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

TL;DR

This paper introduces a new fully implicit Crank-Nicolson discontinuous Galerkin method for the Ginzburg-Landau equation, providing rigorous solvability and optimal error estimates, validated through numerical examples.

Contribution

It develops a novel analytical approach to establish solvability and error bounds for the method applied to the Ginzburg-Landau model.

Findings

Proves unique solvability of the scheme.

Establishes unconditionally optimal error estimates.

Validates results with numerical examples.

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.