Global-in-time $H^1$-stability of L2-1$_\sigma$ method on general nonuniform meshes for subdiffusion equation

TL;DR

This paper establishes the global-in-time $H^1$-stability and sharp $L^2$-norm convergence of the L2-1$_\sigma$ method on nonuniform meshes for subdiffusion equations, under specific time step ratio conditions.

Contribution

It proves the positive semidefiniteness of the bilinear form and derives stability and convergence results for the L2-1$_\sigma$ scheme on general nonuniform meshes.

Findings

Global-in-time $H^1$-stability is established.

Sharp $L^2$-norm convergence is proved.

Stability holds when the time step ratio is at least 0.475329.

Abstract

In this work the L2-1$_\sigma$ method on general nonuniform meshes is studied for the subdiffusion equation. When the time step ratio is no less than $0.475329$, a bilinear form associated with the L2-1$_\sigma$ fractional-derivative operator is proved to be positive semidefinite and a new global-in-time $H^1$-stability of L2-1$_\sigma$ schemes is then derived under simple assumptions on the initial condition and the source term. In addition, the sharp $L^2$-norm convergence is proved under the constraint that the time step ratio is no less than $0.475329$.