Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian

For the discretization of the integral fractional Laplacian $(-\mathrm{\Delta }{)}^s$, $0<s<1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for a lack of $L^2$-regularity of the residual in the regime $3\mathrm{/}4<s<1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an $h$-adaptive algorithm driven by this error estimator. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.