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

TL;DR

This paper analyzes the pointwise convergence of an Alikhanov scheme applied to a 2D nonlinear subdiffusion equation with fractional time derivatives, providing stability results and validating convergence rates through numerical experiments.

Contribution

It introduces a new stability result for the Alikhanov scheme on quasi-graded meshes and establishes convergence orders for nonlinear 2D subdiffusion equations.

Findings

Global L^2-norm convergence order is min{α r, 2}.

Local L^2-norm convergence order is min{r, 2}.

Numerical experiments confirm theoretical convergence rates.

Abstract

In this paper, we discretize the Caputo time derivative of order \alpha \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{\alpha r, 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.