# $hp$-Finite Elements for Fractional Diffusion

## TL;DR

This paper introduces an $hp$-finite element numerical scheme for efficiently solving boundary value problems involving the spectral fractional Laplacian, reducing computational complexity and improving convergence.

## Contribution

It presents a novel $hp$-finite element discretization approach on a truncated semi-infinite cylinder for spectral fractional Laplacian problems, enhancing efficiency and accuracy.

## Key findings

- Significant reduction in degrees of freedom needed for computation
- Slightly improved convergence properties over linear finite element methods
- Validated effectiveness through numerical experiments

## 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 $hp$-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.