Abstract
This paper introduces a numerical scheme for the time-harmonic Maxwell equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the WG approximations in various discrete norms which are either \(H^1\)-like or \(L^2\) and \(L^2\)-like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.
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The authors would like to thank Dr. Wei Cai for a helpful discussion on the Maxwell equations.
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Junping Wang was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Xiu Ye was supported in part by the National Science Foundation under Grant No. DMS-1115097.
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Mu, L., Wang, J., Ye, X. et al. A Weak Galerkin Finite Element Method for the Maxwell Equations. J Sci Comput 65, 363–386 (2015). https://doi.org/10.1007/s10915-014-9964-4
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DOI: https://doi.org/10.1007/s10915-014-9964-4