An implementation of hp-FEM for the fractional Laplacian
<|reference_start|>An implementation of hp-FEM for the fractional Laplacian: 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.<|reference_end|>
@article{bahr2024an,
    title={An implementation of hp-FEM for the fractional Laplacian},
    author={Bj"orn Bahr and Markus Faustmann and Jens Markus Melenk},
    journal={arXiv preprint arXiv:2407.11482},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.11482},
    primaryClass={math.NA cs.NA}
}