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Error Analysis of a Decoupled Finite Element Method for Quad-Curl Problems

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Abstract

Finite element approximation to a decoupled formulation for the quad-curl problem is studied in this paper. The difficulty of constructing elements with certain conformity to the quad–curl problems has been greatly reduced. For convex domains, where the regularity assumption holds for Stokes equation, the approximation to the curl of the true solution has quadratic order of convergence and first order for the energy norm. If the solution shows singularity, an a posterior error estimator is developed and a separate marking adaptive finite element procedure is proposed, together with its convergence proved. Both the a priori and a posteriori error analysis are supported by the numerical examples.

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Acknowledgements

We greatly appreciate the anonymous reviewers’ revising suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Washington University, St. Louis, MO, 63130, USA

    Shuhao Cao

  2. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Long Chen

  3. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Xuehai Huang

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  1. Shuhao Cao
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Correspondence to Xuehai Huang.

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The first author was supported in part by the National Science Foundation under grant DMS-1913080. The second author was supported in part by the National Science Foundation under grants DMS-1913080 and DMS-2012465. The third author was supported by the National Natural Science Foundation of China under grants 11771338 and 12171300, the Natural Science Foundation of Shanghai 21ZR1480500, and the Fundamental Research Funds for the Central Universities 2019110066.

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Cao, S., Chen, L. & Huang, X. Error Analysis of a Decoupled Finite Element Method for Quad-Curl Problems. J Sci Comput 90, 29 (2022). https://doi.org/10.1007/s10915-021-01705-7

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