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arXiv · 1811.08059

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

Jincheng Ren·Hong-lin Liao·Jiwei Zhang·Zhimin Zhang
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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.

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Jincheng Ren, Hong-lin Liao, Jiwei Zhang, Zhimin Zhang. 2018-11-20. Sharp $H^1$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems. https://doi.org/10.1016/j.cam.2020.113352

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