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Well-Posedness and Approximation of Weak Solutions to Time Dependent Maxwell's Equations with $L^2$-Data

Harbir Antil

TL;DR

The paper advances the analysis and numerical approximation of Maxwell's equations with minimal regularity by proving well-posedness for the first-order weak formulation with $L^2$ data and rough coefficients, using interior-time mollification to establish uniqueness and Galerkin methods for existence. It then develops a structure-preserving semi-discrete finite element method based on the Nédélec and Raviart--Thomas de Rham spaces, which preserves a discrete Gauss law and satisfies a time-continuous energy identity, with stability under nonnegative conductivity. The authors prove convergence of the semi-discrete solutions to the unique weak solution as the spatial mesh is refined, assuming divergence-free initialization for the magnetic field and $f\in L^2(0,T;L^2(\\Omega)^3)$, $E_0,B_0\in L^2(\\Omega)^3$. This work thus provides a self-contained analytical framework and a robust, structure-preserving numerical approach for Maxwell systems with rough coefficients and low-regularity data, linking rigorous PDE theory with provable finite element convergence.

Abstract

We study Maxwell's equations in conducting media with perfectly conducting boundary conditions on Lipschitz domains, allowing rough material coefficients and $L^2$-data. Our first contribution is a direct proof of well-posedness of the first-order weak formulation, including solution existence and uniqueness, an energy identity, and continuous dependence on the data. The argument uses interior-in-time mollification to show uniqueness while avoiding reflection techniques. Existence is via the well-known Galerkin method (cf.~Duvaut and Lions \cite[Eqns.~(4.31)--(4.32), p.~346; Thm.~4.1]{GDuvaut_JLLions_1976a}). For completeness, and to make the paper self-contained, a complete proof has been provided. Our second contribution is a structure-preserving semi-discrete finite element method based on the Nédélec/Raviart--Thomas de Rham complex. The scheme preserves a discrete Gauss law for all times and satisfies a continuous-in-time energy identity with stability for nonnegative conductivity. With a divergence-free initialization of the magnetic field (via potential reconstruction or constrained $L^2$ projection), we prove convergence of the semi-discrete solutions to the unique weak solution as the mesh is refined. The analysis mostly relies on projector consistency, weak-* compactness in time-bounded $L^2$ spaces, and identification of time derivatives in dual spaces.

Well-Posedness and Approximation of Weak Solutions to Time Dependent Maxwell's Equations with $L^2$-Data

TL;DR

The paper advances the analysis and numerical approximation of Maxwell's equations with minimal regularity by proving well-posedness for the first-order weak formulation with data and rough coefficients, using interior-time mollification to establish uniqueness and Galerkin methods for existence. It then develops a structure-preserving semi-discrete finite element method based on the Nédélec and Raviart--Thomas de Rham spaces, which preserves a discrete Gauss law and satisfies a time-continuous energy identity, with stability under nonnegative conductivity. The authors prove convergence of the semi-discrete solutions to the unique weak solution as the spatial mesh is refined, assuming divergence-free initialization for the magnetic field and , . This work thus provides a self-contained analytical framework and a robust, structure-preserving numerical approach for Maxwell systems with rough coefficients and low-regularity data, linking rigorous PDE theory with provable finite element convergence.

Abstract

We study Maxwell's equations in conducting media with perfectly conducting boundary conditions on Lipschitz domains, allowing rough material coefficients and -data. Our first contribution is a direct proof of well-posedness of the first-order weak formulation, including solution existence and uniqueness, an energy identity, and continuous dependence on the data. The argument uses interior-in-time mollification to show uniqueness while avoiding reflection techniques. Existence is via the well-known Galerkin method (cf.~Duvaut and Lions \cite[Eqns.~(4.31)--(4.32), p.~346; Thm.~4.1]{GDuvaut_JLLions_1976a}). For completeness, and to make the paper self-contained, a complete proof has been provided. Our second contribution is a structure-preserving semi-discrete finite element method based on the Nédélec/Raviart--Thomas de Rham complex. The scheme preserves a discrete Gauss law for all times and satisfies a continuous-in-time energy identity with stability for nonnegative conductivity. With a divergence-free initialization of the magnetic field (via potential reconstruction or constrained projection), we prove convergence of the semi-discrete solutions to the unique weak solution as the mesh is refined. The analysis mostly relies on projector consistency, weak-* compactness in time-bounded spaces, and identification of time derivatives in dual spaces.
Paper Structure (17 sections, 15 theorems, 156 equations)

This paper contains 17 sections, 15 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Problem setting.
  3. Scope and goals.
  4. Positioning within the literature.
  5. Contributions.
  6. Organization.
  7. Notation and Preliminaries
  8. Well Posedness of Maxwell's Equations
  9. Step 3 (energy identity with a time cutoff and $\delta\downarrow0$).
  10. Nédélec/RT spaces and (discrete) de Rham sequence
  11. Discrete de Rham sequence (exactness).
  12. Global $L^2$ projector onto $\mathcal{RT}_h$.
  13. Riesz projection on $H_0(\mathop{\mathrm{curl}}\nolimits;\Omega)$.
  14. Semi-discrete FE scheme
  15. Discrete unknowns/test spaces.
  16. ...and 2 more sections

Key Result

Lemma 1

Let $v\in L^2(\Omega)^3$. If there exists $g\in L^2(\Omega)^3$ such that then $v\in H(\mathop{\mathrm{curl}}\nolimits;\Omega)$ and $\mathop{\mathrm{curl}}\nolimits v=g$ in $L^2(\Omega)^3$.

Theorems & Definitions (31)

  • Definition 1: Weak solution to Maxwell's Equations
  • Lemma 1: Characterization of $H(\mathop{\mathrm{curl}}\nolimits)$ via distributional curl
  • proof
  • Theorem 1: Solution to \ref{['eq:MaxWeak']} is unique
  • proof
  • Lemma 2: Differentiability of Scalar Pairings
  • Corollary 1: Strong $L^2$–continuity in time and energy identity
  • proof
  • Theorem 2: Well-posedness: Existence, Uniqueness, and Continuous Dependence
  • proof
  • ...and 21 more