Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

TL;DR

This paper establishes weighted Sobolev regularity for solutions to the obstacle problem involving the integral fractional Laplacian, guiding the development of optimal finite element methods on graded meshes.

Contribution

It provides new weighted Sobolev regularity results for the obstacle problem with the fractional Laplacian, aiding in the design of efficient numerical schemes.

Findings

Weighted Sobolev regularity bounds for solutions

Optimal finite element scheme design on graded meshes

Guidance for numerical approximation accuracy

Abstract

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.