\({\boldsymbol{H^1}}\) -Norm Stability and Convergence of an L2-Type Method on Nonuniform Meshes for Subdiffusion Equation
Chaoyu Quan, Xu Wu
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 \(1\lt r\leq 3.2016538\) . In addition, error analysis in the \(H^1\) -norm for general nonuniform meshes is provided, and convergence of order \((5-\alpha )/2\) in the \(H^1\) -norm is proved for modified graded meshes when \(r\gt 5/\alpha -1\) . 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.KeywordsL2-type methodsubdiffusion equationgraded meshpositive semidefiniteness \(H^1\) -norm stability and convergenceMSC codes35R1165M12