Skip to main content
SpringerLink
Log in

A Trefftz-discontinuous Galerkin method for time-harmonic elastic wave problems

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we develop a plane wave discontinuous Galerkin method combined with local spectral element method for the elastic wave propagation in two and three space dimensions. We derive the error estimates of the approximation solutions in the mesh-dependent norm and the mesh-independent norm. Some dependence of the error bounds on the orders q of local spectral elements and the number p of plane wave propagation directions is given. Numerical results assess the validity of the theoretical results and indicate that the resulting approximate solutions generated by the PWDG–LSFE possess high accuracy.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Auld B (1973) Acoustic waves and fields in solids, vol 2. Wiley, New York

    Google Scholar 

  • Buffa A, Monk P (2008) Error estimates for the ultra weak variational formulation of the Helmholtz equation. ESIAM 42:925–940

    Article  MathSciNet  MATH  Google Scholar 

  • Cessenat O (1996) 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

  • Cessenat O, Despres B (1998) Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional helmholtz problem. SIAM J Numer Anal 35(1):255–299

    Article  MathSciNet  MATH  Google Scholar 

  • Cessenat O, Despres B (2003) Using plane waves as basis functions for solving time harmonic equations with the ultra weak variational formulation. J Comput Acous 11(2):227–238

    Article  MATH  Google Scholar 

  • Cummings P, Feng X (2006) Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Math Mod Methods Appl Sci 16:139–160

    Article  MathSciNet  MATH  Google Scholar 

  • Du Y, Wu H (2015) Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number. SIAM J Numer Anal 53(2):782–804

    Article  MathSciNet  MATH  Google Scholar 

  • El Kacimi A, Laghrouche O (2009) Numerical modeling of elastic wave scattering in frequency domain by the partition of unity finite element method. Int J Numer Methods Eng 77:1646–1669

    Article  MATH  Google Scholar 

  • El Kacimi A, Laghrouche O (2010) Improvement of PUFEM for the numerical solution of high-frequency elastic wave scattering on unstructured triangular mesh grids. Int J Numer Methods Eng 84:330–350

    MathSciNet  MATH  Google Scholar 

  • Feng X, Wu H (2011) hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math Comput 80(276):1997–2024

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Graff K (1991) Wave motion in elastic solids. Dover, New York

    MATH  Google Scholar 

  • Guo B, Sun W (2007) The optimal convergence of the h-p version of the finite element method with quasi-uniform meshes. SIAM J Numer Anal 45:698–730

    Article  MathSciNet  MATH  Google Scholar 

  • Hetmaniuk U (2007) Stability estimates for a class of Helmholtz problems. Commun Math Sci 5:665–678

    Article  MathSciNet  MATH  Google Scholar 

  • Hiptmair R, Moiola A, Perugia I (2011) Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the \(p\)-version. SIAM J Numer Anal 49:264–284

    Article  MathSciNet  MATH  Google Scholar 

  • Hiptmair R, Moiola A, Perugia I (2013) Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math Comput 82:247–268

    Article  MathSciNet  MATH  Google Scholar 

  • Hiptmair R, Moiola A, Perugia I (2016) Plane wave discontinuous Galerkin methods: exponential convergence of the \(hp-\)version. Found Comput Math 16:637–675

    Article  MathSciNet  MATH  Google Scholar 

  • Hu Q, Yuan L (2014a) A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations. Int J Numer Anal Model 11:587–607

    MathSciNet  Google Scholar 

  • Hu Q, Yuan L (2014b) A plane wave least-squares method for time-harmonic Maxwell’s equations in absorbing media. SIAM J Sci Comput 36:A1937–A1959

    Article  MathSciNet  MATH  Google Scholar 

  • Hu Q, Yuan L (2018) A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz and time-harmonic Maxwell equations. Adv Comput Math 44:245–275

    Article  MathSciNet  MATH  Google Scholar 

  • Huttunen T, Malinen M, Monk P (2007a) Solving Maxwell’s equations using the ultra weak variational formulation. J Comput Phys 223:731–758

    Article  MathSciNet  MATH  Google Scholar 

  • Huttunen T, Monk P (2007b) The use of plane waves to approximate wave propagation in anisotropic media. J Comput Math 25:350–367

    MathSciNet  Google Scholar 

  • Huttunen T, Monk P, Collino F, Kaipio J (2004) The ultra-weak variational formulation for elastic wave problems. SIAM J Sci Comput 25:1717–1742

    Article  MathSciNet  MATH  Google Scholar 

  • Luostari T (2013) Non-polynomial approximation methods in acoustics and elasticity, Ph.D. thesis, University of Eastern Finland. Available at https://core.ac.uk/download/pdf/19163531.pdf

  • Luostari T, Huttunen T, Monk P (2013) Error estimates for the ultra weak variational formulation in linear elasticity. ESAIM: Model Numer Anal 47:183–211

    Article  MathSciNet  MATH  Google Scholar 

  • Moiola A (2013) Plane wave approximation in linear elasticity. Appl Anal 92:1299–1307

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Monk P (2003) Finite element methods for Maxwell’s equation. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

  • Monk P, Wang D (1999) A least-squares method for the Helmholtz equation. Comput Methods Appl Mech Eng 175:121–136

    Article  MathSciNet  MATH  Google Scholar 

  • Peng J, Wang J, Shu S (2018) Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations, arXiv:1801.08800v2

  • Perrey-Debain E, Trevelyan J, Bettess P (2003) P-wave and S-wave decomposition in boundary integral equation for plane elastodynamics. Commun Numer Methods Eng 19:945–958

    Article  MATH  Google Scholar 

  • Perrey-Debain E, Trevelyan J, Bettess P (2003) Use of wave boundary elements for acoustic computations. J Comput Acoust 11:305–321

    Article  MATH  Google Scholar 

  • Riou H, Ladevèze P, Sourcis B (2008) The multiscale VTCR approach applied to acoustics problems. J Comput Acous 16:487–505

    Article  MATH  Google Scholar 

  • Riou H, Ladevèze P, Sourcis B, Faverjon B, Kovalevsky L (2012) An adaptive numerical strategy for the media-frequency analysis of Helmholtz’s problem. J Comput Acous 20:1–26

    Article  MATH  Google Scholar 

  • Sloane N (2000) Tables of spherical codes (with collaboration of R.H. Hardin, W.D. Smith and others) published electronically at http://www2.research.att.com/njas/packings

  • Yuan L (2019) The plane wave discontinuous Galerkin method combined with local spectral finite elements for the wave propagation in anisotropic media. Numer Math Theory Methods Appl 12:517–546

    Article  MathSciNet  MATH  Google Scholar 

  • Yuan L, Hu Q (2019) Error analysis of the plane wave discontinuous Galerkin method for Maxwell’s equations in anisotropic media. Commun Comput Phys 25:1496–1522

    Article  MathSciNet  Google Scholar 

  • Yuan L, Hu Q (2018) Comparisons of three kinds of plane wave methods for the Helmholtz equation and time-harmonic Maxwell equations with complex wave numbers. J Comput Appl Math 344:323–345

    Article  MathSciNet  MATH  Google Scholar 

  • Yuan L, Hu Q, Hengbin AN (2016) Parallel preconditioners for plane wave Helmholtz and Maxwell systems with large wave numbers. Int J Numer Anal Model 13(5):802–819

    MathSciNet  MATH  Google Scholar 

  • Zhu L, Wu H (2013) Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: \(hp\) version. SIAM J Numer Anal 51:1828–1852

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the anonymous referee for many insightful comments which led to great improvement in the results and the presentation of the paper.

Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Long Yuan

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100094, People’s Republic of China

    Yang Liu

Authors
  1. Long Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Long Yuan.

Additional information

Communicated by Frederic Valentin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

L. Yuan was supported by China NSF under the grant 11501529, Qinddao applied basic research project under grant 17-1-1-9-jch and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents. Y. Liu was supported by China NSF under the Grant 11571196 and the Science Challenge Program (no. TZ2018002).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, L., Liu, Y. A Trefftz-discontinuous Galerkin method for time-harmonic elastic wave problems. Comp. Appl. Math. 38, 137 (2019). https://doi.org/10.1007/s40314-019-0900-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-019-0900-y

Keywords

Mathematics Subject Classification

Search

Navigation