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Residual-Based a Posteriori Error Estimates for the Time-Dependent Ginzburg–Landau Equations of Superconductivity

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Abstract

We propose and analyze residual-based a posteriori error estimator for a new finite element method for the time-dependent Ginzburg–Landau equations with the temporal gauge of superconductivity. The magnetic potential variable is approximated by \(H(\nabla \times ;\Omega )\)-conforming element and the scalar order parameter is approximated by the \(H^1(\Omega )\)-conforming element. Using the dual problem of a linearization of the original problem, we prove the reliability of the a posteriori error estimator, and an adaptive algorithm with the temporal and spatial refining and coarsening steps is then proposed. Numerical results are presented for illustrating the a posteriori error estimator and the adaptive algorithm in convex and nonconvex domains.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their very valuable comments and suggestions which have greatly helped the authors improve the presentation of the paper.

Funding

This work was supported by National Natural Science Foundation of China (11971366,11571266 and 11661161017) and by the Hubei Key Laboratory of Computational Science, Wuhan University, China (2019CFA007).

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  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Huoyuan Duan & Qiuyu Zhang

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  1. Huoyuan Duan
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Correspondence to Huoyuan Duan.

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Duan, H., Zhang, Q. Residual-Based a Posteriori Error Estimates for the Time-Dependent Ginzburg–Landau Equations of Superconductivity. J Sci Comput 93, 79 (2022). https://doi.org/10.1007/s10915-022-02041-0

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