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

Abstract

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.