Two-level error estimation for the integral fractional Laplacian

TL;DR

This paper develops an adaptive finite element method with two-level error indicators for the fractional Laplacian, demonstrating linear convergence and optimal algebraic rates in 2D with mesh refinement.

Contribution

It introduces a novel two-level error estimation approach for the fractional Laplacian and proves convergence properties with adaptive mesh refinement.

Findings

Linear convergence in 2D and 3D

Optimal algebraic convergence rates in 2D

Effective adaptive mesh refinement strategy

Abstract

For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement.