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

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 L² 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≤ ρₖ≤ 1.16, k≥2, 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 L² error analysis. For solutions exhibiting the standard initial weak singularity, we derive an L² 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 O\!(N^-minα,\,3-α\), and hence recovers the optimal order O(N-(3-α)) whenever r≥(3-α)/α. Numerical experiments on random nonmonotone meshes and modified graded meshes confirm the theoretical estimates.