Convergence of the PML method for thermoelastic wave scattering problems

This paper is concerned with the thermoelastic obstacle scattering problem in three dimensions. A uniaxial perfectly matched layer (PML) method is firstly introduced to truncate the unbounded scattering problem, leading to a truncated PML problem in a bounded domain. Under certain constraints on model parameters, the well-posedness for the truncated PML problem is then proved except possibly for a discrete set of frequencies, based on the analytic Fredholm theory. Moreover, the exponential convergence of the uniaxial PML method is established in terms of the thickness and absorbing parameters of PML layer. The proof is based on the PML extension technique and the exponential decay properties of the modified fundamental solution. As far as we know, this is the first convergence result of the PML method for the time-harmonic thermoelastic scattering problem.

💡 Research Summary

The manuscript addresses the three‑dimensional time‑harmonic thermo‑elastic obstacle scattering problem and develops a rigorous convergence theory for a uniaxial perfectly matched layer (PML) truncation. The physical model couples the elastic displacement field u and the temperature field p through the Biot system of linear thermo‑elasticity. The governing equations contain the Lamé operator, mass density, thermal expansion coefficient, and thermal conductivity, leading to three families of waves: two compressional modes (fast and slow) and one shear mode, each with distinct (often complex) wave numbers. The scattering problem is posed in the exterior of a bounded obstacle Ω with homogeneous Dirichlet boundary conditions and Kupradze radiation conditions at infinity. Existence and uniqueness of the exterior solution are recalled via a variational formulation involving the Dirichlet‑to‑Neumann (DtN) map.

To render the problem computationally tractable, the authors introduce a Cartesian (uniaxial) PML. The PML is built by a complex coordinate stretching \