## 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.

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## 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.

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Communicated by Frederic Valentin.

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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).

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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

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

### Keywords

- Elastic waves
- Nonhomogeneous
- Local spectral element
- Plane wave discontinuous Galerkin
- Plane wave basis functions
- Error estimates