Finite element approximations of the nonhomogeneous fractional Dirichlet problem

TL;DR

This paper develops finite element methods for solving the nonhomogeneous fractional Dirichlet problem, using weak boundary conditions and nonlocal derivatives, with proven convergence rates based on regularity estimates.

Contribution

It introduces a novel finite element approach that incorporates nonlocal derivatives as Lagrange multipliers for fractional Laplacian problems.

Findings

Achieved convergence orders for the proposed scheme.

Developed regularity estimates for solutions and nonlocal derivatives.

Implemented a method requiring domain expansion with mesh refinement.

Abstract

We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative as a Lagrange multiplier in the formulation of the problem. In order to obtain convergence orders for our scheme, regularity estimates are developed, both for the solution and its nonlocal derivative. The method we propose requires that, as meshes are refined, the discrete problems be solved in a family of domains of growing diameter.