Sharp Pointwise-in-Time Error Estimate of L1 Scheme for Nonlinear Subdiffusion Equations

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 $σ ∈ (0, 1) ∪ (1, 2).$ Under this general regularity assumption, we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results, i.e., a refined discrete fractional-type Grönwall inequality (DFGI). After that, we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. The present results fill the gap on some interesting convergence results of L1 scheme on $σ ∈ (0, α) ∪ (α, 1) ∪ (1, 2].$ Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.