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A Finite Element Method for the Dynamical Ginzburg–Landau Equations under Coulomb Gauge

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Abstract

This paper is concerned with numerical analysis of a finite element method for the time-dependent Ginzburg–Landau equations under the Coulomb gauge. The main challenge is that the magnetic potential \({{\varvec{A}}}\) is divergence-free and satisfies a Stokes-like structure under the Coulomb gauge. The proposed method uses linear Lagrange element \({\mathcal {P}}_1\) to solve for the order parameter \(\psi \), the lowest order Nédélec edge element \(\mathcal {N}\! \mathcal {D}_{\! 1}\) and linear Lagrange element \({P}_1\) to approximate the magnetic potential \({{\varvec{A}}}\) and the electric potential \(\phi \), respectively. In particular, the proposed method preserves a weakly divergence-free property for \({{\varvec{A}}}\) in the discrete level. The main aim of this work is to establish the second order spatial convergence of the most important variable \(\psi \), though the numerical solutions of \({{\varvec{A}}}\) are only O(h) in space. Our analysis is based on a nonstandard quasi-projection for \(\psi \) and the corresponding \(H^{-1}\)-norm estimates for Maxwell projection. With the quasi-projection, we prove that the lower-order approximation to \({{\varvec{A}}}\) does not pollute the accuracy of \(\psi _h\). An effective one step recovery is also proposed to obtain second order numerical solution for \({{\varvec{A}}}\). Our numerical experiments confirm the optimal second order convergence of \(\psi _h\) and the efficiency of the recovery step.

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Data Availability

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

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Acknowledgements

The authors wish to thank the anonymous referees for many constructive comments that improved the paper.

Funding

This work is partially supported by National Natural Science Foundation of China under Grant Number 12231003.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Huadong Gao & Wen Xie

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Huadong Gao

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HG and WX have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, method, analysis and writing. All authors certify that this material or similar material has not been and will not be submitted to or published in any other publication.

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Correspondence to Wen Xie.

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Gao, H., Xie, W. A Finite Element Method for the Dynamical Ginzburg–Landau Equations under Coulomb Gauge. J Sci Comput 97, 19 (2023). https://doi.org/10.1007/s10915-023-02327-x

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