Mathematics > Numerical Analysis
[Submitted on 12 May 2022 (v1), last revised 22 May 2023 (this version, v3)]
Title:$H^1$-norm stability and convergence of an L2-type method on nonuniform meshes for subdiffusion equation
View PDFAbstract: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.
Submission history
From: Chaoyu Quan [view email][v1] Thu, 12 May 2022 12:50:49 UTC (635 KB)
[v2] Thu, 30 Jun 2022 12:19:12 UTC (693 KB)
[v3] Mon, 22 May 2023 14:33:23 UTC (821 KB)
Current browse context:
math.NA
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.