$hp$-Finite Elements for Fractional Diffusion
It provides a more efficient numerical scheme for solving fractional diffusion problems, which are important in various applications such as anomalous transport and image processing.
The paper introduces an hp-finite element method for solving fractional diffusion problems involving the spectral fractional Laplacian, achieving a drastic reduction in computational complexity and slightly improved convergence compared to standard linear finite element discretizations.
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 $hp$-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.