The Plane Wave Methods for the Time-Harmonic Elastic Wave Problems with the Complex Valued Coefficients

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.