Local convergence of the FEM for the integral fractional Laplacian

TL;DR

This paper establishes sharp local error estimates for first order finite element discretizations of the integral fractional Laplacian, enhancing understanding of their convergence behavior on subdomains.

Contribution

It provides the first sharp local error estimates in both the local $H^1$-norm and energy norm for discretizations of the integral fractional Laplacian.

Findings

Sharp local error estimates in $H^1$-norm and energy norm.

Error bounds combine local best approximation and global weaker norm errors.

Results applicable to first order discretizations of the fractional Laplacian.

Abstract

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.