Convergence Analysis of a High-Order Multiscale Finite Element Method (MsFEM) for Stokes Flows in Heterogeneous Media

Abstract

Abstract. An enriched nonconforming Multiscale 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 (2022), pp. 462–492]. The main feature of this MsFEM is the consideration of high-order sets of weighting functions: for the velocity, they are polynomials of order [Formula: see text] on the faces and of order [Formula: see text] in the volume of the elements; for the pressure they are polynomials of order [Formula: see text] in the element volume. In the previously cited reference, only the case [Formula: see text] was numerically tested. The present paper proposes the first implementation for the case [Formula: see text] in two and three dimensions. Furthermore, a discrete analysis of this MsFEM applied to the Stokes problem in heterogeneous media is performed. In particular, a first error estimate is obtained, proving the convergence of this MsFEM for the Stokes problem in periodic perforated media. In addition, it has been shown in the previously cited reference that the continuous local problems involved in this MsFEM are well-posed. Here, their discrete counterparts are also proved to be well-posed, for any [Formula: see text] in two dimensions and for [Formula: see text] equal to 1 and 2 in three dimensions, with a judicious choice of nonconforming pairs of finite elements.