Superconvergence of mixed finite element approximations to 3-D Maxwell's equations in metamaterials

Authors:
Yunqing Huang;Jichun Li;Wei Yang;Shuyu Sun
Affiliations:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, Hunan, China;Department of Mathematical Sciences, University of Nevada Las Vegas, NV 89154-4020, USA;Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, Hunan, China;Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
Venue:
Journal of Computational Physics
Year:
2011

Citing 6
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Global superconvergence for Maxwell's equations

Mathematics of Computation
Finite Element Methods with Matching and Nonmatching Meshes for Maxwell Equations with Discontinuous Coefficients

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Strong Stability-Preserving High-Order Time Discretization Methods

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Finite Element Analysis for Wave Propagation in Double Negative Metamaterials

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Metamaterials: Theory, Design, and Applications

Metamaterials: Theory, Design, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications

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Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell's equations on Cartesian grids

Journal of Computational Physics

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Abstract

Numerical 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-Nedelec mixed spaces on regular cubic meshes. Then we prove the superconvergence result in the discrete l"2 norm achieved for the lowest-order Raviart-Thomas-Nedelec 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.