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.

翻译：本文讨论了一种多连续介质均匀化的一般框架。多连续介质模型在许多应用中广泛使用，并已建立了相关推导方法。在这些模型中，每个宏观点定义了多个宏观变量，并由此构建了多连续介质方程组。本文提出了一种通用公式及相关构成要素，从而能够实现多连续介质均匀化。我们的推导包含几个主要部分。第一部分提出了一种通用展开式，其中解通过多个宏观变量与相应的胞元问题的乘积来表示。第二部分涉及胞元问题的构建。这些胞元问题被构建为带有各连续约束的鞍点问题。通过用测试函数定义连续体，我们将约束设定为积分表示形式。最后，将展开式代入原始系统，得到多连续介质系统。我们将这一方法应用于椭圆方程的混合形式表述。这是一个具有挑战性的系统，因为该系统不具有对称性。我们讨论了局部问题以及解及其梯度的多种宏观尺度表示。通过使用不同阶次的近似，可以导出不同的方程组。我们探讨了多连续介质均匀化的适用性，并将其与胞元问题中的高对比度联系起来。文中还给出了数值结果。