An Adaptive Finite Element DtN Method for Maxwell's Equations

TL;DR

This paper develops an adaptive finite element method combined with a Dirichlet-to-Neumann boundary condition for efficiently solving 3D electromagnetic scattering problems governed by Maxwell's equations, with proven error estimates and numerical validation.

Contribution

It introduces an a posteriori error estimate-based adaptive finite element DtN method for Maxwell's equations, incorporating truncation error analysis and exponential decay of the truncation error.

Findings

The method achieves accurate solutions with controlled errors.

Truncation error decays exponentially with the number of terms.

Numerical experiments confirm the efficiency and effectiveness.

Abstract

This paper is concerned with a numerical solution to the scattering of a time-harmonic electromagnetic wave by a bounded and impenetrable obstacle in three dimensions. The electromagnetic wave propagation is modeled by a boundary value problem of Maxwell's equations in the exterior domain of the obstacle. Based on the Dirichlet-to-Neumann (DtN) operator, which is defined by an infinite series, an exact transparent boundary condition is introduced and the scattering problem is reduced equivalently into a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is developed to solve the discrete variational problem, where the DtN operator is truncated into a sum of finitely many terms. The a posteriori error estimate takes into account both the finite element approximation error and the truncation error of the DtN operator. The latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.