[Paper Review] Sharp $H^1$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems

Jincheng Ren, Hong-lin Liao+2 more|arXiv (Cornell University)|Nov 20, 2018

Fractional Differential Equations Solutions9 references21 citations

TL;DR

This paper establishes sharp H¹-norm error estimates for two nonuniform time-stepping schemes—L1 and fractional Crank-Nicolson—for reaction-subdiffusion problems with Caputo fractional derivatives. By introducing an improved discrete Grönwall inequality and combining it with a time-space error-splitting technique and global consistency analysis, the authors recover optimal convergence rates despite the initial singularity, achieving H¹-norm error bounds of order 𝒪(τ^{2−α}) for both schemes on graded meshes with optimal grading parameters.

ABSTRACT

Due to the intrinsically initial singularity of solution and the discrete convolution form in numerical Caputo derivatives, the traditional $H^1$-norm analysis (corresponding to the case for a classical diffusion equation) to the time approximations of a fractional subdiffusion problem always leads to suboptimal error estimates (a loss of time accuracy). To recover the theoretical accuracy in time, we propose an improved discrete Grönwall inequality and apply it to the well-known L1 formula and a fractional Crank-Nicolson scheme. With the help of a time-space error-splitting technique and the global consistency analysis, sharp $H^1$-norm error estimates of the two nonuniform approaches are established for a reaction-subdiffusion problems. Numerical experiments are included to confirm the sharpness of our analysis.

Motivation & Objective

• To address the suboptimal H¹-norm error estimates in time for fractional subdiffusion problems due to initial singularity and convolution structure in Caputo derivatives.
• To recover optimal temporal convergence rates in the H¹-norm for nonuniform time discretizations, which are typically lost in classical analysis.
• To extend the theoretical framework of Liao et al. to achieve sharp H¹-norm error estimates for both L1 and fractional Crank-Nicolson schemes on general nonuniform meshes.
• To validate the sharpness of the theoretical estimates through numerical experiments with varying mesh grading parameters and fractional orders.

Proposed method

• Propose an improved discrete Grönwall inequality tailored to handle the initial singularity and convolutional structure of the Caputo derivative in nonuniform time discretizations.
• Apply the time-space error-splitting technique to decouple temporal and spatial errors, enabling independent analysis of each component.
• Employ global consistency analysis to control the local truncation error across the entire time interval, especially near t=0.
• Use a graded time mesh with parameter γ to resolve the initial singularity, where τk ∝ tk−1+γ−1.
• Establish sharp H¹-norm error estimates for both the L1 and fractional Crank-Nicolson schemes by combining the improved discrete Grönwall inequality with stability and consistency arguments.
• Validate the theoretical findings via numerical experiments using finite difference spatial discretization and varying mesh grading parameters.

Experimental results

Research questions

• RQ1 Can optimal H¹-norm error estimates be recovered for nonuniform time discretizations of reaction-subdiffusion problems with Caputo derivatives, despite the initial singularity and convolutional structure?
• RQ2 Does the traditional H¹-norm analysis fail to achieve optimal convergence rates due to the non-smoothness of the solution near t=0, and if so, how can this be corrected?
• RQ3 Can the improved discrete Grönwall inequality be effectively used to restore optimal temporal convergence in H¹-norm for both L1 and fractional Crank-Nicolson schemes?
• RQ4 What is the optimal grading parameter γ for achieving the theoretical convergence rate of 𝒪(τ^{2−α}) in the H¹-norm for these schemes?
• RQ5 Are the theoretical H¹-norm error bounds sharp, as confirmed by numerical experiments with varying mesh parameters and fractional orders?

Key findings

• The paper establishes sharp H¹-norm error estimates of order 𝒪(τ^{2−α}) for both the L1 and fractional Crank-Nicolson schemes on nonuniform time meshes, recovering optimal convergence despite initial singularity.
• The improved discrete Grönwall inequality successfully mitigates the loss of accuracy in H¹-norm analysis, enabling optimal error bounds where classical methods fail.
• Numerical experiments confirm that the convergence rate reaches 𝒪(τ^{2−α}) when the mesh grading parameter γ ≥ γ_opt, validating the sharpness of the theoretical estimates.
• For the L1 scheme, optimal convergence is achieved when γσ ≥ 2−α, and for the fractional Crank-Nicolson scheme, when γσ ≥ 2, with γ_opt determined accordingly.
• The time-space error-splitting technique effectively decouples temporal and spatial errors, allowing precise control and analysis of the H¹-norm error contribution from time discretization.
• Numerical results show that nonuniform meshes significantly improve convergence rates compared to uniform meshes, especially when γ ≥ γ_opt, confirming the theoretical predictions.