Sparse Two-Scale FEM for Homogenization Problems

We analyze two-scale Finite Element Methods for the numerical solution of elliptic homogenization problems with coefficients oscillating at a small length scale epsilon << 1. Based on a refined two-scale regularity on the solutions, two-scale tensor product FE spaces are introduced and error estimates which are robust (i.e., independent of epsilon) are given. We show that under additional two-scale regularity assumptions on the solution, resolution of the fine scale is possible with substantially fewer degrees of freedom and the two-scale full tensor product spaces can be "thinned out'' by means of sparse interpolation preserving at the same time the error estimates.