Determining superconvergence points for $L2-1_\sigma$ scheme of variable-exponent subdiffusion and error estimate

TL;DR

This paper introduces a cost-effective method for identifying superconvergence points in a variable-exponent subdiffusion scheme, employing graded meshes and proving stability and error bounds with numerical validation.

Contribution

It relaxes the superconvergence point selection criterion without loss of accuracy, reducing computational costs in variable-exponent subdiffusion simulations.

Findings

Established stability and error estimates with a convergence rate of $O(N^{- ext{min}\{r ext{ extdelta},2 ight")

Proved that the new approach maintains accuracy while simplifying superconvergence point determination

Numerical results confirm the theoretical error bounds and stability.

Abstract

We develop a numerical scheme for subdiffusion of variable exponent by combining the $L2-1_\sigma$ temporal discretization with finite element spatial approximation. In existing works, determining the superconvergence points requires solving a nonlinear equation related to the variable exponent at each time step. This work relaxes the selection criterion of superconvergence points without affecting the numerical accuracy, which may reduce the cost of determining superconvergence points. To handle the initial singularity of the solution, we employ a graded temporal mesh. Then we prove the stability and error estimates with a convergence rate $O\left(N^{-\min\{r\delta,2\}}+h^{\mu}\right)$ for the $L2-1_\sigma$ scheme of variable-exponent subdiffusion. Numerical results are performed to substantiate the theoretical findings.