An implementation of hp-FEM for the fractional Laplacian

TL;DR

This paper develops and analyzes hp-FEM discretization schemes for the 1D fractional Laplacian, demonstrating exponential convergence with efficient quadrature rules and extending the approach to higher dimensions.

Contribution

The paper introduces quadrature schemes that preserve exponential convergence of hp-FEM for the fractional Laplacian and extends the analysis to higher-dimensional cases.

Findings

Quadrature schemes based on Gauss-Jacobi and Gauss-Legendre rules preserve exponential convergence.

Total algebraic complexity for system setup is O(N^{5/2}).

Numerical examples confirm theoretical convergence rates.

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.