Superconvergence of mixed finite element approximations to 3-D Maxwell's equations in metamaterials
KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Physical Sciences and Engineering (PSE) Division
Environmental Science and Engineering Program
Computational Transport Phenomena Lab
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AbstractNumerical simulation of metamaterials has attracted more and more attention since 2000, after the first metamaterial with negative refraction index was successfully constructed. In this paper we construct a fully-discrete leap-frog type finite element scheme to solve the three-dimensional time-dependent Maxwell's equations when metamaterials are involved. First, we obtain some superclose results between the interpolations of the analytical solutions and finite element solutions obtained using arbitrary orders of Raviart-Thomas-Nédélec mixed spaces on regular cubic meshes. Then we prove the superconvergence result in the discrete l2 norm achieved for the lowest-order Raviart-Thomas-Nédélec space. To our best knowledge, such superconvergence results have never been obtained elsewhere. Finally, we implement the leap-frog scheme and present numerical results justifying our theoretical analysis. © 2011 Elsevier Inc.
SponsorsPartially supported by the NSFC Key Project 11031006 and Hunan Provincial NSF Project 10JJ7001.Supported by National Science Foundation Grant DMS-0810896.Supported by Hunan Education Department Key Project 10A117.Supported by KAUST Faculty Baseline Research Fund.
JournalJournal of Computational Physics