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 \(L^2\)-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.