Finite Elements with weighted bases for the fractional Laplacian

TL;DR

This paper introduces a novel finite element basis for the fractional Laplacian that leverages weighted functions to improve convergence rates in numerical solutions of the Dirichlet problem.

Contribution

A new finite element basis using weighted functions based on distance to boundary, achieving higher convergence rates for fractional Laplacian problems.

Findings

Achieves convergence rate of h^{2-s} under standard smoothness assumptions.

Provides rigorous error analysis with explicit rates.

Numerical experiments validate theoretical improvements.

Abstract

This work presents a numerical study of the Dirichlet problem for the fractional Laplacian $(-\Delta)^s$ with $s\in(0,1)$ using Finite Element methods with non-standard bases. Classical approaches based on piece-wise linear basis yield $h^{\frac 1 2}$ convergence rates in the Sobolev-Slobodeckij norm $H^s$ due to the limited boundary regularity of the solution $u(x)$, which behaves like $\operatorname{dist}(x,\mathbb{R}^d\setminus \Omega)^s$, where $h$ is the diameter of the mesh elements. To overcome this limitation, we propose a novel Finite Element basis of the form $\delta^s \times ($piece-wise linear functions$)$, where $\delta$ is any suitably smooth approximation of $\operatorname{dist}(x,\mathbb{R}^d\setminus \Omega)$. This exploits the improved regularity of $u/\delta^s$, achieving higher convergence rates. Under standard smoothness assumptions the method attains an order $h^{2-s}$ on quasi-uniform meshes, improving the rates with the piece-wise linear basis. We provide a rigorous theoretical error analysis with explicit rates and validate it through numerical experiments.