Wavenumber-explicit $hp$-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media
This work addresses the challenge of ensuring numerical stability and accuracy in electromagnetic simulations for researchers in computational physics and engineering, but it is incremental as it extends existing analysis to more complex media conditions.
The paper tackles the problem of analyzing the quasi-optimality of Galerkin discretizations for Maxwell's equations with impedance boundary conditions in piecewise smooth media, showing that under specific wavenumber-explicit scale resolution conditions, the discretization is quasi-optimal for any wavenumber where the equations are polynomially well-posed.
We consider the time-harmonic Maxwell equations with impedance boundary conditions on a bounded Lipschitz domain $Ω$ with analytic boundary $Î$. We suppose that $Ω$ consists of multiple subdomains, and that the permeability and permittivity tensors are analytic on every subdomain, but may jump across subdomain interfaces. Under these conditions we show that for any wavenumber $k\in\mathbb{C}$ with $|k|\geq 1$ for which Maxwell's equations are polynomially well-posed, a Galerkin discretization based on Nédélec elements of order $p$ on a mesh with mesh width $h$ is quasi-optimal, provided that there holds the wavenumber-explicit scale resolution condition a) that $|k|h/p$ is sufficiently small and b) that $p/\log |k|$ is bounded from below.