A fractional Laplace equation: regularity of solutions and Finite Element approximations

TL;DR

This paper investigates the regularity of solutions to the fractional Laplace equation and establishes optimal finite element approximation rates, supported by theoretical analysis and numerical experiments.

Contribution

It provides new regularity estimates for the fractional Laplace equation and proves optimal finite element convergence rates for both uniform and graded meshes.

Findings

Regularity estimates depend on the Hölder regularity of data.

Optimal convergence rates are achieved with standard linear finite elements.

Numerical results confirm theoretical convergence predictions.

Abstract

This paper deals with the \emph{integral} version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the H\"older regularity of the data. By relying on these results, optimal order of convergence for the standard linear finite element method is proved for quasi-uniform as well as graded meshes. Some numerical examples are given showing results in agreement with the theoretical predictions.