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

Yunqing Huang; Jichun Li; Wei Yang; Shuyu Sun

doi:10.1016/j.jcp.2011.07.025

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

Yunqing Huang, Jichun Li, Wei Yang, Shuyu Sun

Research output: Contribution to journal › Article › peer-review

34 Scopus citations

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-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.

Original language	English (US)
Pages (from-to)	8275-8289
Number of pages	15
Journal	Journal of Computational Physics
Volume	230
Issue number	22
DOIs	https://doi.org/10.1016/j.jcp.2011.07.025
State	Published - Sep 2011

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Partially 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.

ASJC Scopus subject areas

Physics and Astronomy (miscellaneous)
Computer Science Applications

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10.1016/j.jcp.2011.07.025

Cite this

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

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-N{\'e}d{\'e}lec mixed spaces on regular cubic meshes. Then we prove the superconvergence result in the discrete l2 norm achieved for the lowest-order Raviart-Thomas-N{\'e}d{\'e}lec 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. {\textcopyright} 2011 Elsevier Inc.",

author = "Yunqing Huang and Jichun Li and Wei Yang and Shuyu Sun",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Partially 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.",

year = "2011",

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AU - Huang, Yunqing

AU - Li, Jichun

AU - Yang, Wei

AU - Sun, Shuyu

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Partially 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.

PY - 2011/9

Y1 - 2011/9

N2 - 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-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.

AB - 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-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.

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JO - Journal of Computational Physics

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ER -

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

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