Finite Elements with weighted bases for the fractional Laplacian

2025/11/03 by del Teso, Félix, Fronzoni, Stefano, Gómez-Castro, David
#35K55 #35R11 #65N30 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2511.01727

Abstract

This work presents a numerical study of the Dirichlet problem for the fractional Laplacian (-Δ)^s with s∈(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 dist(x,ℝ^d∖ 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 × (piece-wise linear functions), where δ is any suitably smooth approximation of dist(x,ℝ^d∖ 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.

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• Weighted Sobolev regularity and rate of approximation of the obstacle\n problem for the integral fractional Laplacian