A High-order Multi-scale Finite Element Method (MsFEM) for solving Oseen problems in heterogeneous media

An enriched non-conforming Multi-scale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media was proposed in [Q. Feng, G. Allaire, and P. Omnes, Multiscale Model. Simul., 20(1):462–492, 2022] and further studied in [L. Balazi, G. Allaire, and P. Omnes, preprint \url{https://hal.science/hal-05198860}, 2025]. The main feature of this MsFEM is the consideration of high-order sets of weighting functions: for the velocity, they are polynomials of order $n$ on the faces and of order $n-1$ in the volume of the elements; for the pressure they are polynomials of order $n$ in the element volume. The present paper proposes to extend this method to the Oseen problem in heterogeneous porous media. Through numerical simulations, specially for the case $n=2$, we show that the developed MsFEM is able to deal with high Reynolds numbers.