A strong approximation in L2 for the solutions of the Maxwell system with highly oscillating periodic coefficients

TL;DR

This paper establishes a strong L2 approximation for solutions of the Maxwell system with highly oscillating periodic coefficients, revealing oscillations in both space and time, and extends two-scale convergence theory to this context.

Contribution

It introduces a novel corrector involving plane waves for the Maxwell system, improving approximation accuracy in the strong topology, and connects two-scale convergence with classical Block decomposition.

Findings

Proves strong L2 approximation for Maxwell solutions with oscillating coefficients.

Develops a corrector combining elliptic correctors and plane waves.

Extends two-scale convergence theory to include oscillations in space and time.

Abstract

We consider a Maxwell system on $\mathbb{R}^3$ with periodic and highly oscillating coefficients. It is known that the solutions converge in the weak-$\ast$ topology of $L^\infty(0,T;\,L^2(\mathbb{R}^3))$ to the solution of a similar problem with constant coefficients given as the $H$-limits of the electric permittivity and the magnetic permeability respectively, i.e. the limit in the sense of the homogenization of linear elliptic equations with varying coefficients. However, it is not true that the elliptic corrector also provides a corrector for the solution of the Maxwell system, i.e. an approximation of the solutions in the strong topology of $L^2$. We shall prove that the oscillations in the space variable also produce oscillations in the time variable. We get a corrector consisting of adding to the elliptic corrector the sum of infinitely plane waves in the fast variable. Note that related results have been previously proved for the wave equation. Our proof is based on the two-scale convergence theory for almost periodic functions. One of the novelties is to show how this contains the classical Block decomposition.