Pointwise-in-time convergence analysis of an Alikhanov scheme for a 2D nonlinear subdiffusion equation

Abstract: In this paper, we discretize the Caputo time derivative of order α\in (0,1) using the Alikhanov scheme on a quasi-graded temporal mesh, and employ the Newton linearization method to approximate the nonlinear term. This yields a linearized fully discrete scheme for the two-dimensional nonlinear time fractional subdiffusion equation with weakly singular solutions. For the purpose of conducting a pointwise convergence analysis using the comparison principle, we develop a new stability result. The global L^2-norm convergence order is min{αr, 2}, and the local L^2-norm convergence order is min{r, 2} under appropriate conditions and assumptions. Ultimately, the rates of convergence demonstrated by the numerical experiments serve to validate the analytical outcomes.