. Sloan, R.S. Womersley, Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21 (1–2), 107–125 (2004)MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    Y.S. Smyrlis, Density results with linear combinations of translates of fundamental solutions. J. Approx. Theory 161 (2), 617–633 (2009)MathSciNetMATHCrossRefGoogle Scholar
  105. 105.
    E.A. Spence, Wavenumber-explicit bounds in time-harmonic acoustic scattering. SIAM J. Math. Anal. 46 (4), 2987–3024 (2014)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    E. Spence, “When all else fails, integrate by parts”: an overview of new and old variational formulations for linear elliptic PDEs, in Unified Transform Method for Boundary Value Problems: Applications and Advances, ed. by A. Fokas, B. Pelloni (SIAM, Philadelphia, 2015), pp. 93–159Google Scholar
  107. 107.
    M. Stojek, Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Numer. Methods Eng. 41 (5), 831–849 (1998)MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    T. Strouboulis, I. Babuška, R. Hidajat, The generalized finite element method for Helmholtz equation: theory, computation, and open problems. Comput. Methods Appl. Mech. Eng. 37–40, 4711–4731 (2006)MathSciNetMATHCrossRefGoogle Scholar
  109. 109.
    K.Y. Sze, G.H. Liu, H. Fan, Four- and eight-node hybrid-Trefftz quadrilateral finite element models for Helmholtz problem. Comput. Methods Appl. Mech. Eng. 199, 598–614 (2010)MathSciNetMATHCrossRefGoogle Scholar
  110. 110.
    R. Tezaur, L. Zhang, C. Farhat, A discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid/solid problems. Comput. Methods Appl. Mech. Eng. 197 (19–20), 1680–1698 (2008)MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    R. Tezaur, I. Kalashnikova, C. Farhat, The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber. Comput. Methods Appl. Mech. Eng. 268, 126–140 (2014)MathSciNetMATHCrossRefGoogle Scholar
  112. 112.
    E. Trefftz, Ein Gegenstuck zum Ritzschen Verfahren, in Proceedings of the 2nd International Congress for Applied Mechanics, Zurich, 1926, pp. 131–137Google Scholar
  113. 113.
    I. Tsukerman, A class of difference schemes with flexible local approximation. J. Comput. Phys. 211 (2), 659–699 (2006)MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    I.N. Vekua, New Methods for Solving Elliptic Equations (North Holland, Amsterdam, 1967); Translation from Russian edition (1948)Google Scholar
  115. 115.
    D. Wang, R. Tezaur, J. Toivanen, C. Farhat, Overview of the discontinuous enrichment method, the ultra-weak variational formulation, and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons. Int. J. Numer. Methods Eng. 89 (4), 403–417 (2012)MathSciNetMATHCrossRefGoogle Scholar
  116. 116.
    R.S. Womersley, I.H. Sloan, Interpolation and cubature on the sphere. http://web.maths.unsw.edu.au/~rsw/Sphere
  117. 117.
    S.F. Wu, The Helmholtz Equation Least Squares Method. Modern Acoustics and Signal Processing (Springer, New York, 2015)CrossRefGoogle Scholar
  118. 118.
    L. Yuan, Q. Hu, A solver for Helmholtz system generated by the discretization of wave shape functions. Adv. Appl. Math. Mech. 5 (6), 791–808 (2013)MathSciNetMATHCrossRefGoogle Scholar
  119. 119.
    E. Zheng, F. Ma, D. Zhang, A least-squares non-polynomial finite element method for solving the polygonal-line grating problem. J. Math. Anal. Appl. 397 (2), 550–560 (2013)MathSciNetMATHCrossRefGoogle Scholar
  120. 120.
    E. Zheng, F. Ma, D. Zhang, A least-squares finite element method for solving the polygonal-line arc-scattering problem. Appl. Anal. 93 (6), 1164–1177 (2014)MathSciNetMATHCrossRefGoogle Scholar
  121. 121.
    O. Zienkiewicz, Trefftz type approximation and the generalized finite element method- history and development. Comput. Assist. Mech. Eng. Sci. 4 (3), 305–316 (1997)MATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Seminar for Applied MathematicsETH ZürichZürichSwitzerland
  2. 2.Department of Mathematics and StatisticsUniversity of ReadingReadingUK
  3. 3.Faculty of MathematicsUniversity of ViennaViennaAustria
  4. 4.Department of MathematicsUniversity of PaviaPaviaItaly

Personalised recommendations