An L1 approximation for a fractional reaction-diffusion equation, a second-order error analysis over time-graded meshes

TL;DR

This paper introduces a second-order accurate L1 time-stepping scheme for a fractional reaction-diffusion equation with Riemann-Liouville derivatives, providing sharp error estimates on nonuniform meshes and combining it with finite element spatial discretization.

Contribution

It is the first to demonstrate second-order accuracy of the L1 scheme on nonuniform meshes for this type of fractional PDE, with comprehensive convergence analysis.

Findings

The L1 scheme achieves second-order accuracy on nonuniform meshes.

Numerical tests confirm the theoretical error estimates.

The mesh assumption can potentially be relaxed based on numerical evidence.

Abstract

A time-stepping L1 scheme for subdiffusion equation with a Riemann--Liouville time-fractional derivative is developed and analyzed. This is the first paper to show that the L1 scheme for the model problem under consideration is second-order accurate (sharp error estimate) over nonuniform time-steps. The established convergence analysis is novel, innovative and concise. For completeness, the L1 scheme is combined with the standard Galerkin finite elements for the spatial discretization, which will then define a fully-discrete numerical scheme. The error analysis for this scheme is also investigated. To support our theoretical contributions, some numerical tests are provided at the end. The considered (typical) numerical example suggests that the imposed time-graded meshes assumption can be further relaxed.