# An adaptive finite element PML method for the acoustic-elastic interaction in three dimensions

## TL;DR

This paper develops an adaptive finite element PML method with an a posteriori error estimate for 3D acoustic-elastic interaction problems, ensuring well-posedness, exponential convergence, and demonstrating its effectiveness through numerical experiments.

## Contribution

It introduces a novel adaptive finite element PML approach with error estimation for 3D acoustic-elastic scattering, improving accuracy and efficiency.

## Key findings

- The method achieves exponential convergence.
- Numerical results validate the method's effectiveness.
- The approach is competitive for complex scattering problems.

## Abstract

Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable, and isotropic elastic solid, which is immersed in a homogeneous compressible air or fluid. The paper concerns the numerical solution for such an acoustic-elastic interaction problem in three dimensions. An exact transparent boundary condition (TBC) is developed to reduce the problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by using a PML equivalent TBC. An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.