Well-Posedness and Approximation of Weak Solutions to Time Dependent Maxwell's Equations with $L^2$-Data
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arXiv
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| Format: | Preprint |
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2025
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| author | Antil, Harbir |
| author_facet | Antil, Harbir |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_20752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-Posedness and Approximation of Weak Solutions to Time Dependent Maxwell's Equations with $L^2$-Data Antil, Harbir Numerical Analysis Analysis of PDEs Computational Physics 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. |
| title | Well-Posedness and Approximation of Weak Solutions to Time Dependent Maxwell's Equations with $L^2$-Data |
| topic | Numerical Analysis Analysis of PDEs Computational Physics |
| url | https://arxiv.org/abs/2510.20752 |