Coefficient Recombination and Error Estimates for the L2 Method for Subdiffusion on Nonuniform Meshes

Abstract

The L2 discretization of the Caputo derivative is attractive for its high-order accuracy, but its analysis on nonuniform temporal meshes is complicated due to the lack of a monotone structure in the discrete coefficients. This difficulty is particularly significant for deriving sharp $L^2$ error estimates in the presence of the typical weak singularity of subdiffusion solutions near the initial time. In this work, we develop a coefficient-recombination framework for the L2 method on nonuniform meshes. Under the adjacent step-ratio condition $0.86\le ρ_k\le 1.16$, $k\ge2$, we prove that the recombined L2 coefficients are positive and monotone. A key ingredient is a cross-row recombination technique: with the recombination parameter associated with a fixed time level, all coefficient rows retain positivity and monotonicity, which allows us to close an induction for the $L^2$ error analysis. For solutions exhibiting the standard initial weak singularity, we derive an $L^2$ error estimate on nonuniform meshes satisfying the step-ratio condition. On a modified graded mesh with grading parameter $r>1$, the estimate yields the convergence rate $\mathcal{O}\!\left(N^{-\min\{rα,\,3-α\}}\right)$, and hence recovers the optimal order $\mathcal{O}(N^{-(3-α)})$ whenever $r\ge(3-α)/α$. Numerical experiments on random nonmonotone meshes and modified graded meshes confirm the theoretical estimates.