SHARP POINTWISE-IN-TIME ERROR ESTIMATE OF L1 SCHEME FOR NONLINEAR SUBDIFFUSION EQUATIONS

An essential feature of the subdiffusion equations with the alpha-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 sigma is an element of (0, 1) boolean OR (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 Gro center dot nwall 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 sigma is an element of (0, alpha) boolean OR (alpha, 1) boolean OR (1, 2]. Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.