Multicontinuum homogenization. General theory and applications

TL;DR

This paper develops a comprehensive framework for multicontinuum homogenization, deriving general equations and cell problems, and demonstrating applications to complex elliptic systems with numerical validation.

Contribution

It introduces a novel general formulation for multicontinuum homogenization, including new expansion methods and saddle point cell problems, applicable to high-contrast systems.

Findings

Effective multicontinuum models derived for complex elliptic equations

Various macroscale representations demonstrated for solutions and gradients

Numerical results validate the applicability of the proposed homogenization approach

Abstract

In this paper, we discuss a general framework for multicontinuum homogenization. Multicontinuum models are widely used in many applications and some derivations for these models are established. In these models, several macroscopic variables at each macroscale point are defined and the resulting multicontinuum equations are formulated. In this paper, we propose a general formulation and associated ingredients that allow performing multicontinuum homogenization. Our derivation consists of several main parts. In the first part, we propose a general expansion, where the solution is expressed via the product of multiple macro variables and associated cell problems. The second part consists of formulating the cell problems. The cell problems are formulated as saddle point problems with constraints for each continua. Defining the continua via test functions, we set the constraints as an integral representation. Finally, substituting the expansion to the original system, we obtain multicontinuum systems. We present an application to the mixed formulation of elliptic equations. This is a challenging system as the system does not have symmetry. We discuss the local problems and various macroscale representations for the solution and its gradient. Using various order approximations, one can obtain different systems of equations. We discuss the applicability of multicontinuum homogenization and relate this to high contrast in the cell problem. Numerical results are presented.