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 $L^2$-norm and the energy norm. The core of the proof hinges on the $L^2$-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.
Citation: Zhen Guan, Xianxian Cao, Junjun Wang. A new analytical technique of the fully implicit Crank–Nicolson discontinuous Galerkin method for the Ginzburg–Landau Model[J]. Communications in Analysis and Mechanics, 2026, 18(3): 710-731. doi: 10.3934/cam.2026029
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 $L^2$-norm and the energy norm. The core of the proof hinges on the $L^2$-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.
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Zhen Guan, Xianxian Cao, Junjun Wang. A new analytical technique of the fully implicit Crank–Nicolson discontinuous Galerkin method for the Ginzburg–Landau Model[J]. Communications in Analysis and Mechanics, 2026, 18(3): 710-731. doi: 10.3934/cam.2026029
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