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

Chaoyu Quan, Xu Wu|arXiv (Cornell University)|May 12, 2022

Differential Equations and Numerical Methods4 citations

TL;DR

This paper establishes H¹-norm stability and convergence for an L2-type method on general nonuniform meshes for the subdiffusion equation. Under mild time step ratio constraints (0.457 ≤ ρₖ ≤ 3.562), it proves the positive semidefiniteness of a key bilinear form, enabling long-time H¹-stability and convergence of order (5−α)/2 in H¹-norm for modified graded meshes with r > 5/α − 1.

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 $ρ_k$, such as $0.4573328\leq ρ_k\leq 3.5615528$ 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 $15/α-1$. 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.

Motivation & Objective

• To establish H¹-norm stability and convergence for L2-type schemes on general nonuniform time meshes for the subdiffusion equation.
• To address the lack of H¹-stability analysis for L2 methods on nonuniform meshes, especially under low-regularity solutions.
• To prove the positive semidefiniteness of the bilinear form associated with the L2 fractional-derivative operator under mild time step ratio constraints.
• To extend stability and convergence results to graded and modified graded meshes with specific grading parameters.
• To provide rigorous H¹-norm error analysis and confirm convergence rates numerically.

Proposed method

• Proves positive semidefiniteness of the bilinear form $\mathcal{B}_n(v,w) = \sum_{k=1}^n \langle L_k^\alpha v, \delta_k w \rangle$ under time step ratio constraints $\rho_k \in [0.4573328, 3.5615528]$ for $k \geq 2$.
• Uses a novel analysis framework based on discrete fractional Grönwall inequalities and energy techniques to derive long-time H¹-stability.
• Applies the analysis to standard graded meshes with grading parameter $r \leq 3.2016538$ to establish H¹-stability.
• Derives H¹-norm error estimates for general nonuniform meshes and proves convergence of order $(5 - \alpha)/2$ for modified graded meshes when $r > 5/\alpha - 1$.
• Employs spectral collocation in space with Chebyshev–Gauss–Lobatto points to achieve high spatial accuracy.
• Validates theoretical findings via numerical experiments on modified graded meshes with varying $\alpha$, $r$, and $N$, measuring maximum H¹-errors.

Experimental results

Research questions

• RQ1 Is the bilinear form associated with the L2 fractional-derivative operator positive semidefinite on general nonuniform meshes under mild time step ratio constraints?
• RQ2 Can long-time H¹-stability be established for the implicit L2 scheme on nonuniform meshes for the subdiffusion equation?
• RQ3 What is the optimal convergence rate in the H¹-norm for the L2 method on modified graded meshes, and how does it depend on the grading parameter $r$?
• RQ4 Does the positive semidefiniteness result for the bilinear form remain sharp under numerical testing?
• RQ5 Can the theoretical H¹-norm convergence rate of $(5 - \alpha)/2$ be observed in practice, and does it align with observed convergence orders?

Key findings

• The bilinear form $\mathcal{B}_n$ is proven positive semidefinite under the mild constraint $0.4573328 \leq \rho_k \leq 3.5615528$ for $k \geq 2$, enabling H¹-stability analysis.
• Long-time H¹-stability of the L2 scheme is established for the subdiffusion equation with homogeneous Dirichlet boundary conditions on general nonuniform meshes.
• For standard graded meshes, H¹-stability holds when the grading parameter satisfies $1 < r \leq 3.2016538$
• Convergence of order $(5 - \alpha)/2$ in the H¹-norm is proved for modified graded meshes when $r > 5/\alpha - 1$
• Numerical results confirm that the observed convergence orders in the H¹-norm are approximately $\min\{r\alpha, 3 - \alpha\}$, consistent with theoretical expectations.
• The positive definiteness bound $\lambda_{\min} \geq C N^{\alpha}$ is numerically verified to be optimal, with $\lambda_{\max} \geq C N^{r\alpha}$