Homogenization and Numerical Upscaling for Spectral Fractional Diffusion
Viet Ha Hoang, Chen Hui Pang, Christoph Schwab · Multiscale Modeling and Simulation · 2026
Abstract. We consider two-scale, linear spectral fractional diffusion of order [Formula: see text] with homogeneous Dirichlet boundary condition and locally periodic, two-scale coefficients in a bounded domain [Formula: see text], with fundamental period [Formula: see text]. We derive a local limiting two-scale homogenized equation for the so-called Caffarelli–Silvestre (CS) extension (we refer to the corresponding boundary value problem simply as the “extended boundary value problem” or “extended BVP”) in the tensorized domain [Formula: see text], by applying the two-scale convergence approach of Nguetseng [ SIAM J. Math. Anal., 20 (1989), pp. 608–623] and Allaire [ SIAM J. Math. Anal., 23 (1992), pp. 1482–1518], to the local, elliptic PDE in the extended BVP. Based on the two-scale homogenized equation of the CS extension, we show that the homogenized equation of the nonlocal two-scale spectral fractional diffusion problem is the spectral fractional diffusion corresponding to the limiting diffusion operator from classical homogenization theory for local, elliptic diffusion in [Formula: see text]. We study anisotropic regularity of the solution of the local, limiting two-scale homogenized equation in [Formula: see text]. Using this, we develop the essentially optimal sparse tensor product finite element discretizations using continuous, piecewise linear Lagrangian finite elements for each, the slow variable [Formula: see text], the fast variable [Formula: see text], and the extended variable [Formula: see text]. As the solution of the two-scale homogenized equation is analytic with respect to the extended variable [Formula: see text] in weighted Sobolev spaces, we develop a second, likewise essentially optimal approach using the full tensor product of [Formula: see text] finite element spaces in [Formula: see text] and a sparse tensor product finite element space in [Formula: see text] using continuous, piecewise linear Lagrangian finite element basis functions for the variables [Formula: see text] and [Formula: see text]. 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 nonlocal two-scale spectral fractional diffusion problem. Error estimates in terms of the microscopic scale [Formula: see text] and the macroscopic finite element mesh size [Formula: see text] are rigorously derived for these numerical correctors. Numerical experiments confirm the theoretical error estimates of the sparse tensor product finite element schemes.