Approximation of singular solutions and singular data for Maxwell’s equations by Lagrange elements

Huoyuan Duan; Jiwei Cao; Ping Lin; Roger C. E. Tan

doi:10.1093/imanum/draa090

Approximation of singular solutions and singular data for Maxwell’s equations by Lagrange elements

Huoyuan Duan (Lead / Corresponding author), Jiwei Cao, Ping Lin, Roger C. E. Tan

Mathematics

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Abstract

A Lagrange finite element method is proposed for Maxwell’s equations in Lipschitz domains. The method is suitable for the approximation of singular solutions lying outside (H¹(Ω))³⁠, with nonhomogeneous singular boundary data in the tangential trace space of H(curl;Ω) and a singular right-hand-side source term in (H₀(curl;Ω))′ (the dual space of H₀(curl;Ω))⁠. Numerical results are presented to illustrate performance and the theoretical results.

Original language	English
Pages (from-to)	771-816
Number of pages	46
Journal	IMA Journal of Numerical Analysis
Volume	42
Issue number	1
Early online date	7 Jan 2021
DOIs	https://doi.org/10.1093/imanum/draa090
Publication status	Published - Jan 2022

Keywords

Maxwell equations
Lagrange element
finite element method

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10.1093/imanum/draa090

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title = "Approximation of singular solutions and singular data for Maxwell{\textquoteright}s equations by Lagrange elements",

abstract = "A Lagrange finite element method is proposed for Maxwell{\textquoteright}s equations in Lipschitz domains. The method is suitable for the approximation of singular solutions lying outside (H1(Ω))3⁠, with nonhomogeneous singular boundary data in the tangential trace space of H(curl;Ω) and a singular right-hand-side source term in (H0(curl;Ω))′ (the dual space of H0(curl;Ω))⁠. Numerical results are presented to illustrate performance and the theoretical results.",

keywords = "Maxwell equations, Lagrange element, finite element method",

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note = "Funding Information: National Natural Science Foundation of China (11971366, 11571266 and 11661161017 to H.D); Collaborative Innovation Centre of Mathematics (to H.D.); Hubei Key Laboratory of Computational Science (Wuhan University) (to H.D.); Natural Science Foundation of Hubei Province (2019CFA007 to H.D.). Publisher Copyright: {\textcopyright} 2021 The Author(s).",

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doi = "10.1093/imanum/draa090",

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AU - Cao, Jiwei

AU - Lin, Ping

AU - Tan, Roger C. E.

N1 - Funding Information: National Natural Science Foundation of China (11971366, 11571266 and 11661161017 to H.D); Collaborative Innovation Centre of Mathematics (to H.D.); Hubei Key Laboratory of Computational Science (Wuhan University) (to H.D.); Natural Science Foundation of Hubei Province (2019CFA007 to H.D.). Publisher Copyright: © 2021 The Author(s).

PY - 2022/1

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N2 - A Lagrange finite element method is proposed for Maxwell’s equations in Lipschitz domains. The method is suitable for the approximation of singular solutions lying outside (H1(Ω))3⁠, with nonhomogeneous singular boundary data in the tangential trace space of H(curl;Ω) and a singular right-hand-side source term in (H0(curl;Ω))′ (the dual space of H0(curl;Ω))⁠. Numerical results are presented to illustrate performance and the theoretical results.

AB - A Lagrange finite element method is proposed for Maxwell’s equations in Lipschitz domains. The method is suitable for the approximation of singular solutions lying outside (H1(Ω))3⁠, with nonhomogeneous singular boundary data in the tangential trace space of H(curl;Ω) and a singular right-hand-side source term in (H0(curl;Ω))′ (the dual space of H0(curl;Ω))⁠. Numerical results are presented to illustrate performance and the theoretical results.

KW - Maxwell equations

KW - Lagrange element

KW - finite element method

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DO - 10.1093/imanum/draa090

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JO - IMA Journal of Numerical Analysis

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ER -

Approximation of singular solutions and singular data for Maxwell’s equations by Lagrange elements

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