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arXiv · 2407.11482

An implementation of hp-FEM for the fractional Laplacian

Björn Bahr·Markus Faustmann·Jens Markus Melenk
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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.

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Björn Bahr, Markus Faustmann, Jens Markus Melenk. 2024-07-16. An implementation of hp-FEM for the fractional Laplacian. https://arxiv.org/abs/2407.11482

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