Expanded mixed multiscale finite element methods and their applications for flows in porous media

TL;DR

This paper introduces expanded mixed multiscale finite element methods for modeling flows in porous media, providing a more comprehensive approach that captures multiple unknowns simultaneously and improves accuracy in complex multiscale environments.

Contribution

The paper develops a new family of expanded mixed MsFEMs that solve four unknowns simultaneously, with rigorous convergence analysis and applicability to non-separable scales.

Findings

Enhanced accuracy with global information in non-scale-separated cases

Rigorous convergence proofs for conforming and nonconforming methods

Numerical results demonstrate efficiency in porous media flow simulations

Abstract

We develop a family of expanded mixed Multiscale Finite Element Methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity and Lagrange multipliers. We use multiscale basis functions for the both velocity and gradient of pressure. In the expanded mixed MsFEM framework, we consider both cases of separable-scale and non-separable spatial scales. We specifically analyze the methods in three categories: periodic separable scales, $G$- convergence separable scales, and continuum scales. When there is no scale separation, using some global information can improve accuracy for the expanded mixed MsFEMs. We present rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes both conforming and nonconforming expanded mixed MsFEM. Numerical results are presented for various multiscale models and flows in porous media with shales to illustrate the efficiency of the expanded mixed MsFEMs.