A PDE approach to fractional diffusion: a posteriori error analysis

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.