Approximation of the spectral fractional powers of the Laplace-Beltrami Operator

TL;DR

This paper develops and analyzes numerical algorithms for approximating spectral fractional Laplace-Beltrami problems on surfaces, achieving near-optimal convergence rates using integral representations, quadrature, and finite element methods.

Contribution

It introduces a new numerical approach combining Balakrishnan integral representation, sinc quadrature, and finite element methods for fractional Laplace-Beltrami operators on surfaces.

Findings

Optimal convergence rates observed and derived analytically.

Algorithms perform well in approximating Gaussian fields on surfaces.

Logarithmic factors may appear in convergence estimates.

Abstract

We consider numerical approximations of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consist of a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rates of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in $L^2$ and $H^1$. The performances of the algorithms are illustrated in different settings including the approximation of Gaussian fields on surfaces.