$hp$-Finite Elements for Fractional Diffusion
Abstract
The purpose of this work is to introduce and analyze a numerical scheme to efficiently solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a semi-infinite cylinder in one more spatial dimension. After a suitable truncation of this cylinder, the resulting problem is discretized with linear finite elements in the original domain and with -finite elements in the extended direction. The proposed approach yields a drastic reduction of the computational complexity in terms of degrees of freedom and even has slightly improved convergence properties compared to a discretization using linear finite elements for both the original domain and the extended direction. The performance of the method is illustrated by numerical experiments.
Cite
@article{arxiv.1706.04066,
title = {$hp$-Finite Elements for Fractional Diffusion},
author = {Dominik Meidner and Johannes Pfefferer and Klemens Schürholz and Boris Vexler},
journal= {arXiv preprint arXiv:1706.04066},
year = {2018}
}