The Multiscale Finite Element Method (MsFEM) is a powerful numerical method in the context of multiscale analysis. It uses basis functions which encode details of the fine scale description, and performs in a two-stage procedure: (i) offline stage in which basis functions are computed solving local fine scale problems; (ii) online stage in which a cheap Galerkin approximation problem is solved using a coarse mesh. However, as in other numerical methods, a crucial issue is to certify that a prescribed accuracy is obtained for the numerical solution. In the present work, we propose an a posteriori error estimate for MsFEM using the concept of Constitutive Relation Error (CRE) based on dual analysis. It enables to effectively address global or goal-oriented error estimation, to assess the various error sources, and to drive robust adaptive algorithms. We also investigate the additional use of model reduction inside the MsFEM strategy in order to further decrease numerical costs. We particularly focus on the use of the Proper Generalized Decomposition (PGD) for the computation of multiscale basis functions. PGD is a suitable tool that enables to explicitly take into account variations in geometry,
material coefficients, or boundary conditions. In many configurations, it can thus be efficiently employed to solve with low computing cost the various local fine-scale problems associated with MsFEM. In addition to showing performances of the coupling between PGD and MsFEM, we introduce dedicated estimates on PGD model reduction error, and use these to certify the quality of the overall MsFEM solution.
