Convergence of the PML method for scattering problems in poroelastic media

Abstract

This paper is concerned with the time-harmonic wave scattering problems in three dimensional poroelastic media. By introducing an intermediate variable $p$, the original $\mathbf{u}-\mathbf{w}$ system is equivalently transformed into a $\mathbf{u}-p$ system with fewer degrees of freedom, which facilitates the derivation of the fundamental solution, Green's identity and positivity of the complex wave numbers. A perfectly matched layer (PML) method is then introduced in the spherical coordinates to truncate the unbounded scattering problem. Under certain assumptions on the poroelastic and PML parameters, we prove the existence and uniqueness of solutions to the PML problems both in the truncated domain and layer. Moreover, the exponential convergence of the PML method is established in terms of the thickness and parameters of the PML layer. The proof is based on the PML extension and the exponential decay properties of the stretched fundamental solution. As far as we know, this is the first convergence result of the PML method for poroelastic scattering problems.