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Superconvergence Analysis of Curlcurl-Conforming Elements on Rectangular Meshes

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Abstract

In our recent work (Hu et al. in SIAM J Sci Comput 42(6):A3859–A3877, 2020), we observed numerically some superconvergence phenomena of the curlcurl-conforming finite elements on rectangular domains. In this paper, we provide a theoretical justification for our numerical observation and establish a superconvergence theory for the curlcurl-conforming elements on rectangular meshes. For the elements with parameters r (\(r=k-1\), k, \(k+1\)) and k (\(k\ge 2\)), we show that the first (second) component of the numerical solution \(\varvec{u}_h\) converges with rate \(r+1\) at r vertical (horizontal) Gaussian lines in each element when \(r=k-1,k\) with \(k\ge 3\), \(\nabla \times \varvec{u}_h\) converges with rate \(k+1\) at \(k^2\) Lobatto points in each element when \(k\ge 3\), and the first (second) component of \(\nabla \times \nabla \times \varvec{u}_h\) converges with rate k at \((k-1)\) horizontal (vertical) Gaussian lines when \(k\ge 2\). They are all one-order higher than the related optimal rates. More numerical experiments are provided to confirm our theoretical results.

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The code used in this work will be made available upon a reasonable request.

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Acknowledgements

We would like to thank Dr. Waixiang Cao for the helpful discussions.

Funding

The work is supported in part by the National Natural Science Foundation of China (NSAF U1930402, NSFC 12131005, NSFC 11871092, and NSFC 12101036) and Fundamental Research Funds for the Central Universities FRF-TP-22-096A1.

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Authors and Affiliations

  1. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Lixiu Wang

  2. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Qian Zhang

  3. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

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  1. Lixiu Wang
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  2. Qian Zhang
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Correspondence to Qian Zhang.

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Wang, L., Zhang, Q. & Zhang, Z. Superconvergence Analysis of Curlcurl-Conforming Elements on Rectangular Meshes. J Sci Comput 95, 62 (2023). https://doi.org/10.1007/s10915-023-02182-w

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