Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian
It provides a theoretical guarantee of optimal convergence for adaptive methods solving fractional PDEs, addressing a known bottleneck in the regime 3/4 < s < 1.
The paper presents a weighted residual a posteriori error estimator for the integral fractional Laplacian discretized with piecewise linear functions, and proves optimal convergence rates for an h-adaptive algorithm driven by this estimator.
For the discretization of the integral fractional Laplacian $(-Δ)^s$, $0 < s < 1$, 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 $3/4 < s < 1$, 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.