Volume 17, Issue 2
Mixed Spectral Element Method for 2D Maxwell's Eigenvalue Problem

Na Liu, Luis Tobón, Yifa Tang & Qing Huo Liu

Commun. Comput. Phys., 17 (2015), pp. 458-486.

Published online: 2018-04

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  • Abstract

It is well known that conventional edge elements in solving vector Maxwell's eigenvalue equations by the finite element method will lead to the presence of spurious zero eigenvalues. This problem has been addressed for the first order edge element by Kikuchi by the mixed element method. Inspired by this approach, this paper describes a higher order mixed spectral element method (mixed SEM) for the computation of two-dimensional vector eigenvalue problem of Maxwell's equations. It utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element framework with a weak divergence condition. It is shown that this method can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous analysis of the convergence of the mixed SEM is presented, based on the higher order edge element interpolation error estimates, which fully confirms the robustness of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape, coaxial and dual-inner-conductor cavities to verify the merits of the proposed method.

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@Article{CiCP-17-458, author = {Na Liu , and Luis Tobón , and Yifa Tang , and Qing Huo Liu , }, title = {Mixed Spectral Element Method for 2D Maxwell's Eigenvalue Problem}, journal = {Communications in Computational Physics}, year = {2018}, volume = {17}, number = {2}, pages = {458--486}, abstract = {

It is well known that conventional edge elements in solving vector Maxwell's eigenvalue equations by the finite element method will lead to the presence of spurious zero eigenvalues. This problem has been addressed for the first order edge element by Kikuchi by the mixed element method. Inspired by this approach, this paper describes a higher order mixed spectral element method (mixed SEM) for the computation of two-dimensional vector eigenvalue problem of Maxwell's equations. It utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element framework with a weak divergence condition. It is shown that this method can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous analysis of the convergence of the mixed SEM is presented, based on the higher order edge element interpolation error estimates, which fully confirms the robustness of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape, coaxial and dual-inner-conductor cavities to verify the merits of the proposed method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.230113.140814a}, url = {http://global-sci.org/intro/article_detail/cicp/10965.html} }
TY - JOUR T1 - Mixed Spectral Element Method for 2D Maxwell's Eigenvalue Problem AU - Na Liu , AU - Luis Tobón , AU - Yifa Tang , AU - Qing Huo Liu , JO - Communications in Computational Physics VL - 2 SP - 458 EP - 486 PY - 2018 DA - 2018/04 SN - 17 DO - http://doi.org/10.4208/cicp.230113.140814a UR - https://global-sci.org/intro/article_detail/cicp/10965.html KW - AB -

It is well known that conventional edge elements in solving vector Maxwell's eigenvalue equations by the finite element method will lead to the presence of spurious zero eigenvalues. This problem has been addressed for the first order edge element by Kikuchi by the mixed element method. Inspired by this approach, this paper describes a higher order mixed spectral element method (mixed SEM) for the computation of two-dimensional vector eigenvalue problem of Maxwell's equations. It utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element framework with a weak divergence condition. It is shown that this method can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous analysis of the convergence of the mixed SEM is presented, based on the higher order edge element interpolation error estimates, which fully confirms the robustness of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape, coaxial and dual-inner-conductor cavities to verify the merits of the proposed method.

Na Liu, Luis Tobón, Yifa Tang & Qing Huo Liu. (2020). Mixed Spectral Element Method for 2D Maxwell's Eigenvalue Problem. Communications in Computational Physics. 17 (2). 458-486. doi:10.4208/cicp.230113.140814a
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