Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zhen Guan, Xianxian Cao, Junjun Wang

TL;DR

The paper addresses solving the coupled generalized Ginzburg-Landau equations with general nonlinearities by developing a semi-implicit DLN-based Galerkin finite element method. It introduces a fully discrete scheme that avoids additional time-discrete systems, and proves unconditionally optimal $L^2$ and $H^1$ error estimates along with $L^{\infty}$-boundedness of the numerical solution through a novel two-case analysis of $\tau$ and $h$, using inverse and discrete Agmon inequalities. The authors validate their theoretical results with three numerical experiments in 2D/3D, demonstrating robust performance, mesh-independence, and applicability to high-nonlinearity regimes. The work provides a solid, extensible framework for nonlinear complex GL systems and paves the way for applying polygonal-mesh FE methods and broader nonlinear PDE models.

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 $τ\leq h$ and $τ>h$, respectively. Three numerical examples covering both two- and three-dimensional scenarios are eventually provided for the validation of the theoretical findings.

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

TL;DR

The paper addresses solving the coupled generalized Ginzburg-Landau equations with general nonlinearities by developing a semi-implicit DLN-based Galerkin finite element method. It introduces a fully discrete scheme that avoids additional time-discrete systems, and proves unconditionally optimal and error estimates along with -boundedness of the numerical solution through a novel two-case analysis of and , using inverse and discrete Agmon inequalities. The authors validate their theoretical results with three numerical experiments in 2D/3D, demonstrating robust performance, mesh-independence, and applicability to high-nonlinearity regimes. The work provides a solid, extensible framework for nonlinear complex GL systems and paves the way for applying polygonal-mesh FE methods and broader nonlinear PDE models.

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 and -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 and , respectively. Three numerical examples covering both two- and three-dimensional scenarios are eventually provided for the validation of the theoretical findings.
Paper Structure (11 sections, 8 theorems, 89 equations, 2 figures, 12 tables)

This paper contains 11 sections, 8 theorems, 89 equations, 2 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Derivation of the fully discrete DLN numerical scheme
  3. The variational formulation
  4. Finite element terminology
  5. Numerical method
  6. Some important results
  7. Theoretical analysis of the fully discrete numerical scheme
  8. Stability of the fully discrete DLN scheme
  9. Boundedness of the numerical solutions in $L^{\infty}$-norm
  10. Numerical examples
  11. Conclusions

Key Result

Lemma 2.1

Let $V$ be a complex inner product space, where $(\cdot,\cdot)$ denotes its inner product and $\|\cdot\|$ is the norm induced by this inner product. Then, for any collection of vectors $v^0, v^1, \cdots, v^N$ in $V$, the following statement holds: where

Figures (2)

  • Figure 1: Trend of the $L^2$-norm errors of $u$ and $v$ as the mesh size $h$ decreases for Example \ref{['example1']} with parameters $k=1,\theta = 0.5$.
  • Figure 2: Trend of the $L^2$-norm errors of $u$ and $v$ as the mesh size $h$ decreases for Example \ref{['example2']} with parameters $k=1,\theta = 0.75$.

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 8 more