A PDE approach to fractional diffusion: a posteriori error analysis

TL;DR

This paper develops a reliable a posteriori error estimator for fractional diffusion problems using PDE techniques, enabling effective adaptive algorithms with demonstrated numerical efficiency.

Contribution

It introduces a new a posteriori error estimator for fractional elliptic problems based on PDE methods, with proven reliability and efficiency.

Findings

Estimator is equivalent to the energy error up to data oscillation.

Adaptive algorithm shows competitive numerical performance.

Estimator relies on small discrete problems on anisotropic cylindrical stars.

Abstract

We derive a computable a posteriori error estimator for the $\alpha$-harmonic extension problem, which localizes the fractional powers of elliptic operators supplemented with Dirichlet boundary conditions. Our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars. It exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation, under suitable assumptions. We design a simple adaptive algorithm and present numerical experiments which reveal a competitive performance.