A semi-implicit DLN Galerkin finite element method for coupled Ginzburg-Landau equations with general nonlinearity

TL;DR

This paper introduces a semi-implicit Galerkin finite element method based on the DLN scheme for coupled Ginzburg-Landau equations, providing unconditional error estimates and simplifying the analysis with a novel technique.

Contribution

It develops a new analytical approach for error estimation in semi-implicit Galerkin methods without extra temporal discretization, applicable to general nonlinear Ginzburg-Landau equations.

Findings

Unconditionally optimal error estimates in L2 and H1 norms.

Numerical validation in 2D and 3D scenarios.

Simplified theoretical analysis avoiding additional time discretization.

Abstract

In this paper, based on the two-step discretization scheme proposed by Dahlquist, Liniger and Nevanlinna (DLN), we develop a semi-implicit Galerkin finite element method for solving the coupled generalized Ginzburg-Landau equations. By virtue of a novel analytical technique, the boundedness of the numerical solution in the infinity norm is established, upon which the unconditionally optimal error estimates in the $L^2$ and $H^1$-norms are further derived. Compared with the space-time error splitting technique commonly adopted in the literature, the analytical method proposed in this paper does not require the introduction of an additional temporal discretization system, thus greatly simplifying the theoretical argument. The core point of the argument lies in the skillful application of the inverse inequality and discrete Agmon inequality to analyze the two cases, namely $\tau \leq h$ and $\tau>h$, respectively. Three numerical examples covering both two- and three-dimensional scenarios are eventually provided for the validation of the theoretical findings.