Volume 55, Number 6, November-December 2021
|Page(s)||2921 - 2947|
|Published online||06 December 2021|
An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, P.R. China
2 Faculty of Science, Beijing University of Technology, Beijing 100124, P.R. China
3 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
4 School of Mathematics, Jilin University, Changchun, Jilin 130012, China
* Corresponding author: email@example.com
Accepted: 3 November 2021
Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an unbounded domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.
Mathematics Subject Classification: 78A45 / 65N30 / 65N12 / 65N50
Key words: Elastic wave equation / scattering by biperiodic structures / adaptive finite element method / transparent boundary condition / DtN map / a posteriori error estimate
© The authors. Published by EDP Sciences, SMAI 2021
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