Constraint-preserving hybrid finite element methods for Maxwell's equations

28 Jun 2019  ·  Berchenko-Kogan Yakov, Stern Ari ·

Maxwell's equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux densities. These constraints correspond to fundamental physical laws: the nonexistence of magnetic monopoles and the conservation of charge, respectively... However, one or both of these constraints may be violated when one applies a finite element method to discretize in space. This is a well-known and longstanding problem in computational electromagnetics. We use domain decomposition to construct a family of primal hybrid finite element methods for Maxwell's equations, where the Lagrange multipliers are shown to correspond to a numerical trace of the magnetic field and a numerical flux of the electric flux density. Expressing the charge-conservation constraint in terms of this numerical flux, we show that both constraints are strongly preserved. As a special case, these methods include a hybridized version of N\'ed\'elec's method, implying that it preserves the constraints more strongly than previously recognized. These constraint-preserving properties are illustrated using numerical experiments in both the time domain and frequency domain. Additionally, we observe a superconvergence phenomenon, where hybrid post-processing yields an improved estimate of the magnetic field. read more

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Numerical Analysis Numerical Analysis