A three-field Multiscale Method

TL;DR

This paper introduces the MH²M, a multiscale finite element method for Darcy flow, based on a three-field domain decomposition approach, providing stability, convergence, and connections to existing methods.

Contribution

It proposes the MH²M, a novel multiscale finite element method for Darcy equations, leveraging a three-field formulation for improved stability and efficiency.

Findings

The method is stable and convergent for various finite element spaces.

It produces a symmetric positive definite system depending only on the trace.

Connections with existing multiscale methods are established.

Abstract

\emph{A Three-Field Domain Decomposition Method} is the title of a seminal paper by F. Brezzi and L. D. Marini which introduces a three-field formulation for elliptic partial differential equations. Based on that, we propose the Multiscale-Hybrid-Hybrid Method (MH$^2$M) for the Darcy model, a multiscale finite element method that yields, after a series of formal manipulations, a symmetric positive definite formulation that depends only on the trace of the solution. We show stability and convergence results for a family of finite element spaces and establish relationships with other multiscale finite element methods.