[Paper Review] Unconditional superconvergence analysis of a linearized Crank-Nicolson Galerkin FEM for generalized Ginzburg-Landau equation

Meng Li, Dongyang Shi+1 more|arXiv (Cornell University)|Jun 24, 2018

Numerical methods for differential equations3 citations

TL;DR

This paper presents an unconditional superconvergence analysis for a linearized Crank-Nicolson Galerkin finite element method (FEM) applied to the generalized Ginzburg-Landau equation (GLE). By combining time-discrete and space-time discrete schemes using bilinear elements, the authors derive optimal error estimates: $O(\tau^2 + h^2)$ in the $H^1$-norm and global superconvergence via interpolated postprocessing, without time-step restrictions, confirming robust convergence across large time steps.

ABSTRACT

In this paper, a linearized Crank-Nicolson Galerkin finite element method (FEM) for generalized Ginzburg-Landau equation (GLE) is considered, in which, the difference method in time and the standard Galerkin FEM are employed. Based on the linearized Crank-Nicolson difference method in time and the standard Galerkin finite element method with bilinear element in space, the time-discrete and space-time discrete systems are both constructed. We focus on a rigorous analysis and consideration of unconditional superconvergence error estimates of the discrete schemes. Firstly, by virtue of the temporal error results, the regularity for the time-discrete system is presented. Secondly, the classical Ritz projection is used to obtain the spatial error with order $O(h^2)$ in the sense of $L^2-$norm. Thanks to the relationship between the Ritz projection and the interpolated projection, the superclose estimate with order $O(τ^2 + h^2)$ in the sense of $H^1-$norm is derived. Thirdly, it follows from the interpolated postprocessing technique that the global superconvergence result is deduced. Finally, some numerical results are provided to confirm the theoretical analysis.

Motivation & Objective

• To develop a linearized Crank-Nicolson Galerkin FEM for the generalized Ginzburg-Landau equation (GLE) with improved stability and convergence.
• To establish unconditional superconvergence error estimates for the discrete scheme, removing time-step restrictions common in nonlinear problems.
• To rigorously analyze the temporal and spatial error components using Ritz projection and interpolated postprocessing techniques.
• To confirm theoretical results with numerical experiments under varying time steps and mesh sizes.

Proposed method

• A linearized Crank-Nicolson scheme is applied in time, decoupling the nonlinear term to avoid iterative solvers.
• The standard Galerkin FEM with bilinear finite elements is used in space, forming a fully discrete system.
• The Ritz projection operator $R_h$ is employed to derive spatial error estimates of order $O(h^2)$ in the $L^2$-norm.
• The relationship between the Ritz projection and the interpolation operator $I_h$ enables superclose estimates of order $O(\tau^2 + h^2)$ in the $H^1$-norm.
• Interpolated postprocessing via $I_{2h}^2$ is applied to achieve global superconvergence in the $H^1$-norm.
• A time-space error splitting technique isolates temporal and spatial errors, enabling unconditional analysis without grid ratio constraints.

Experimental results

Research questions

• RQ1 Can unconditional superconvergence be achieved for the generalized Ginzburg-Landau equation using a linearized Crank-Nicolson Galerkin FEM?
• RQ2 What is the convergence rate of the discrete solution in the $H^1$-norm under optimal time and space discretization?
• RQ3 How does the Ritz projection and interpolated postprocessing contribute to achieving superconvergence without time-step restrictions?
• RQ4 Can the method maintain stability and accuracy for large time steps, as confirmed numerically?

Key findings

• The temporal error is bounded by $O(\tau^2)$ in the $H^2$-norm, enabling robust superconvergence analysis.
• The spatial error using the Ritz projection achieves $O(h^2)$ convergence in the $L^2$-norm.
• A superclose estimate of $O(\tau^2 + h^2)$ is derived in the $H^1$-norm between the discrete solution and the Ritz projection.
• The interpolated postprocessing technique yields global superconvergence with error $\|u^n - I_{2h}^2 U_h^n\|_1 = O(\tau^2 + h^2)$.
• Numerical results confirm the theoretical convergence rates across multiple time steps, including large $\tau = 20h$, validating unconditional stability.
• The method achieves optimal convergence rates without requiring time-step constraints such as $\tau = O(h^d)$, demonstrating unconditional superconvergence.