Reduced-basis approach for homogenization beyond the periodic setting

We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parameterized by the macroscopic scale is required at the microscopic scale. This is a framework very well suited for model reduction attempts. The purpose of this work is to show how the reduced-basis approach allows one to speedup the computation of a large number of cell problems without any loss of precision. The essential components of this reduced-basis approach are the a posteriori error estimation, which provides sharp error bounds for the outputs of interest, and an approximation process divided into offline and online stages, which decouples the generation of the approximation space and its use for Galerkin projections.