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Adaptive refinement for hp–Version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem

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Abstract

In this article, we develop an hp-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h–refinement) and local basis enrichment (p–refinement), but also incorporates local directional adaptivity, whereby the elementwise plane wave basis is aligned with the dominant scattering direction. Numerical experiments based on employing an empirical a posteriori error indicator clearly highlight the efficiency of the proposed approach for various examples.

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References

  1. Agrawal, A., Hoppe, R.H.W.: Optimization of plane wave directions in plane wave discontinuous Galerkin methods for the Helmholtz equation. Port Math. 74, 69–89 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amara, M., Chaudhry, S., Diaz, J., Djellouli, R., Fiedler, S.L.: A local wave tracking strategy for efficiently solving mid- and high-frequency Helmholtz problems. Comput. Methods Appl. Mech. Engrg. 276, 473–508 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amara, M., Djellouli, R., Farhat, C.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, F., Ihlenburg, F., Strouboulis, T., Gangaraj, S.K.: A posteriori error estimation for finite element solutions of Helmholtz’ equation I. The quality of local indicators and estimators. Internat. J. Numer Methods Engrg. 40(18), 3443–3462 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Babuška, F., Ihlenburg, F., Strouboulis, T., Gangaraj, S.K.: A posteriori error estimation for finite element solutions of Helmholtz’ equation II. Estimation of the pollution error. Internat. J. Numer Methods Engrg. 40(21), 3883–3900 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Betcke, T., Phillips, J.: Adaptive plane wave discontinuous Galerkin methods for Helmholtz problems. In: Proceedings of the 10th International Conference on the Mathematical and Numerical Aspects of Waves, pp. 261–264 (2011)

  7. Betcke, T., Phillips, J.: Approximation by dominant wave directions in plane wave methods. Technical report, UCL, 2012. Available at http://discovery.ucl.ac.uk/1342769/

  8. Braess, D.: Finite elements. Theory, fast solvers, and applications in solid mechanics, 2nd edn. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  9. Cessenat, O., Després, B.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Congreve, S., Gedicke, J., Perugia, I.: Numerical investigation of the conditioning for plane wave discontinuous Galerkin methods. In: Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2017, the 12th European Conference on Numerical Mathematics and Advanced Applications. Springer, Voss (2018)

  11. Dörfler, W., Sauter, S.: A posteriori error estimation for highly indefinite Helmholtz problems. Comp. Meth. Appl Math. 13(3), 333–347 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Formaggia, L., Perotto, S.: New anisotropic a priori error estimates. Numer. Math. 89(4), 641–667 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Formaggia, L., Perotto, S.: Anisotropic error estimates for elliptic problems. Numer. Math. 94(1), 67–92 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Georgoulis, E., Hall, E., Houston, P.: Discontinuous Galerkin methods for advection–diffusion–reaction problems on anisotropically refined meshes. SIAM J. Sci Comput. 30(1), 246–271 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gittelson, C.J.: Plane wave discontinuous Galerkin methods. Master’s thesis, ETH Zurich, (2008) http://www.sam.math.ethz.ch/hiptmair/StudentProjects/Gittelson/thesis.pdf

  16. Gittelson, C.J., Hiptmair, R., Perugia, I.: Plane wave discontinuous Galerkin methods: analysis of the h-version. ESAIM Math. Model. Numer. Anal. 43 (2), 297–331 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hall, E.J.C.: Anisotropic Adaptive Refinement For Discontinuous Galerkin Methods. PhD thesis, Department of Mathematics University of Leicester (2007)

  18. Hiptmair, R., Moiola, A., Perugia, I.: Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes. Appl. Numer. Math. 79, 79–91 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hiptmair, R., Moiola, A., Perugia, I.: A survey of Trefftz methods for the Helmholtz equation. In: Barrenechea, G.R., Brezzi, F., Cangiani, A., Georgoulis, E.H. (eds.) Building bridges: connections and challenges in modern approaches to numerical partial differential equations, pp 237–279. Springer, Cham (2016)

  20. Houston, P.: AptoFEM finite element analysis software (2017) http://www.aptofem.com [Online]

  21. Houston, P., Süli, E.: A note on the design of hp-adaptive finite element methods for elliptic partial differential equations. Comput. Methods Appl. Mech Engrg. 194 (2–5), 229–243 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huttunen, T., Monk, P., Kaipio, J.P.: Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182(1), 27–46 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kapita, S., Monk, P., Warburton, T.: Residual-based adaptivity and PWDG methods for the Helmholtz equation. SIAM J. Sci. Comput., 37(3) (2015)

  24. Luostari, T., Huttunen, T., Monk, P.: Improvements for the ultra weak variational formulation. Internat. J. Numer Methods Engrg. 94(6), 598–624 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Melenk, J.M., Wohlmuth, B.I.: On residual-based a posteriori error estimation in hp-FEM. Adv. Comp. Math. 15(1–4), 311–331 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mitchell, W.F., McClain, M.A.: A comparison of hp-adaptive strategies for elliptic partial differential equations. Technical Report NISTIR 7824 National Institute of Standards and Technology (2011)

  27. Mitchell, W.F., McClain, M.A.: A comparison of hp-adaptive strategies for elliptic partial differential equations. ACM Trans. Math Softw. 41(1), 2,1–2,39 (2014)

    Article  MathSciNet  Google Scholar 

  28. Moiola, A., Hiptmair, R., Perugia, I.: Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math Phys. 62(5), 809–837 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sauter, S., Zech, J.: A posteriori error estimation of hp-dG, finite element methods for highly indefinite Helmholtz problems. SIAM J. Numer. Anal. 53, 2414–2440 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21(1), 107–125 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Womersley, R.S.: Extremal (maximum determinant) points on the sphere S 2. http://web.maths.unsw.edu.au/rsw/Sphere/Extremal/New/index.html [Online] (2007)

  32. Zech, J.: A posteriori error estimation of hp-DG finite element methods for highly indefinite Helmholtz problems. Master’s thesis, Universität Zürich. http://www.math.uzh.ch/compmath/index.php?id=dipl (2014)

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Funding

S. Congreve and I. Perugia have been funded by the Austrian Science Fund (FWF) through the project P29197-N32. I. Perugia has also been funded by the FWF through the project F 65.

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Correspondence to Scott Congreve.

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Communicated by: Jan Hesthaven

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Congreve, S., Houston, P. & Perugia, I. Adaptive refinement for hp–Version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem. Adv Comput Math 45, 361–393 (2019). https://doi.org/10.1007/s10444-018-9621-9

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  • DOI: https://doi.org/10.1007/s10444-018-9621-9

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