On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 2: A Heterogeneous Multiscale Method for Maxwell's equations

Patrick Ciarlet (1) , Sonia Fliss (1) , Christian Stohrer (2)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
2 KIT - Karlsruhe Institute of Technology

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 homog-enization scheme for the Maxwell's equation 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.

Mots clés: Numerical Homogenization; Maxwell's equation; Heterogeneous Multiscale Method; Edge Finite Elements; Two-Scale Convergence; T -coercivity

Domaines: Analyse numérique [math.NA]

Patrick Ciarlet, Sonia Fliss, Christian Stohrer. On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 2: A Heterogeneous Multiscale Method for Maxwell's equations. 2016. ⟨hal-01364782v1⟩