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

## TL;DR

This paper develops and analyzes an adaptive finite element method with a novel weighted residual error estimator for the integral fractional Laplacian, achieving quasi-optimal convergence rates even in challenging regularity regimes.

## Contribution

It introduces a weighted residual a posteriori error estimator for the fractional Laplacian and proves its effectiveness in driving an adaptive algorithm with optimal convergence rates.

## Key findings

- The error estimator is reliable and effective for $0<s<1$.
- The adaptive algorithm achieves quasi-optimal convergence rates.
- The analysis includes new local inverse estimates for the fractional Laplacian.

## Abstract

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 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.