Sharp $H^1$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems
This work provides a rigorous theoretical framework for achieving optimal convergence rates in numerical methods for fractional subdiffusion equations, benefiting researchers in numerical analysis and computational mathematics.
The authors addressed the suboptimal time accuracy in $H^1$-norm error estimates for fractional subdiffusion problems caused by initial singularity and discrete convolution. They proposed an improved discrete Grönwall inequality and used a time-space error-splitting technique to achieve sharp $H^1$-norm error estimates for the L1 and fractional Crank-Nicolson schemes, confirmed by numerical experiments.
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.