On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous multiscale method for Maxwell’s equations

Abstract

In the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these difficulties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell's equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell's equations in a periodic setting and give numerical experiments in accordance to the stated results.

Calculation of the Electromagnetic Field of a Rectangular Waveguide With Chiral Medium

“…One of the ways to deal with non-physical solutions is to use such formulations of the initial problem, or such systems of basic functions, which would exclude the appearance of fictitious solutions. The approach based on the application of the mixed finite element method [32][33][34] is appropriate. This method was used in [35] to calculate a plane-parallel waveguide with a non-chiral insert.…”

Mathematical analysis of transmission properties of electromagnetic meta-materials

“…Another related work is the multiscale asymptotic expansion for Maxwell's equations [12]. For further recent contributions to HMM approximations for Maxwell's equations we refer to [16,26]. Sparse tensor product finite elements for multiscale Maxwell-type equations are analyzed in [14] and an adaptive generalized multiscale finite element method is studied in [15].…”

Heterogeneous Multiscale Method for Maxwell's Equations

“…To our knowledge, all previously presented FE-HMMs for Maxwell's equations were based on the second order formulation, also known as the curl-curl problem. In [11] and [15], Heterogeneous Multiscale Methods for time-harmonic Maxwell's equations in second order formulation are introduced. An HMM for time-dependent Maxwell's equations is analyzed in [19].…”