# Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures

## TL;DR

This paper develops an adaptive finite element method with a truncated Dirichlet-to-Neumann map for elastic wave scattering by periodic structures, providing error estimates and demonstrating effectiveness through numerical experiments.

## Contribution

It introduces a new a posteriori error estimate and an adaptive algorithm for elastic wave scattering problems using a truncated DtN map.

## Key findings

- The method effectively reduces scattering problems to bounded domains.
- The a posteriori error estimate guides adaptive mesh refinement.
- Numerical results confirm the method's accuracy and efficiency.

## Abstract

Consider the scattering of a time-harmonic elastic plane wave by a periodic rigid surface. The elastic wave propagation is governed by the two-dimensional Navier equation. Based on a Dirichlet-to-Neumann (DtN) map, a transparent boundary condition (TBC) is introduced to reduce the scattering problem into a boundary value problem in a bounded domain. By using the finite element method, the discrete problem is considered, where the TBC is replaced by the truncated DtN map. A new duality argument is developed to derive the a posteriori error estimate, which contains both the finite element approximation error and the DtN truncation error. An a posteriori error estimate based adaptive finite element algorithm is developed to solve the elastic surface scattering problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.