Numerical Homogenization of Heterogeneous Fractional Laplacians

TL;DR

This paper introduces a multiscale numerical method for efficiently solving heterogeneous fractional Laplacian problems by localizing the nonlocal operator and employing advanced quasi-interpolation techniques, achieving optimal convergence.

Contribution

The paper develops a localized multiscale scheme for heterogeneous fractional Laplacians using a novel projective quasi-interpolation operator with proven stability and approximation properties.

Findings

Achieves optimal convergence rates for heterogeneous fractional Laplacian problems.

Demonstrates effectiveness of the method through numerical experiments in 2+1 dimensions.

Provides a computationally efficient scheme by truncating basis function corrections.

Abstract

In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form, however the Caffarelli-Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a sub-grid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is of main interest, we construct a projective quasi-interpolation that has both $d$ and $d+1$ dimensional averages over subsets in the spirit of the Scott-Zhang operator. We show that this operator satisfies local stability and local approximation properties in weighted Sobolev spaces. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, and utilizing a global $L^2$ projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension $2+1$ for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method.