An implementation of hp-FEM for the fractional Laplacian

2024/07/16 by Björn Bahr, Bahr, Björn, Markus Faustmann +3 · 1 citation
Engineering · #Aluminum Alloys Composites Properties #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2407.11482

openalex publication_date 2024/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

Cited by

• A note on spectral Monte-Carlo method for fractional Poisson equation on high-dimensional ball

Related

• Finite Elements with weighted bases for the fractional Laplacian
• A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations
• On explicit form of the FEM stiffness matrix for the integral fractional Laplacian on non-uniform meshes
• Linear and Nonlinear Fractional Elliptic Problems
• FEM for 1D-problems involving the logarithmic Laplacian: error estimates and numerical implementation