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A high order unfitted finite element method for time-Harmonic Maxwell interface problems

Zhiming Chen, Ke Li, Maohui Lyu, Xueshuang Xiang

TL;DR

The unfitted finite element method is based on a mixed formulation in the discontinuous Galerkin framework on a Cartesian mesh with possible hanging nodes and the regularity of the solution to Maxwell interface problems with $H^2$ interfaces in each subdomain is proved.

Abstract

We propose a high order unfitted finite element method for solving timeharmonic Maxwell interface problems. The unfitted finite element method is based on a mixed formulation in the discontinuous Galerkin framework on a Cartesian mesh with possible hanging nodes. The $H^2$ regularity of the solution to Maxwell interface problems with $C^2$ interfaces in each subdomain is proved. Practical interface-resolving mesh conditions are introduced under which the hp inverse estimates on three-dimensional curved domains are proved. Stability and hp a priori error estimate of the unfitted finite element method are proved. Numerical results are included to illustrate the performance of the method.

A high order unfitted finite element method for time-Harmonic Maxwell interface problems

TL;DR

The unfitted finite element method is based on a mixed formulation in the discontinuous Galerkin framework on a Cartesian mesh with possible hanging nodes and the regularity of the solution to Maxwell interface problems with interfaces in each subdomain is proved.

Abstract

We propose a high order unfitted finite element method for solving timeharmonic Maxwell interface problems. The unfitted finite element method is based on a mixed formulation in the discontinuous Galerkin framework on a Cartesian mesh with possible hanging nodes. The regularity of the solution to Maxwell interface problems with interfaces in each subdomain is proved. Practical interface-resolving mesh conditions are introduced under which the hp inverse estimates on three-dimensional curved domains are proved. Stability and hp a priori error estimate of the unfitted finite element method are proved. Numerical results are included to illustrate the performance of the method.
Paper Structure (9 sections, 18 theorems, 114 equations, 5 figures, 1 table)

This paper contains 9 sections, 18 theorems, 114 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Maxwell interface problem
  3. The unfitted finite element method
  4. Notation and the inverse trace estimate
  5. The unfitted finite element method
  6. The finite element convergence analysis
  7. The projection operator for coercive Maxwell equations
  8. The time-harmonic Maxwell equations
  9. Numerical results

Key Result

Lemma 2.1

Let $D$ be a bounded Lipschitz domain, $\boldsymbol{F}\in{\boldsymbol{H}}^1(D)'{\color{black}{\cap{\boldsymbol{H}}_0(\mathrm{curl};D)'}}$, and $\mathrm{div}\,\boldsymbol{F}\in H^{-1}(D)$. Then there exists a constant $C$ depending only on the domain $D$ such that $\|\boldsymbol{F}\|_{{\boldsymbol{H}

Figures (5)

  • Figure 3.1: Illustration of proper intersections of the interface $\Gamma$ and an element $K$. From (a) to (d), $K$ has $1,2,3,4$ vertices in one of the domains $\Omega_i$, $i=1,2$.
  • Figure 3.2: Examples of improper intersections of the interface $\Gamma$ and an element $K$, for which local refinements are required to resolve the interface.
  • Figure 3.3: Examples of merging a small element $K$ with 2 or 8 of its neighboring elements to form a large element.
  • Figure 3.4: Left: The figure used in the proof of Lemma \ref{['lem:3.1new']}. Right: The figure used in the proof of Lemma \ref{['lem:3.4']}. The tetrahedron with at most two intersection points of $\Gamma$ and the edges of $K$.
  • Figure 5.1: The induced mesh across $\Gamma$ in $\Omega_1$ when $h_0=1/8, p=1$ (left), the cross-sections of the induced mesh at $x_3=0$ (middle) and $x_1=0$ (right) when $h_0=1/8$, $p=3$.

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 21 more