Long time $H^1$-stability of fast L2-1$_\sigma$ method on general nonuniform meshes for subdiffusion equations

In this work, the global-in-time $H^1$-stability of a fast L2-1$_\sigma$ method on general nonuniform meshes is studied for subdiffusion equations, where the convolution kernel in the Caputo fractional derivative is approximated by sum of exponentials. Under some mild restrictions on time stepsize, a bilinear form associated with the fast L2-1$_\sigma$ formula is proved to be positive semidefinite for all time. As a consequence, the uniform global-in-time $H^1$-stability of the fast L2-1$_\sigma$ schemes can be derived for both linear and semilinear subdiffusion equations, in the sense that the $H^1$-norm is uniformly bounded as the time tends to infinity. To the best of our knowledge, this appears to be the first work for the global-in-time $H^1$-stability of fast L2-1$_\sigma$ scheme on general nonuniform meshes for subdiffusion equations. Moreover, the sharp finite time $H^1$-error estimate for the fast L2-1$_\sigma$ schemes is reproved based on more delicate analysis of coefficients where the restriction on time step ratios is relaxed comparing to existing works.