Volume 13, Issue 5
The Plane Wave Methods for the Time-Harmonic Elastic Wave Problems with the Complex Valued Coefficients

Adv. Appl. Math. Mech., 13 (2021), pp. 1169-1202.

Published online: 2021-06

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• Abstract

In this paper the plane wave methods are discussed for solving the time-harmonic elastic wave propagation problems with the complex valued coefficients in two and three space dimensions. The plane wave least-squares method and the ultra-weak variational formulation are developed for the elastic wave propagation. The error estimates of the approximation solutions generated by the PWLS method are derived. Moreover, Combined with local spectral elements, the plane wave methods  are generalized to solve the nonhomogeneous elastic wave problems. Numerical results verify the validity of the theoretical results and indicate that the resulting approximate solution generated by the PWLS method is generally more accurate than that generated by a new variant of the ultra-weak variational formulation method when the Lamé constants $\lambda$ and $\mu$ are complex valued.

• Keywords

Elastic waves, nonhomogeneous, plane wave least-squares, ultra-weak variational formulation, plane wave basis functions, error estimates, local spectral elements, preconditioner.

65N30, 65N55

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@Article{AAMM-13-1169, author = {Long and Yuan and and 16676 and and Long Yuan and Shuai and Xi and and 16677 and and Shuai Xi and Binlin and Zhang and and 16678 and and Binlin Zhang}, title = {The Plane Wave Methods for the Time-Harmonic Elastic Wave Problems with the Complex Valued Coefficients}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {5}, pages = {1169--1202}, abstract = {

In this paper the plane wave methods are discussed for solving the time-harmonic elastic wave propagation problems with the complex valued coefficients in two and three space dimensions. The plane wave least-squares method and the ultra-weak variational formulation are developed for the elastic wave propagation. The error estimates of the approximation solutions generated by the PWLS method are derived. Moreover, Combined with local spectral elements, the plane wave methods  are generalized to solve the nonhomogeneous elastic wave problems. Numerical results verify the validity of the theoretical results and indicate that the resulting approximate solution generated by the PWLS method is generally more accurate than that generated by a new variant of the ultra-weak variational formulation method when the Lamé constants $\lambda$ and $\mu$ are complex valued.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0350}, url = {http://global-sci.org/intro/article_detail/aamm/19258.html} }
TY - JOUR T1 - The Plane Wave Methods for the Time-Harmonic Elastic Wave Problems with the Complex Valued Coefficients AU - Yuan , Long AU - Xi , Shuai AU - Zhang , Binlin JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1169 EP - 1202 PY - 2021 DA - 2021/06 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0350 UR - https://global-sci.org/intro/article_detail/aamm/19258.html KW - Elastic waves, nonhomogeneous, plane wave least-squares, ultra-weak variational formulation, plane wave basis functions, error estimates, local spectral elements, preconditioner. AB -

In this paper the plane wave methods are discussed for solving the time-harmonic elastic wave propagation problems with the complex valued coefficients in two and three space dimensions. The plane wave least-squares method and the ultra-weak variational formulation are developed for the elastic wave propagation. The error estimates of the approximation solutions generated by the PWLS method are derived. Moreover, Combined with local spectral elements, the plane wave methods  are generalized to solve the nonhomogeneous elastic wave problems. Numerical results verify the validity of the theoretical results and indicate that the resulting approximate solution generated by the PWLS method is generally more accurate than that generated by a new variant of the ultra-weak variational formulation method when the Lamé constants $\lambda$ and $\mu$ are complex valued.

Long Yuan, Shuai Xi & Binlin Zhang. (1970). The Plane Wave Methods for the Time-Harmonic Elastic Wave Problems with the Complex Valued Coefficients. Advances in Applied Mathematics and Mechanics. 13 (5). 1169-1202. doi:10.4208/aamm.OA-2020-0350
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