[Paper Review] Error analysis for a fractional-derivative parabolic problem on quasi-graded meshes using barrier functions

Natalia Kopteva, Xiangyun Meng|arXiv (Cornell University)|May 17, 2019

Fractional Differential Equations Solutions21 references4 citations

TL;DR

This paper presents a novel stability and error analysis for fractional-derivative parabolic problems with Caputo time derivatives using barrier functions on quasi-graded temporal meshes. It establishes sharp pointwise-in-time error bounds for both L1 and Alikhanov-type schemes, showing that milder mesh grading than optimal still yields optimal convergence rates in positive time, a result of practical significance and theoretical novelty.

ABSTRACT

An initial-boundary value problem with a Caputo time derivative of fractional order $α\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semi-discretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.

Motivation & Objective

• To develop a general and simple numerical stability analysis for time-discretized fractional-derivative parabolic problems with Caputo derivatives.
• To establish sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary grading degree.
• To demonstrate that milder (non-optimal) grading can still yield optimal convergence rates in positive time, a finding of practical importance.
• To extend the barrier function methodology to both L1-type and Alikhanov-type time discretizations, ensuring broad applicability.
• To validate theoretical findings with numerical experiments confirming sharpness and convergence behavior.

Proposed method

• Utilizes barrier functions to analyze the stability of discrete fractional-derivative operators satisfying the discrete maximum principle.
• Derives a key stability estimate (1.2) bounding the solution of a discrete problem in terms of the source term's decay rate, with explicit dependence on mesh grading.
• Applies the barrier function approach to L1-type schemes, defined via piecewise linear reconstruction and convolution quadrature.
• Extends the analysis to Alikhanov-type schemes, which achieve higher-order accuracy in time for fractional subdiffusion problems.
• Employs inverse-monotonicity of the discrete operator matrix to ensure the validity of the barrier function construction.
• Validates theoretical bounds through numerical experiments on initial-value test problems with varying fractional order α and mesh grading parameters.

Experimental results

Research questions

• RQ1 Can a general and simple stability analysis be developed for fractional-derivative parabolic problems on quasi-graded temporal meshes?
• RQ2 What is the sharp pointwise-in-time error behavior of L1 and Alikhanov-type schemes on meshes with arbitrary grading?
• RQ3 Does milder mesh grading (compared to the optimal) still yield optimal convergence rates in positive time?
• RQ4 How do the theoretical error bounds compare to numerical results in terms of sharpness and convergence rates?
• RQ5 Can the barrier function method be systematically applied to various discrete fractional-derivative operators beyond the L1 scheme?

Key findings

• The proposed barrier function approach yields sharp pointwise-in-time error bounds for both L1 and Alikhanov-type schemes on quasi-graded meshes with arbitrary grading degree.
• The analysis shows that milder grading than the optimal (e.g., r < (2−α)/α for L1 or r < (3−α)/α for Alikhanov) still achieves optimal convergence rates in positive time.
• For the L1 method, numerical experiments confirm convergence rates of approximately 1.0 for r=1, 1.4 for r=(2−α)/0.9, and 1.6 for r=(2−α)/α, consistent with theoretical predictions.
• For the Alikhanov method, convergence rates of approximately 2.0 are observed for r=2 and 2.5 for r=(3−α)/0.95, matching the theoretical bound M^{-min{r,3−α}}.
• Maximum nodal errors for the Alikhanov method decrease as M^{-min{αr,3−α}}, confirming the global error bound from Theorem 18 and Remark 20.
• Numerical results demonstrate that the error bounds are sharp, with pointwise error behavior closely matching the theoretical estimate (4.5) across different α and mesh parameters.