Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Error analysis of DGTD for linear Maxwell equations with inhomogeneous interface conditions

Benjamin Dörich, Julian Dörner, Marlis Hochbruck

TL;DR

The work analyzes discontinuous Galerkin time-domain discretizations for linear Maxwell equations with inhomogeneous interface conditions across a two-domain split, where a surface current J_surf and surface charge rho_surf reside on the interface. A robust analytical framework is developed by enlarging the state space to allow piecewise curl operators and by lifting the interface data, enabling a unitary C0-semigroup approach to prove well-posedness and stability; the surface current lifting is carefully integrated into the discrete scheme. The authors prove rigorous error bounds for both the semi-discrete and fully discrete schemes: a spatial error of order h^{r_*} with r_* = min{s,k} (where s is the solution regularity and k the polynomial degree) and a temporal error of order tau^2, yielding a total bound of the form C(h^{r_*} + tau^2); a nodal interface interpolation variant provides additional guidance for efficient computation. Numerical experiments in two dimensions validate the theory, reveal how surface current regularity affects convergence, and illustrate second-order time accuracy under the CFL condition, highlighting the practicality of the approach for simulating graphene-like interface phenomena.

Abstract

In the present paper we consider linear and isotropic Maxwell equations with inhomogeneous interface conditions. We discretize the problem with the discontinuous Galerkin method in space and with the leapfrog scheme in time. An analytical setting is provided in which we show wellposedness of the problem, derive stability estimates, and exploit this in the error analysis to prove rigorous error bounds for both the spatial and full discretization. The theoretical findings are confirmed with numerical experiments.

Error analysis of DGTD for linear Maxwell equations with inhomogeneous interface conditions

TL;DR

The work analyzes discontinuous Galerkin time-domain discretizations for linear Maxwell equations with inhomogeneous interface conditions across a two-domain split, where a surface current J_surf and surface charge rho_surf reside on the interface. A robust analytical framework is developed by enlarging the state space to allow piecewise curl operators and by lifting the interface data, enabling a unitary C0-semigroup approach to prove well-posedness and stability; the surface current lifting is carefully integrated into the discrete scheme. The authors prove rigorous error bounds for both the semi-discrete and fully discrete schemes: a spatial error of order h^{r_*} with r_* = min{s,k} (where s is the solution regularity and k the polynomial degree) and a temporal error of order tau^2, yielding a total bound of the form C(h^{r_*} + tau^2); a nodal interface interpolation variant provides additional guidance for efficient computation. Numerical experiments in two dimensions validate the theory, reveal how surface current regularity affects convergence, and illustrate second-order time accuracy under the CFL condition, highlighting the practicality of the approach for simulating graphene-like interface phenomena.

Abstract

In the present paper we consider linear and isotropic Maxwell equations with inhomogeneous interface conditions. We discretize the problem with the discontinuous Galerkin method in space and with the leapfrog scheme in time. An analytical setting is provided in which we show wellposedness of the problem, derive stability estimates, and exploit this in the error analysis to prove rigorous error bounds for both the spatial and full discretization. The theoretical findings are confirmed with numerical experiments.
Paper Structure (5 sections, 18 theorems, 165 equations, 6 figures)

This paper contains 5 sections, 18 theorems, 165 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Spatial discretization
  4. Full discretization
  5. Numerical experiments

Key Result

Corollary 2.1

Let $\vvV\in PH(\mathop{\mathrm{curl}}\nolimits, Q)$. It holds $\vvV\in H([){}]{\operatorname{curl}, {Q}}$ if and only if

Figures (6)

  • Figure 1: Sketches of different domain configurations $Q$.
  • Figure 2:
  • Figure 3: Error in the $L^2$-norm of the cavity solution \ref{['eq:CavityAnsatz']} plotted against the mesh size for a fixed time-step width $\tau = 1\cdot 10^{-4}$. The dotted lines correspond to the lift defined in \ref{['eq:LiftOp']}, and the circled lines to the interpolation in \ref{['eq:InterpolateSemiDiscreteMaxwell']}.
  • Figure 4: Illustration of discrete regularity: the norm $\|f_\alpha\|_{H^{\eta}}$, for $f_\alpha$ given in \ref{['eq:TrigonometricPolynomial']}, for different values of $\alpha$ and $\eta$ plotted against the number $M$ of Fourier modes.
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (36)

  • Remark 1.1
  • Corollary 2.1
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of \ref{['thm:UniqueExistence']}
  • Remark 3.2
  • ...and 26 more