Free Access
Issue
ESAIM: M2AN
Volume 47, Number 1, January-February 2013
Page(s) 183 - 211
DOI https://doi.org/10.1051/m2an/2012025
Published online 31 August 2012
  1. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. [CrossRef] [MathSciNet] [Google Scholar]
  2. A.H. Barnett and T. Betcke, An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons. SIAM J. Sci. Comput. 32 (2010) 1417–1441. [CrossRef] [Google Scholar]
  3. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3rd edition. Springer (2008). [Google Scholar]
  4. A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation. ESAIM : M2AN 42 (2008) 925–940. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  5. 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 (1996). [Google Scholar]
  6. O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35 (1998) 255–299. [CrossRef] [MathSciNet] [Google Scholar]
  7. P. Cummings and X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Math. Mod. Methods Appl. Sci. 16 (2006) 139–160. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. El Kacimi and O. Laghrouche, Numerical modeling of elastic wave scattering in frequency domain by partition of unity finite element method. Int. J. Numer. Methods Eng. 77 (2009) 1646–1669. [CrossRef] [Google Scholar]
  9. C. Farhat, I. Harari and L.P. Franca, A discontinuous enrichment method. Comput. Methods Appl. Mech. Eng. 190 (2001) 6455–6479. [CrossRef] [Google Scholar]
  10. C. Farhat, I. Harari and 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 (2003) 1389–1429. [CrossRef] [Google Scholar]
  11. G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225 (2007) 1961–1984. [CrossRef] [MathSciNet] [Google Scholar]
  12. R. Hardin, N. Sloane and W. Smith, Spherical coverings. Available on http://www.research.att.com/˜njas/coverings/index.html (1994). [Google Scholar]
  13. 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. [CrossRef] [MathSciNet] [Google Scholar]
  14. R. Hiptmair, A. Moiola and I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. In press. [Google Scholar]
  15. T. Huttunen, P. Monk and J.P. Kaipio, Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182 (2002) 27–46. [CrossRef] [MathSciNet] [Google Scholar]
  16. T. Huttunen, P. Monk, F. Collino and J.P. Kaipio, The ultra weak variational formulation for elastic wave problems. SIAM J. Sci. Comput. 25 (2004) 1717–1742. [CrossRef] [MathSciNet] [Google Scholar]
  17. T. Huttunen, P. Monk and J.P. Kaipio, The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation. Int. J. Numer. Methods Eng. 61 (2004) 1072–1092. [CrossRef] [Google Scholar]
  18. T. Huttunen, M. Malinen and P. Monk, Solving Maxwell’s equations using the ultra weak variational formulation. J. Comput. Phys. 223 (2007) 731–758. [CrossRef] [MathSciNet] [Google Scholar]
  19. T. Huttunen, J.P. Kaipio and P. Monk,An ultra-weak method for acoustic fluid-solid interaction. J. Comput. Appl. Math. 213 (2008) 1667–1685. [CrossRef] [Google Scholar]
  20. V.D. Kupradze, Potential methods in the theory of elasticity. Israel Program for Scientific Translations (1965). [Google Scholar]
  21. T. Luostari, T. Huttunen and P. Monk, The ultra weak variational formulation for 3D elastic wave problems, in Proc. 20th International Congress on Acoustics, ICA (2010).Available in 2010 on http://www.acoustics.asn.au. [Google Scholar]
  22. P. Massimi, R. Tezaur and C. Farhat, A discontinuous enrichment method for three-dimensional multiscale harmonic wave propagation problems in multi-fluid and fluid-solid media. Int. J. Numer. Methods Eng. 76 (2008) 400–425. [CrossRef] [Google Scholar]
  23. M.M. Melenk and I. Babuška, The partition of unity finite element method : basic theory and applications. Comput. Methods Appl. Mech. Eng. 139 (1996) 289–314. [CrossRef] [MathSciNet] [Google Scholar]
  24. A. Moiola, Trefftz-Discontinuous Galerkin Methods for Time-Harmonic Wave Problems. Ph.D. thesis, ETH Zürich (2011). [Google Scholar]
  25. A. Moiola, Plane wave approximation in linear elasticity. To appear in Appl. Anal. [Google Scholar]
  26. A. Moiola, R. Hiptmair and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 65 (2011) 809–837. [CrossRef] [MathSciNet] [Google Scholar]
  27. P. Monk and D.-Q. Wang, A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 175 (1999) 121–136. [CrossRef] [MathSciNet] [Google Scholar]
  28. Y.-H. Pao, Betti’s identity and transition matrix for elastic waves. J. Acoust. Soc. Am. 64 (1978) 302–310. [CrossRef] [Google Scholar]
  29. E. Perrey-Debain, Plane wave decomposition in the unit disc : convergence estimates and computational aspects. J. Comput. Appl. Math. 193 (2006) 140–156. [CrossRef] [Google Scholar]
  30. I. Sloan and R. Womersley, Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21 (2004) 107–125. [CrossRef] [MathSciNet] [Google Scholar]
  31. D. Wang, J. Toivanen, R. Tezaur and 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 (2012) 403–417. [CrossRef] [Google Scholar]
  32. R. Womersley and I. Sloan, Interpolation and cubature on the sphere. Available on http://web.maths.unsw.edu.au/˜rsw/Sphere. [Google Scholar]

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