This work establishes $H^1$-norm stability and convergence for an L2 method on general nonuniform meshes when applied to the subdiffusion equation. Under mild 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 crucial bilinear form associated with the L2 fractional-derivative operator is proved. This result enables us to derive long time $H^1$-stability of L2 schemes. These positive semidefiniteness and $H^1$-stability properties hold for standard graded meshes with grading parameter $1<r\leq 3.2016538$. In addition, error analysis in the $H^1$-norm for general nonuniform meshes is provided, and convergence of order $(5-\alpha)/2$ in $H^1$-norm is proved for modified graded meshes when $r>5/\alpha-1$. To the best of our knowledge, this study is the first work on $H^1$-norm stability and convergence of L2 methods on general nonuniform meshes for the subdiffusion equation.

翻译：摘要：本文建立了在一般非均匀网格上应用L2方法求解次扩散方程时的$H^1$范数稳定性与收敛性。在时间步长比$\rho_k$的温和约束条件下（如当$k\geq 2$时，$0.4573328\leq \rho_k\leq 3.5615528$），证明了与L2分数阶导数算子相关的一个关键双线性形式的正半定性。这一结果使我们能够推导出L2格式的长时间$H^1$稳定性。对于分级参数$1<r\leq 3.2016538$的标准分级网格，这些正半定性和$H^1$稳定性性质同样成立。此外，本文还提供了针对一般非均匀网格的$H^1$范数误差分析，并证明在修正分级网格上当$r>5/\alpha-1$时，$H^1$范数收敛阶为$(5-\alpha)/2$。据我们所知，本研究是首次探讨次扩散方程一般非均匀网格上L2方法$H^1$范数稳定性与收敛性的工作。