A posteriori error analysis for approximations of time-fractional subdiffusion problems

TL;DR

This paper develops new a posteriori error estimates for time-fractional subdiffusion problems using L1 and Convolution Quadrature schemes, enabling adaptive mesh refinement and improved solution accuracy.

Contribution

It introduces novel interpretations of numerical schemes as perturbed evolution equations, leading to optimal-order a posteriori error estimates for fractional subdiffusion problems.

Findings

Error estimators are reliable and optimal in $L^2(H)$ and $L^ abla(H)$ norms.

Numerical experiments confirm the effectiveness of the estimators for adaptive mesh refinement.

The approach successfully locates singularities and improves approximation accuracy.

Abstract

In this paper we consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpretations of the numerical schemes which lead to a posteriori error estimates. Our approach is based on appropriate pointwise representations of the numerical schemes as perturbed evolution equations and on stability estimates for the evolution equation. A posteriori error estimates in $L^2(H)$ and $L^\infty (H)$ norms of optimal order are derived. Extensive numerical experiments indicate the reliability and the optimality of the estimators for the schemes considered, as well as their efficiency as error indicators driving adaptive mesh selection locating singularities of the problem.