Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

TL;DR

This paper develops an efficient finite element method for the integral fractional Laplacian on polygonal domains, leveraging boundary integral operators to handle dense matrices and improve computational performance.

Contribution

It introduces a novel approach that generalizes boundary integral techniques to fractional orders without domain shape restrictions or special matrix structures.

Findings

Effective approximation of the fractional Laplacian on polygonal domains.

Sparse matrix assembly and storage techniques improve computational efficiency.

Numerical examples demonstrate the method's flexibility and performance.

Abstract

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples.