A Semi-Analytic Diagonalization FEM for the Spectral Fractional Laplacian

TL;DR

This paper introduces a semi-analytic finite element method for efficiently approximating solutions to the spectral fractional Laplacian, leveraging eigenvalue problem solutions and quadrature schemes.

Contribution

It presents a novel semi-analytic diagonalization FEM approach that combines eigenvalue solutions with quadrature for spectral fractional Laplacian approximation.

Findings

Method achieves accurate approximations in numerical tests.

The approach relates to and improves upon existing quadrature schemes.

Numerical examples demonstrate the method's effectiveness.

Abstract

We present a technique for approximating solutions to the spectral fractional Laplacian, which is based on the Caffarelli-Silvestre extension and diagonalization. Our scheme uses the analytic solution to the associated eigenvalue problem in the extended dimension. We show its relation to a quadrature scheme. Numerical examples demonstrate the performance of the method.