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Numerical Investigation of the Conditioning for Plane Wave Discontinuous Galerkin Methods

  • Scott Congreve
  • Joscha Gedicke
  • Ilaria Perugia
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

We present a numerical study to investigate the conditioning of the plane wave discontinuous Galerkin discretization of the Helmholtz problem. We provide empirical evidence that the spectral condition number of the plane wave basis on a single element depends algebraically on the mesh size and the wave number, and exponentially on the number of plane wave directions; we also test its dependence on the element shape. We show that the conditioning of the global system can be improved by orthogonalization of the local basis functions with the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES iterations for solving the discrete problem iteratively.

Notes

Acknowledgements

The authors have been funded by the Austrian Science Fund (FWF) through the project P 29197-N32. The third author has also been funded by the FWF through the project F 65.

References

  1. 1.
    F. Bassi, L. Botti, A. Colombo, D.A. Di Pietro, P. Tesini, On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231(1), 45–65 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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
  3. 3.
    O. Cessenat, B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35(1), 255–299 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    O. Cessenat, B. Després, Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. J. Comput. Acoust. 11(2), 227–238 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    S.C. Eisenstat, H.C. Elman, M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2), 345–357 (1983)MathSciNetCrossRefGoogle Scholar
  6. 6.
    C. Gittelson, Plane wave discontinuous Galerkin methods, Master’s thesis, SAM, ETH Zurich, Switzerland, 2008Google Scholar
  7. 7.
    C.J. Gittelson, R. Hiptmair, I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version. M2AN Math. Model. Numer. Anal. 43(2), 297–331 (2009)Google Scholar
  8. 8.
    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(1), 264–284 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version. Found. Comput. Math. 16(3), 637–675 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    T. Huttunen, P. Monk, J.P. Kaipio, Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182(1), 27–46 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    T. Luostari, T. Huttunen, P. Monk, Improvements for the ultra weak variational formulation. Int. J. Numer. Methods Eng. 94(6), 598–624 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    L. Mascotto, Ill-conditioning in the virtual element method: stabilizations and bases. Numer. Methods Partial Differ. Equ. 34(4), 1258–1281 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Melenk, On generalized finite element methods, Ph.D. thesis, University of Maryland, 1995Google Scholar
  14. 14.
    M.A. Schweitzer, Stable enrichment and local preconditioning in the particle-partition of unity method. Numer. Math. 118(1), 137–170 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    G.W. Stewart, Matrix Algorithms. Vol. I: Basic Decompositions (Society for Industrial and Applied Mathematics, Philadelphia, 1998)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Scott Congreve
    • 1
  • Joscha Gedicke
    • 1
  • Ilaria Perugia
    • 1
  1. 1.University of ViennaFaculty of MathematicsViennaAustria

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