Heterogeneous Multiscale Method for Maxwell's Equations
Marlis Hochbruck, Bernhard Maier, Christian Stohrer · Multiscale Modeling and Simulation · 2019
We present a finite element heterogeneous multiscale method for time-dependent Maxwell's equations in first-order formulation in highly oscillatory materials using Nédélec's edge elements. Based on a uniform approach for the error analysis of nonconforming space discretizations [D. Hipp, M. Hochbruck, and C. Stohrer, IMA J. Numer. Anal., 39 (2019), pp. 1206--1245], we prove an error bound for the semidiscrete scheme. We further present error bounds for the fully discrete scheme, where we consider time discretization using algebraically stable Runge--Kutta methods, the Crank--Nicolson method, and the leapfrog method. These error bounds are confirmed by numerical experiments.