Unconditional convergence of a fast two-level linearized algorithm for semilinear subdiffusion equations

TL;DR

This paper introduces a fast, efficient two-level linearized algorithm with unequal time-steps for semilinear subdiffusion equations, achieving significant computational savings and providing rigorous error analysis.

Contribution

The paper develops a novel fast two-level linearized scheme with nonuniform time meshes for semilinear subdiffusion equations, reducing computational complexity and establishing sharp error estimates.

Findings

Computational cost reduced to O(MN log N)

Error estimates reflect solution regularity

Numerical examples confirm effectiveness

Abstract

A fast two-level linearized scheme with unequal time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations because it significantly reduces the computational cost $O(MN^2)$ and storage $O(MN)$ for the standard L1 formula to $O(MN\log N)$ and $O(M\log N)$, respectively, for $M$ grid points in space and $N$ levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time $t=0$, and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a new discrete fractional Gr\"{o}nwall inequality, a global consistency analysis and a discrete $H^2$ energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.