Pointwise-in-time a posteriori error control for time-fractional parabolic equations

TL;DR

This paper develops pointwise-in-time a posteriori error bounds for time-fractional parabolic equations, enabling adaptive mesh refinement that achieves optimal convergence rates despite solution singularities.

Contribution

It introduces a novel a posteriori error estimation technique for time-fractional parabolic equations and applies it to improve the L1 method with adaptive mesh refinement.

Findings

Achieves optimal convergence rate of 2−α with adaptive mesh

Provides pointwise-in-time error bounds in L2 and L∞ norms

Enhances the L1 method for fractional parabolic equations

Abstract

For time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$, we give pointwise-in-time a posteriori error bounds in the spatial $L_2$ and $L_\infty$ norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates $2-\alpha$ in the presence of solution singularities.