A new analytical technique of the fully implicit Crank–Nicolson discontinuous Galerkin method for the Ginzburg–Landau Model

Abstract

In this paper, a fully implicit Crank–Nicolson discontinuous Galerkin method is proposed for solving the Ginzburg–Landau equation. By leveraging a novel analytical technique, we rigorously establish the unique solvability of the constructed numerical scheme as well as its unconditionally optimal error estimates under both the L²-norm and the energy norm. The core of the proof hinges on the L²-norm boundedness of the numerical solution and the refined estimation of the cubic nonlinear term. Finally, four numerical examples are presented to validate the theoretical findings.

Keywords

Crank–Nicolson / discontinuous Galerkin method / Ginzburg–Landau equation / unconditionally optimal error estimates / cubic nonlinear term

References

[1] B. Guo, M. Jiang, Y. Li, Ginzburg-Landau equations, Science Press, Beijing, 2020.

[2] J. Zhang, X. Yang, A three-layer finite difference scheme for the nonlinear Kuramoto-Tsuzuki complex equation with variable coefficients, Math. Methods Appl. Sci., 49 (2026), 108-118. https://doi.org/10.1002/mma.70138

[3] D. Li, W. Cao, C. Zhang, Z. Zhang, Optimal error estimates of a linearized Crank-Nicolson Galerkin FEM for the Kuramoto-Tsuzuki equations, Commun. Comput. Phys., 26 (2019), 838-854. https://doi.org/10.4208/cicp.OA-2018-0208

[4] X. Li, X. Cui, S. Zhang, Analysis of a Crank-Nicolson fast element-free Galerkin method for the nonlinear complex Ginzburg-Landau equation, J. Comput. Appl. Math., 457 (2025), 116323. https://doi.org/10.1016/j.cam.2024.116323

[5] Z. Guan, X. Cao, Weighted implicit-explicit discontinuous Galerkin methods for two-dimensional Ginzburg-Landau equations on general meshes, Netw. Heterog. Media, 20 (2025), 1367-1391. https://doi.org/10.3934/nhm.2025059

[6] B. Li, W. Sun, Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations, Int. J. Numer. Anal. Model., 10 (2013), 622-633.

[7] W. Sun, J. Wang, Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear Schrödinger system in 3D, J. Comput. Appl. Math., 317 (2017), 685-699. https://doi.org/10.1016/j.cam.2016.12.004

[8] X. Feng, B. Li, S. Ma, High-order mass- and energy-conserving SAV-Gauss collocation finite element methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 59 (2021), 1566-1591. https://doi.org/10.1137/20M1344998

[9] J. Guo, Y. Chen, J. Zhou, Q. Liang, Optimal error estimates of conservative virtual element method for the coupled nonlinear Schrödinger-Helmholtz equation, Commun. Nonlinear Sci. Numer. Simul., 145 (2025), 108680. https://doi.org/10.1016/j.cnsns.2025.108680

[10] D. Li, X. Li, H. Sun, Optimal error estimates of SAV Crank-Nicolson finite element method for the coupled nonlinear Schrödinger equation, J. Sci. Comput., 97 (2023), 71. https://doi.org/10.1007/s10915-023-02384-2

[11] T. Wang, T. Wang, Optimal pointwise error estimates of two conservative finite difference schemes for the coupled Gross-Pitaevskii equations with angular momentum rotation terms, J. Comput. Appl. Math., 425 (2023), 115056. https://doi.org/10.1016/j.cam.2022.115056

[12] D. Di Pietro, A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations, Math. Comput., 79 (2010), 1303-1330. https://doi.org/10.1090/S0025-5718-10-02333-1

[13] P. A. Gazca-Orozco, A. Kaltenbach, On the stability and convergence of discontinuous Galerkin schemes for incompressible flows, IMA J. Numer. Anal., 45 (2025), 243-282. https://doi.org/10.1093/imanum/drae004

[14] F. Chave, D. A. Di Pietro, F. Marche, F. Pigeonneau, A hybrid high-order method for the Cahn-Hilliard problem in mixed form, SIAM J. Numer. Anal., 54 (2016), 1873-1898. https://doi.org/10.1137/15M1041055

[15] G. D. Akrivis, V. A. Dougalis, O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53. https://doi.org/10.1007/BF01385769

[16] J. G. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. Part IV: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384. https://doi.org/10.1137/0727022

[17] Z. Z. Sun, Q. Zhang, G. Gao, Finite difference methods for nonlinear evolution equations, de Gruyter, Berlin, 2023. https://doi.org/10.1515/9783110796018

[18] J. Wang, A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput., 60 (2014), 390-407. https://doi.org/10.1007/s10915-013-9799-4

[19] W. Liu, Y. Chen, Q. Gu, Y. Huang, Virtual element method for nonlinear Sobolev equation on polygonal meshes, Numer. Algorithms, 94 (2023), 1731-1761. https://doi.org/10.1007/s11075-023-01553-6