QUASI-OPTIMAL CONVERGENCE RATE FOR AN ADAPTIVE METHOD FOR THE INTEGRAL FRACTIONAL LAPLACIAN

For the discretization of the integral fractional Laplacian (-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 the 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.

Automated summary

This study introduces a reliable weighted residual a posteriori error estimator for the discretization of the integral fractional Laplacian, and proves optimal convergence rates for an h-adaptive algorithm driven by this error estimator. The key lies in the local inverse estimates for the fractional Laplacian, which are crucial for the analysis of the adaptive algorithm.