Skip to main content
SpringerLink
Log in

Discontinuous Galerkin Methods for Acoustic Wave Propagation in Polygons

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We analyze space semi-discretizations of linear, second-order wave equations by discontinuous Galerkin methods in polygonal domains where solutions exhibit singular behavior near corners. To resolve these singularities, we consider two families of locally refined meshes: graded meshes and bisection refinement meshes. We prove that for appropriately chosen refinement parameters, optimal asymptotic rates of convergence with respect to the total number of degrees of freedom are obtained, both in the energy norm errors and the \(\mathcal {L}^2\)-norm errors. The theoretical convergence orders are confirmed in a series of numerical experiments which also indicate that analogous results hold for incompatible data which is not covered by the currently available regularity theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Adler, J., Nistor, V.: Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2D. Math. Comput. 84(295), 2191–2220 (2015)

    Article  MathSciNet  Google Scholar 

  2. Arnold, D., Brezzi, F., Cockburn, B., Marini, D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)

    Article  MathSciNet  Google Scholar 

  3. Babuška, I., Guo, B.Q.: The \(h\)-\(p\) version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal. 25(4), 837–861 (1988)

    Article  MathSciNet  Google Scholar 

  4. Babuška, I., Guo, B.Q.: Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order. SIAM J. Math. Anal. 19(1), 172–203 (1988)

    Article  MathSciNet  Google Scholar 

  5. Babuška, I., Kellogg, R.B., Pitkäranta, J.: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33(4), 447–471 (1979)

    Article  MathSciNet  Google Scholar 

  6. Băcuţă, C., Nistor, V., Zikatanov, L.T.: Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. Numer. Math. 100(2), 165–184 (2005)

    Article  MathSciNet  Google Scholar 

  7. Băcuţă, C., Li, H., Nistor, V.: Differential operators on domains with conical points: precise uniform regularity estimates. Rev. Roum. Math. Pures Appl. 62(3), 383–411 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Cohen, G., Joly, P., Tordjman, N.: Construction and analysis of higher order finite elements with mass lumping for the wave equation. In: Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993), SIAM, Philadelphia, PA, pp. 152–160 (1993)

  9. Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology, vol. 5, Springer, Berlin. Evolution problems. I, With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan Craig (1992)

  10. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, Berlin (2004)

    Book  Google Scholar 

  11. Gaspoz, F.D., Morin, P.: Convergence rates for adaptive finite elements. IMA J. Numer. Anal. 29(4), 917–936 (2009)

    Article  MathSciNet  Google Scholar 

  12. Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)

    MATH  Google Scholar 

  13. Grote, M., Schneebeli, A., Schötzau, D.: Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2431 (2006)

    Article  MathSciNet  Google Scholar 

  14. Grote, M., Schötzau, D.: Optimal error estimates for the fully discrete interior penalty DG method for the wave equation. J. Sci. Comput. 40, 257–272 (2009)

    Article  MathSciNet  Google Scholar 

  15. Guo, B.Q., Babuška, I.: The \(hp\)-version of the finite element method. Part I: the basic approximation results. Comput. Mech. 1, 21–41 (1986)

    Article  Google Scholar 

  16. Hochbruck, M., Sturm, A.: Error analysis of a second-order locally implicit method for linear Maxwell’s equations. SIAM J. Numer. Anal. 54(5), 3167–3191 (2016)

    Article  MathSciNet  Google Scholar 

  17. Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: Open Source Scientific Tools for Python (2001). https://www.scipy.org. Accessed 05 Apr 2018

  18. Köcher, U., Bause, M.: Variational space–time methods for the wave equation. J. Sci. Comput. 61(2), 424–453 (2014)

    Article  MathSciNet  Google Scholar 

  19. Kokotov, A.Y., Plamenevskiĭ, B.A.: On the asymptotic behavior of solutions of the Neumann problem for hyperbolic systems in domains with conical points. Algebra i Analiz 16(3), 56–98 (2004)

    MathSciNet  Google Scholar 

  20. Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč 16, 209–292 (1967)

    MathSciNet  Google Scholar 

  21. Kozlov, V.A., Mazya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Mathematical Surveys and Monographs, vol. 85. American Mathematical Society, Providence (2001)

    Google Scholar 

  22. Li, H., Nistor, V.: LNG\(_{-}\)FEM: graded meshes on domains of polygonal structures. In: Li, J., Yang, H., Machorro, E.A. (eds.) Recent Advances in Scientific Computing and Applications, Contemporary Mathematics, vol. 586, pp. 239–246. American Mathematical Society, Providence (2013)

    Chapter  Google Scholar 

  23. Luong, V.T., Tung, N.T.: The Dirichlet–Cauchy problem for nonlinear hyperbolic equations in a domain with edges. Nonlinear Anal. 125, 457–467 (2015)

    Article  MathSciNet  Google Scholar 

  24. Matyukevich, S.I., Plamenevskiĭ, B.A.: On dynamic problems in the theory of elasticity in domains with edges. Algebra i Analiz 18(3), 158–233 (2006)

    MathSciNet  Google Scholar 

  25. Maz’ya, V., Rossmann, J.: Elliptic Equations in Polyhedral Domains, Mathematical Surveys and Monographs, vol. 162. American Mathematical Society, Providence (2010)

    Book  Google Scholar 

  26. Müller, F., Schwab, C.: Finite elements with mesh refinement for wave equations in polygons. J. Comput. Appl. Math. 283, 163–181 (2015)

    Article  MathSciNet  Google Scholar 

  27. Müller, F., Schwab, C.: Finite elements with mesh refinement for elastic wave propagation in polygons. Math. Methods Appl. Sci. 39, 5027–5042 (2016)

    Article  MathSciNet  Google Scholar 

  28. Müller, F.: Numerical analysis of finite element methods for second order wave equations in polygons. Ph.D. thesis, ETH Zürich, Diss. ETH No. 24385 (2017)

  29. Müller, F., Schötzau, D., Schwab, C.: Symmetric interior penalty discontinuous Galerkin methods for elliptic problems in polygons. SIAM J. Numer. Anal. 55(5), 2490–2521 (2017)

    Article  MathSciNet  Google Scholar 

  30. Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of Adaptive Finite Element Methods: An Introduction, Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)

    Book  Google Scholar 

  31. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, Mathematics and Applications, vol. 69. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  32. Plamenevskiĭ, B.A.: On the wave equation in a cylinder with edges. Funktsional. Anal. i Prilozhen. 32(1), 81–84 (1998)

    Article  MathSciNet  Google Scholar 

  33. Raviart, P.-A., Thomas, J.-M.: Introduction à l’analyse numérique des equations aux derivées partielles. Dunod, Paris (1998). (in French)

    MATH  Google Scholar 

  34. Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Problems: Theory and Implementation, Frontiers in Applied Mathematics. SIAM, Philadelphia (2008)

    Book  Google Scholar 

  35. Schötzau, D., Schwab, C.: Exponential convergence of \(hp\)-FEM for elliptic problems in polyhedra: Mixed boundary conditions and anisotropic polynomial degrees. Found. Comput. Math. 2017. https://doi.org/10.1142/S0218202515500438

    Article  MathSciNet  Google Scholar 

  36. Schwab, C.: \(p\)- and \(hp\)-FEM—Theory and Application to Solid and Fluid Mechanics. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  37. Shewchuk, J.R.: Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin, M.C., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering. Lecture Notes in Computer Science, vol. 1148, pp. 203–222. Springer, Berlin (1996)

    Chapter  Google Scholar 

  38. Wihler, T.P.: Discontinuous Galerkin FEM for Elliptic Problems in Polygonal Domains. Ph.D. thesis, Swiss Federal Institute of Technology Zurich, Diss. ETH No. 14973 (2002)

  39. Wloka, J.: Partielle Differentialgleichungen: Sobolevräume und Randwertaufgaben. Mathematische Leitfäden. B. G. Teubner, Stuttgart (1982). (in German)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Fabian Müller & Christoph Schwab

  2. Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Dominik Schötzau

Authors
  1. Fabian Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominik Schötzau
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christoph Schwab
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dominik Schötzau.

Additional information

Research supported by the Swiss National Science Foundation under Grant No. SNF 200021_149819/1 and by the Natural Sciences and Engineering Research Council of Canada (NSERC). This paper benefitted from helpful discussions at the Mathematical Research Institute Oberwolfach (MFO) during meeting No. 1711, March 13–17, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Müller, F., Schötzau, D. & Schwab, C. Discontinuous Galerkin Methods for Acoustic Wave Propagation in Polygons. J Sci Comput 77, 1909–1935 (2018). https://doi.org/10.1007/s10915-018-0706-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0706-x

Keywords

Mathematics Subject Classification

Search

Navigation