[Paper Review] Sharp pointwise-in-time error estimate of L1 scheme for nonlinear subdiffusion equations

Dongfang Li, Hongyu Qin+1 more|arXiv (Cornell University)|Jan 12, 2021

Fractional Differential Equations Solutions17 references4 citations

TL;DR

This paper establishes sharp pointwise-in-time error estimates for the L1 scheme applied to nonlinear subdiffusion equations with time-fractional Caputo derivatives. By introducing a refined discrete fractional-type Grönwall inequality and rigorously analyzing truncation errors under general solution regularity $\sigma \in (0,1) \cup (1,2)$, it proves optimal convergence rates: $\mathcal{O}(\tau^{\sigma+1-\alpha} t_n^{\alpha-1})$ for $\sigma < \alpha$ and $\mathcal{O}(\tau t_n^{\alpha-1})$ for $\sigma = \alpha$, confirmed by numerical experiments.

ABSTRACT

An essential feature of the subdiffusion equations with the $α$-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter $σ\in (0,1)\cup(1,2)$. Under this general regularity assumption, we here obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. To the end, we present a refined discrete fractional-type Grönwall inequality and a rigorous analysis for the truncation errors. Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

Motivation & Objective

• To establish sharp pointwise-in-time error estimates for the widely used L1 scheme in solving nonlinear subdiffusion equations with $\alpha$-order time fractional derivatives.
• To address the challenge of non-smooth solutions with weak singularity at $t=0$, characterized by a regularity parameter $\sigma \in (0,1) \cup (1,2)$.
• To overcome difficulties arising from variable truncation errors in discrete convolution and non-monotone error evolution due to nonlinearity.
• To provide the first rigorous pointwise error analysis for the L1 scheme in nonlinear subdiffusion problems under general regularity assumptions.
• To validate theoretical findings with numerical experiments demonstrating convergence orders matching theoretical predictions.

Proposed method

• Formulates the nonlinear subdiffusion problem with Caputo time derivative $\partial_t^\alpha u - \Delta u = f(u)$ on a domain $\Omega \subset \mathbb{R}^d$.
• Applies the L1 scheme on uniform time meshes for temporal discretization and central finite differences for spatial derivatives.
• Develops a refined discrete fractional-type Grönwall inequality to control error propagation under weak solution regularity.
• Performs a detailed truncation error analysis that accounts for time-level dependence due to the non-smooth nature of the solution.
• Uses energy-like estimates combined with the new Grönwall inequality to derive pointwise-in-time error bounds in the maximum norm.
• Validates the theoretical results through numerical experiments with varying $\sigma$, $\alpha$, and mesh sizes.

Experimental results

Research questions

• RQ1 What is the sharp pointwise-in-time convergence rate of the L1 scheme for nonlinear subdiffusion equations when the solution has limited regularity characterized by $\sigma \in (0,1) \cup (1,2)$?
• RQ2 Can a refined discrete fractional-type Grönwall inequality be constructed to handle the non-monotonic error evolution and variable truncation errors in the L1 scheme?
• RQ3 How do the temporal convergence orders behave as $t \to 0$ and at the final time $T$, especially when $\sigma < \alpha$?
• RQ4 Does the L1 scheme achieve optimal convergence for nonlinear subdiffusion under the same regularity assumptions as for linear problems?
• RQ5 Are the theoretical convergence rates confirmed by numerical experiments across different spatial dimensions and nonlinearities?

Key findings

• For $\sigma \in (0,\alpha)$, the pointwise-in-time error is bounded by $\|u^n - U^n\|_{\infty} \lesssim \tau^{\sigma+1-\alpha} t_n^{\alpha-1} + t_n^{\alpha} h^2$, which is the first such estimate even for linear problems.
• For $\sigma = \alpha$, the error bound is $\|u^n - U^n\|_{\infty} \lesssim \tau t_n^{\alpha-1} + t_n^{\alpha} h^2$, establishing the first sharp pointwise estimate for nonlinear subdiffusion equations.
• Numerical experiments confirm that the temporal convergence order at $t_n \to 0$ is $\sigma$ when $\sigma < 1$, and $\alpha$ when $\sigma = \alpha$, matching theoretical predictions.
• At the final time $t_N = T$, the convergence order tends to $\sigma + 1 - \alpha$ for $\sigma < 1$ and $2 - \alpha$ for $1 < \sigma < 2$, consistent with theory.
• Spatial convergence order is consistently $2$, confirming the expected second-order accuracy of the central finite difference scheme.
• The refined discrete fractional-type Grönwall inequality is essential for controlling error accumulation and enables the sharp error estimate under weak regularity.