A short FE implementation for a 2d homogeneous Dirichlet problem of a Fractional Laplacian

TL;DR

This paper provides a simple, efficient MATLAB finite element implementation for solving 2D homogeneous Dirichlet problems involving the fractional Laplacian, facilitating numerical analysis of nonlocal phenomena.

Contribution

It offers a concise, adaptable MATLAB code for 2D fractional Laplacian problems, filling a gap in accessible computational tools for researchers.

Findings

The code efficiently solves 2D fractional Laplacian problems.

It is easily modifiable for other kernels and time-dependent problems.

Provides a practical tool for scientists studying nonlocal phenomena.

Abstract

In \cite{AcostaBorthagaray}, a complete $n$-dimensional finite element analysis of the homogeneous Dirichlet problem associated to a fractional Laplacian was presented. Here we provide a comprehensive and simple 2D {\it MATLAB}\textsuperscript{\textregistered} finite element code for such a problem. The code is accompanied with a basic discussion of the theory relevant in the context. The main program is written in about 80 lines and can be easily modified to deal with other kernels as well as with time dependent problems. The present work fills a gap by providing an input for a large number of mathematicians and scientists interested in numerical approximations of solutions of a large variety of problems involving nonlocal phenomena in two-dimensional space.