Finite Elements with weighted bases for the fractional Laplacian

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 $\mathrm{dist}(x,{\mathbb{R}}^d\setminus \Omega {)}^s$, 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 $\mathrm{dist}(x,{\mathbb{R}}^d\setminus \Omega )$. 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.