An implementation of hp-FEM for the fractional Laplacian

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 $O\left(N^{5/2}\right)$, where N is the problem size. Numerical examples 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.