Homogenization and Numerical Upscaling for Spectral Fractional Diffusion

Abstract

We consider two-scale, linear spectral fractional diffusion of order \(2s\in (0,2)\) with homogeneous Dirichlet boundary condition and locally periodic, two-scale coefficients in a bounded domain \(D \subset \mathbb{R}^d\), with fundamental period \(Y=(0,1)^d \subset \mathbb{R}^d\). We derive a local limiting two-scale homogenized equation for the so-called Caffarelli-Sylvestre (CS) extension in the tensorized domain \(D\times Y \times (0,\infty)\subset \mathbb{R}^{2d+1}\), by applying the two-scale convergence approach of Nguetseng and Allaire, to the local, elliptic PDE furnished by the CS extension. Based on the two-scale homogenized equation of the CS extension, we show that the homogenized equation of the non-local two-scale %the two-scale limiting homogenization of the spectral fractional diffusion problem is the spectral fractional diffusion corresponding to the limiting diffusion operator from classical homogenization theory for local, elliptic diffusion in \(D\). We study anisotropic regularity of the solution of the local, limiting two-scale homogenized equation in \(D\times Y \times (0,\infty)\). Using this, we develop the essentially optimal sparse tensor product finite element discretizations using continuous, piecewise linear Langrangean Finite Elements for each, the slow variable \(x'\in D\), the fast variable \(y\in Y\) and the extended variable \(z\in (0,\infty)\). As the solution of the two-scale homogenized equation is analytic with respect to the extended variable \(z\) in weighted Sobolev spaces, we develop a second, likewise essentially optimal approach using the full tensor product of \(hp\) finite element spaces in \(z\in (0,\infty)\) and a sparse tensor product finite element space in \(D\times Y\) using continuous, piecewise linear Langrangean Finite Element basis functions for the variables \(x\) and \(y\). From the finite element solution of this extended two-scale homogenized equation, we construct novel numerical correctors for the two-scale CS extended equation. This results in novel numerical correctors for the solution of the original non-local two-scale spectral fractional diffusion problem. Error estimates in terms of the microscopic scale \(\varepsilon\) and the macroscopic finite element mesh size \(h\) are rigorously derived for these numerical correctors. Numerical experiments confirm the theoretical error estimates of the sparse tensor product finite element schemes.