H^1-stability of an L2-type method on general nonuniform meshes for subdiffusion equation

05/12/2022

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by   Chaoyu Quan, et al.

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In this work the H^1-stability of an L2 method on general nonuniform meshes is established for the subdiffusion equation. Under some mild constraints on the time step ratio ρ_k, for example 0.4573328≤ρ_k≤ 3.5615528 for all k≥ 2, a crucial bilinear form associated with the L2 fractional-derivative operator is proved to be positive semidefinite. As a consequence, the H^1-stability of L2 schemes can be derived for the subdiffusion equation. In the special case of graded mesh, such positive semidefiniteness holds when the grading parameter 1<r≤ 3.2016538 and therefore the H^1-stability of L2 schemes holds. To the best of our knowledge, this is the first work on the H^1-stability of L2 method on general nonuniform meshes for subdiffusion equation.