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

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.2010 Mathematics Subject Classification. 78A45, 65N12, 65N15, 65N30.

An Adaptive Finite Element DtN Method for the Open Cavity Scattering Problems

“…Remark 3.1. We notice that the result and proof of Lemma 3.4 is different from those for the scattering problems in periodic structures [19,27,32]. For the latter problems, the DtN operators are defined on a straight line or plane surface and have only finitely many terms when acting on the incident fields.…”

An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures

“…Compared to the PML method, the DtN method can reduce the size of the computational domain. As a viable alternative to the PML method, the adaptive finite element DtN methods have also been developed recently to solve many two-and three-dimensional scattering problems, such as the acoustic scattering problems [26,28,39], the three-dimensional electromagnetic scattering problem [29], and the two-dimensional elastic wave scattering problems [32,33].…”

“…This paper concerns the numerical solution of the elastic wave scattering by biperiodic structures in three dimensions. It is a non-trivial extension of the elastic wave scattering by periodic structures in two dimensions [32]. There are two challenges for the three-dimensional problem.…”

An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

“…It was shown that the truncation error decays exponentially with respect to the truncation parameter N . The adaptive finite element DtN method has also been applied to solve the diffraction grating problems [32] as well as the elastic wave equation in periodic structures [26]. The numerical results show that the adaptive finite element DtN method is competitive with the adaptive finite element PML method.…”