High-dimensional finite elements for elliptic problems with multiple scales

Elliptic homogenization problems in a domain Omega subset of R-d with n + 1 separated scales are reduced to elliptic one-scale problems in dimension (n + 1)d. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work, and memory requirements comparable to those in a standard FEM for single-scale problems in Omega, while it gives numerical approximations of the correct homogenized limit as well as of all first-order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples for model diffusion problems with two and three scales confirm our results.