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

TL;DR

This paper develops sharp $H^1$-norm error estimates for two time-stepping schemes applied to reaction-subdiffusion problems, overcoming traditional suboptimal bounds caused by initial singularities and discrete convolution effects.

Contribution

It introduces an improved discrete Grönwall inequality and applies it to L1 and fractional Crank-Nicolson schemes for better error analysis in reaction-subdiffusion problems.

Findings

Established sharp $H^1$-norm error estimates for the schemes.

Validated theoretical results with numerical experiments.

Achieved optimal temporal accuracy despite initial singularities.

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 Gr\"{o}nwall 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.