Multiscale Finite Element methods for advection-dominated problems in perforated domains

We consider an advection-diffusion equation that is advection-dominated and posed on a perforated domain. On the boundary of the perforations, we set either homogeneous Dirichlet or homogeneous Neumann conditions. The purpose of this work is to investigate the behavior of several variants of multiscale finite element type methods, all of them based upon local basis functions satisfying weak continuity conditions in the Crouzeix-Raviart sense on the boundary of mesh elements. In the spirit of our previous works [C. Le Bris, F. Legoll, and A. Lozinski, Chin. Ann. Math. Ser. B, 34 (2013), pp. 113-138; Multiscale Model. Simul., 12 (2014), pp. 1046-1077] introducing such multiscale basis functions, and of [C. Le Bris, F. Legoll, and F. Madiot, Math. Model. Numer. Anal., 51 (2017), pp. 851-888] assessing their interest in advection-diffusion problems, we present, study, and compare various options in terms of choice of basis elements, adjunction of bubble functions, and stabilized formulations.