Finite Elements with weighted bases for the fractional Laplacian

Abstract

This work presents a numerical study of the Dirichlet problem for the fractional Laplacian $(-Δ)^s$ 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 $\operatorname{dist}(x,\mathbb{R}^d\setminus Ω)^s$, where $h$ is the diameter of the mesh elements. To overcome this limitation, we propose a novel Finite Element basis of the form $δ^s \times ($piece-wise linear functions$)$, where $δ$ is any suitably smooth approximation of $\operatorname{dist}(x,\mathbb{R}^d\setminus Ω)$. This exploits the improved regularity of $u/δ^s$, 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.