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Finite Elements with weighted bases for the fractional Laplacian

Félix del Teso, Stefano Fronzoni, David Gómez-Castro

TL;DR

The authors address the Dirichlet problem for the fractional Laplacian $(-\Delta)^s$ with $s\in(0,1)$ by introducing a weighted finite element basis $\delta^s\times$(piecewise linear) to exploit enhanced regularity of $u/\delta^s$ near the boundary. They prove $H^s$-convergence rates up to $h^{2-s}$ (with a $|\log h|^{1/2}$ factor) under suitable smoothness of the right-hand side and domain, and obtain $L^2$-rates via an Aubin–Nitsche argument, under stronger regularity assumptions. The method relies on over-triangulations, a regularized distance $\delta$, and a Céa-type bound with a tailored interpolant, achieving significantly improved performance over standard $PL$ bases for fractional problems. Numerical experiments in 1D corroborate the theoretical rates and illustrate the boundary-accurate behavior of the weighted basis. Overall, the work provides a rigorous, implementable framework for high-accuracy FEM approximations of nonlocal Dirichlet problems in bounded domains, with potential extensions to broader nonlocal operators.

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.

Finite Elements with weighted bases for the fractional Laplacian

TL;DR

The authors address the Dirichlet problem for the fractional Laplacian with by introducing a weighted finite element basis (piecewise linear) to exploit enhanced regularity of near the boundary. They prove -convergence rates up to (with a factor) under suitable smoothness of the right-hand side and domain, and obtain -rates via an Aubin–Nitsche argument, under stronger regularity assumptions. The method relies on over-triangulations, a regularized distance , and a Céa-type bound with a tailored interpolant, achieving significantly improved performance over standard bases for fractional problems. Numerical experiments in 1D corroborate the theoretical rates and illustrate the boundary-accurate behavior of the weighted basis. Overall, the work provides a rigorous, implementable framework for high-accuracy FEM approximations of nonlocal Dirichlet problems in bounded domains, with potential extensions to broader nonlocal operators.

Abstract

This work presents a numerical study of the Dirichlet problem for the fractional Laplacian with using Finite Element methods with non-standard bases. Classical approaches based on piece-wise linear basis yield convergence rates in the Sobolev-Slobodeckij norm due to the limited boundary regularity of the solution , which behaves like , where is the diameter of the mesh elements. To overcome this limitation, we propose a novel Finite Element basis of the form piece-wise linear functions, where is any suitably smooth approximation of . This exploits the improved regularity of , achieving higher convergence rates. Under standard smoothness assumptions the method attains an order 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.

Paper Structure

This paper contains 31 sections, 10 theorems, 89 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and functional setting
  3. Sobolev and Hölder spaces.
  4. Weak formulation of the fractional Dirichlet problem.
  5. Regularity of the domain and regularized distance functions.
  6. Generic regularity of the solution to the fractional Dirichlet problem.
  7. Over-triangulations.
  8. Weighted elements.
  9. WFEM of the fractional Dirichlet problem.
  10. Main results
  11. Convergence rates in Hs.
  12. L2-rates via Aubin--Nitsche duality
  13. Comments on related literature
  14. Proofs of the main results
  15. Proof of convergence rates in Hs
  16. ...and 16 more sections

Key Result

Lemma 1.7

Let $s\in(0,1)$, $\partial\Omega\in C^{1,1}$ and $\delta$ satisfy as:delta with $\sigma=1$. Then $V_h \subset V$.

Figures (5)

  • Figure 1: Example of over-triangulation.
  • Figure 2: FEM vs. WFEM basis function on $\Omega=(-1,1)$ for $s=0.2$ and $h=0.25$
  • Figure 3: Comparison between explicit solution \ref{['eq:explicitSol']}, piece-wise linear interpolation $I_h u^\ast$ and $J_h u^\ast$, defined in \ref{['eq:competitor']}, for different choices of the function $\delta$. $\Omega=\Omega_h=(-1,1)$, uniform meshes with $h=|\Omega|/2^4$.
  • Figure 4: WFEM computational errors. $\Omega = \Omega_h = (-1,1)$, $f=1$, and $\delta(x)=1-|x|^4$.
  • Figure 5: WFEM computational errors. $\Omega = (-1+\varepsilon,1-\varepsilon)$, $\varepsilon=10^{-10}$, $f$ given by \ref{['eq:RHS parabola']}, $\delta(x)=1-|x|^4$, and $u^* = (1-|x|^2)_+$.

Theorems & Definitions (28)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3: Explicit solution for $f = 1$ and $\Omega$ a ball
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 1.7
  • Remark 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 18 more