Finite element approximation of fractional Neumann problems

TL;DR

This paper develops and analyzes finite element methods for solving Neumann problems involving the fractional Laplacian, providing theoretical insights and numerical validation of the approach.

Contribution

It introduces a finite element approximation scheme for fractional Neumann problems, analyzing convergence, well-posedness, and implementation details.

Findings

Convergence of the finite element discretization is established.

Numerical experiments demonstrate the method's effectiveness.

Properties of solutions are illustrated through simulations.

Abstract

In this paper we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation of such problems, including their well-posedness and asymptotic behavior of solutions. We address the convergence of the finite element discretizations and discuss the implementation of the method. Finally, we present several numerical experiments in one- and two-dimensional domains that illustrate the method's performance as well as certain properties of solutions.