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This work establishes $H^1$-norm stability and convergence for an L2 methodon general nonuniform meshes when applied to the subdiffusion equation. Undermild constraints on the time step ratio $\rho_k$, such as $0.4573328\leq\rho_k\leq 3.5615528$ for $k\geq 2$, the positive semidefiniteness of a crucialbilinear form associated with the L2 fractional-derivative operator is proved.This result enables us to derive long time $H^1$-stability of L2 schemes. Thesepositive semidefiniteness and $H^1$-stability properties hold for standardgraded meshes with grading parameter $1<r\leq 3.2016538$. In addition, erroranalysis in the $H^1$-norm for general nonuniform meshes is provided, andconvergence of order $(5-\alpha)/2$ in $H^1$-norm is proved for modified gradedmeshes when $r>5/\alpha-1$. To the best of our knowledge, this study is thefirst work on $H^1$-norm stability and convergence of L2 methods on generalnonuniform meshes for the subdiffusion equation.

$H^1$-norm stability and convergence of an L2-type method on nonuniform meshes for subdiffusion equation