Numerical Approximation of the Integral Fractional Laplacian

TL;DR

This paper introduces a novel finite element algorithm for approximating solutions to elliptic problems involving the fractional Laplacian, combining integral representation, sinc quadrature, domain truncation, and finite element methods.

Contribution

It presents a new nonconforming finite element approach with detailed error analysis for fractional Laplacian problems, integrating multiple numerical techniques.

Findings

The proposed method accurately approximates the fractional Laplacian.

Error behavior depends on mesh size, truncation, and quadrature parameters.

Numerical results validate the effectiveness of the algorithm.

Abstract

We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (i) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (ii) truncate each elliptic problem to a bounded domain, (iii) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.