Non-Conforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media. Part II: error estimates for periodic microstructure

TL;DR

This paper provides rigorous error estimates for a multiscale finite element method applied to Stokes flows in heterogeneous media with periodic microstructure, enhancing understanding of its accuracy and convergence.

Contribution

It extends the multiscale finite element method for Stokes flows to arbitrary weighting functions and offers error bounds in periodic microstructures, improving upon previous work.

Findings

Error bounds established for periodic microstructure case

Numerical experiments show improved accuracy over previous methods

Method generalizes to various weighting functions

Abstract

This paper is dedicated to the rigorous numerical analysis of a Multiscale Finite Element Method (MsFEM) for the Stokes system, when dealing with highly heterogeneous media, as proposed in [B.P.~Muljadi et al., arXiv:1404.2837]. The method is in the vein of the classical Crouzeix-Raviart approach. It is generalized here to arbitrary sets of weighting functions used to enforced continuity across the mesh edges. We provide error bounds for a particular set of weighting functions in a periodic setting, using an accurate estimate of the homogenization error. Numerical experiments demonstrate an improved accuracy of the present variant with respect to that of Part I, both in the periodic case and in a broader setting.