An implementation of hp-FEM for the fractional Laplacian

TL;DR

This work develops quadrature-augmented hp-FEM for the 1D integral Dirichlet fractional Laplacian, proving that root-exponential convergence is preserved when the quadrature uses at least $n\ge p+1$ points per integral, and it analyzes the resulting $\mathcal{O}(N^{5/2})$ assembly cost on geometric meshes. The authors design and analyze Gauss-Legendre and Gauss-Jacobi quadrature schemes for the singular bilinear form and analytic load, leveraging a regularizing transformation to align singularities with coordinate axes and applying the First Strang Lemma to bound errors. They provide detailed consistency-error estimates for both the linear and bilinear forms, prove uniform coercivity under a quadrature refinement condition, and demonstrate exponential convergence in numerical experiments. The paper also sketches a rigorous extension to higher dimensions on shape-regular meshes, using Chernov–Schwab-type quadrature on simplex pairs and discussing the need for anisotropic, boundary-concentrated meshes for similar rates, thereby laying groundwork for practical nonlocal PDE solvers in multiple dimensions.

Abstract

We consider the discretization of the $1d$-integral Dirichlet fractional Laplacian by $hp$-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of $hp$-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial degree is enough to preserve root exponential convergence. The total number of algebraic operations to set up the system is $\mathcal{O}(N^{5/2})$, where $N$ is the problem size. Numerical example illustrate the analysis. We also extend our analysis to the fractional Laplacian in higher dimensions for $hp$-finite element spaces based on shape regular meshes.

Figures (4)

• Figure 1: Three different methods (see Example \ref{['example_energy_error_approx']}) to calculate the energy norm error of $hp$-FEM with $n = \mathcal{O}(p)$ quadrature points on a geometric mesh with grading factor $\sigma = 0.172$, polynomial degree $p = L$, $s = 3/4$.
• Figure 2: Exponential convergence in the energy norm (approximation (\ref{['energy_norm_approx_triangle']}) with $m = 6p$) of $hp$-FEM on geometric mesh with grading factor $\sigma = 0.25$, polynomial degree $p = L$, $n:= \lfloor1.2 \; p \rfloor$ quadrature points and different fractional parameters $s$.
• Figure 3: Exponential convergence in the energy norm (approximation (\ref{['energy_norm_approx_triangle']}) with $m = 6p$) of $hp$-FEM with $n = \mathcal{O}(p)$ quadrature points on geometric mesh, polynomial degree $p = L$, $s = 3/4$. Left: grading factor $\sigma = 0.172$. Right: grading factor $\sigma = 0.5$.
• Figure 4: Exponential convergence of the elementwise contributions $| I_{T,T'}(v,w) - Q^n_{T,T'}(v,w) |$ for the integrated Legendre polynomials (\ref{['def_int_legendre_pol']}) on geometric meshes $\mathcal{T}^{L}_{geo,\sigma}$ with $L=2$ layers and different grading parameters $\sigma$. Left: adjacent elements. Right: separated elements.

Theorems & Definitions (50)

• Definition 2.1: Shape regular meshes and spline spaces
• Definition 2.2: Geometric mesh ${\mathcal{T}}^{L}_{geo,\sigma}$ and basis ${\mathcal{B}}^{geo}$ of the spline space ${\color{black} S^{p,1}_0}({\mathcal{T}}^{L}_{geo,\sigma})$
• Proposition 2.3: bahr2023exponential
• Theorem 2.4: Exponential convergence including quadrature
• Remark 3.1
• Lemma 3.2
• proof
• Remark 3.3
• Corollary 3.4
• proof