On stability and convergence of L2-1$_σ$ method on general nonuniform meshes for subdiffusion equation

In this work the L2-1$_\sigma$ method on general nonuniform meshes is studied for the subdiffusion equation. Under some constraints on the time step ratio $\rho_k$, for example $\rho_k\geq 0.475329$ for all $k\geq 2$, a crucial bilinear form associated with the L2-1$_\sigma$ fractional-derivative operator is proved to be positive semidefinite and the $H^1$-stability of L2-1$_\sigma$ schemes is then derived for all time under simple assumptions on the initial condition and the source term. In addition, we prove the sharp convergence when $\rho_k\geq 0.475329$, which reduces the restriction $\rho_k\geq 4/7$ proposed by Liao, McLean and Zhang in [SIAM J. Numer. Anal. 57 (2019), no. 1, 218-237].