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Wavenumber-explicit $hp$-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media

Jens Markus Melenk, David Wörgötter

Abstract

We consider the time-harmonic Maxwell equations with impedance boundary conditions on a bounded Lipschitz domain $Ω$ with analytic boundary $Γ$. 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.

Wavenumber-explicit $hp$-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media

Abstract

We consider the time-harmonic Maxwell equations with impedance boundary conditions on a bounded Lipschitz domain with analytic boundary . We suppose that 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 with for which Maxwell's equations are polynomially well-posed, a Galerkin discretization based on Nédélec elements of order on a mesh with mesh width is quasi-optimal, provided that there holds the wavenumber-explicit scale resolution condition a) that is sufficiently small and b) that is bounded from below.
Paper Structure (22 sections, 18 theorems, 185 equations, 2 figures)

This paper contains 22 sections, 18 theorems, 185 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General notation and main results
  3. ${\mathcal{A}}$-partitions and classes of piecewise analytic functions
  4. Traces of ${\bf H}(\operatorname{curl}, \Omega)$ and variational formulation of Maxwell's equations
  5. Main results
  6. Frequency splitting operators
  7. Frequency splitting operators in piecewise smooth media
  8. Frequency splitting operators on analytic surfaces
  9. The auxiliary Maxwell problem
  10. Well-posedness of the auxiliary Maxwell problem
  11. Proof of Theorem \ref{['Mainresult1']}
  12. Discretization
  13. Regular triangulations and Nédélec-type I elements
  14. Element-by-element approximation operators
  15. Wavenumber-explicit finite element analysis
  16. ...and 7 more sections

Key Result

Proposition 2.8

Let $\Omega\subseteq\mathbb{R}^3$ be a bounded Lipschitz domain with boundary $\Gamma$ and outer unit normal ${\bf n}$. We consider the maps $\Pi_{T}$ and $\Pi_{t}$, which for vector fields ${\bf v}\in\left(\mathcal{C}^{\infty}(\overline{\Omega})\right)^3$ are defined as These maps extend to bounded and surjective operators $\Pi_{T}:{\bf H}(\operatorname{curl}, \Omega)\rightarrow\operatorname{{\b

Figures (2)

  • Figure 1: FEM convergence for piecewise constant coefficients, left: $p=1$, right: $p=2$.
  • Figure 2: FEM convergence for smooth coefficients, left: $p=1$, right: $p=2$.

Theorems & Definitions (31)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Remark 2.9
  • Remark 2.11
  • ...and 21 more