A new analytical technique of the fully implicit Crank-Nicolson discontinuous Galerkin method for the Ginzburg-Landau Model
Xianxian Cao, Zhen Guan, Junjun Wang
TL;DR
The paper develops a fully implicit Crank-Nicolson discontinuous Galerkin method for the two-dimensional Ginzburg-Landau equation with cubic nonlinearity. It proves global unique solvability and unconditional optimal error estimates in both the $L^2$-norm and the energy norm by establishing $L^2$-norm boundedness of the numerical solution and refining nonlinear term estimates, using a Brouwer fixed-point argument for existence and discrete Gronwall arguments for stability. Numerical experiments in FreeFEM validate the theoretical results, demonstrating expected spatial convergence orders $O(h^{k+1})$ in $L^2$ and $O(h^{k})$ in the DG norm, as well as second-order temporal accuracy. The approach offers a robust framework for fully implicit schemes for nonlinear PDEs and suggests potential extensions to 3D and higher-order time discretizations.
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 \(L^2\)-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, two numerical examples are presented to validate the theoretical findings.