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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 isapproximated by sum of exponentials. Under some mild restrictions on timestepsize, a bilinear form associated with the fast L2-1$_\sigma$ formula isproved to be positive semidefinite for all time. As a consequence, the uniformglobal-in-time $H^1$-stability of the fast L2-1$_\sigma$ schemes can be derivedfor 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 ofour 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 forsubdiffusion equations. Moreover, the sharp finite time $H^1$-error estimatefor the fast L2-1$_\sigma$ schemes is reproved based on more delicate analysisof coefficients where the restriction on time step ratios is relaxed comparingto existing works.

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