Edge Element Methods for Maxwell's Equations with Strong Convergence for Gauss' Laws

Patrick Ciarlet 1 Haijun Wu 2 Jun Zou 3
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : In this paper we propose and investigate some edge element approximations for three Maxwell systems in three dimensions: the stationary Maxwell equations, the time-harmonic Maxwell equations and the time-dependent Maxwell equations. These approximations have three novel features. First, the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of finite element meshes, including the stationary Maxwell equations. Second, they ensure the optimal strong convergence of the Gauss' laws in some appropriate norm, in addition to the standard optimal convergence in energy-norm, under the general weak regularity assumptions that hold for both convex and non-convex polyhedral domains and for the discontinuous coefficients that may have large jumps across the interfaces between different media. Finally, no saddle-point discrete systems are needed to solve for the stationary Maxwell equations, unlike most existing edge element schemes.
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Patrick Ciarlet, Haijun Wu, Jun Zou. Edge Element Methods for Maxwell's Equations with Strong Convergence for Gauss' Laws. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2014, 52 (2), pp.779-807. ⟨10.1137/120899856⟩. ⟨hal-01112201⟩

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