A Family of Immersed Finite Element Spaces and Applications to Three-Dimensional \(\bf{H}(\operatorname{curl})\) Interface Problems
Abstract.
Efficient and accurate computation of \({\bf H}(\operatorname{curl})\) interface problems is of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in three-dimensional (3D) computation as they can circumvent generating complex 3D interface-fitted meshes. However, many unfitted mesh methods rely on nonconforming approximation spaces, which may cause a loss of accuracy for solving Maxwell-type equations, and the widely used penalty techniques in the literature may not help in recovering the optimal convergence. In this article, we provide a remedy by developing Nédélec-type immersed finite element (IFE) spaces with a Petrov–Galerkin scheme that is able to produce optimal-convergent solutions. To establish a systematic framework, we analyze all the \(H^1\), \({\bf H}(\operatorname{curl})\), and \({\bf H}(\operatorname{div})\) IFE spaces and form a discrete de Rham complex. Based on these fundamental results, we further develop a fast solver using a modified Hiptmair–Xu preconditioner which works for both the generalized minimal residual (GMRES) and conjugate gradient (CG) methods for solving the nonsymmetric linear algebraic system. The approximation capabilities of the proposed IFE spaces will be also established.
Reproducibility of computational results.
This paper has been awarded the “SIAM Reproducibility Badge: code and data available”, as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available in https://github.com/lyc102/ifem OR in the Supplementary Materials.
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Acknowledgment.
We would like to thank the anonymous reviewers for their constructive comments and suggestions on improving this article.
Supplementary Materials
Index of Supplementary Materials
Title of paper: A Family of Immersed Finite Element Spaces and Applications to Three Dimensional H(curl) Interface Problems
Authors: Long Chen, Ruchi Guo, and Jun Zou
File: 120943_1_supp_515743_rq18d5_sc.pdf
Type: PDF
Contents: SM1: Proof of unisolvence for edge elements for a regular tetrahedron. SM2. Sobolev-type inequalities on interface elements. SM3. Proof of a norm equivalence involving jump conditions. SM4. Some figures for illustration of Assumptions A2 and A3.
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Published In
SIAM Journal on Scientific Computing
Pages: A3121 - A3149
DOI: 10.1137/22M1505360
ISSN (online): 1095-7197
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© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 27 June 2022
Accepted: 11 April 2023
Published online: 15 December 2023
Keywords
MSC codes
Reproducibility of computational results.
This paper has been awarded the “SIAM Reproducibility Badge: code and data available”, as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available in https://github.com/lyc102/ifem OR in the Supplementary Materials.
Authors
Funding Information
HKSAR: 15302120
Hong Kong RGC General Research Fund: 14306921, 14306719
National Science Foundation (NSF): DMS-2012465
Funding: The work of the first author was funded by NSF DMS-2012465 and DMS-2309785. The second author was funded by NSF DMS-2309777. The work of the second author was supported in part by HKSAR grant 15302120. The work of the third author was substantially supported by Hong Kong RGC General Research Fund (projects 14306921 and 14306719).
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