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

Abstract

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.

Two-Grid Methods for Maxwell’s Equations in a Cole-Cole Dispersive Medium

“…Besides the finite-difference method, the finite element method [3,19] and the weak Galerkin finite element method [44] are also useful tools for solving electromagnetic field problems. With the inspiration of the design of the FDTD scheme which was based on the ADE method, there were some publications reported on the dispersion modeling within the time-domain finite element method (TDFEM) framework.…”

A posteriori error estimates for the Brinkman–Darcy–Forchheimer problem

“…A posteriori analysis was first introduced by I. Babuška [5], developed by R. Verfürth [37], and has been the object of a large number of publications. A posteriori error estimations have been studied for several types of partial differential equations such that the Stokes or Navier-Stokes equation (see for instance [37,10,4,1,12,13,19]), the Maxwell and Lamé equations [30,8]. Many works have been established for the Darcy flow, see for instance [2,11,14,29].…”