A three-field Multiscale Method

Abstract

Inspired by the seminal paper \emph{A Three-Field Domain Decomposition Method} by F. Brezzi and L. D. Marini, we propose the Multiscale-Hybrid-Hybrid Method (MH${}^2$M) for the Darcy problem. After a sequence of formal manipulations, the resulting multiscale finite element method leads to a symmetric positive definite formulation posed solely in terms of the trace variable. We establish stability and convergence results for a family of finite element spaces and highlight connections with other multiscale finite element methods.