In this paper, we discuss multiscale methods for nonlinear problems. The main idea of these approaches is to use local constraints and solve problems in oversampled regions for constructing macroscopic equations. These techniques are intended for problems without scale separation and high contrast, which often occur in applications. For linear problems, the local solutions with constraints are used as basis functions. This technique is called Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). GMsFEM identifies macroscopic quantities based on rigorous analysis. In corresponding upscaling methods, the multiscale basis functions are selected such that the degrees of freedom have physical meanings, such as averages of the solution on each continuum. This paper extends the linear concepts to nonlinear problems, where the local problems are nonlinear. The main concept consists of: (1) identifying macroscopic quantities; (2) constructing appropriate oversampled local problems with coarse-grid constraints; (3) formulating macroscopic equations. We consider two types of approaches. In the first approach, the solutions of local problems are used as basis functions (in a linear fashion) to solve nonlinear problems. This approach is simple to implement; however, it lacks the nonlinear interpolation, which we present in our second approach. In this approach, the local solutions are used as a nonlinear forward map from local averages (constraints) of the solution in oversampling region. This local fine-grid solution is further used to formulate the coarse-grid problem. Both approaches are discussed on several examples and applied to single-phase and two-phase flow problems, which are challenging because of convection-dominated nature of the concentration equation.

中文翻译：

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非线性非局部多连续体升级框架及其应用

在本文中，我们讨论非线性问题的多尺度方法。这些方法的主要思想是使用局部约束并解决过采样区域中的问题以构建宏观方程。这些技术旨在解决应用中经常出现的没有刻度分离和高对比度的问题。对于线性问题，带有约束的局部解被用作基函数。这种技术称为约束能量最小化广义多尺度有限元方法 (CEM-GMsFEM)。GMsFEM 基于严格的分析识别宏观量。在相应的放大方法中，选择多尺度基函数使得自由度具有物理意义，例如每个连续体上的解的平均值。本文将线性概念扩展到非线性问题，其中局部问题是非线性的。主要概念包括： (1) 识别宏观量；(2) 用粗网格约束构造适当的过采样局部问题；(3) 制定宏观方程。我们考虑两种类型的方法。在第一种方法中，局部问题的解被用作解决非线性问题的基函数（以线性方式）。这种方法很容易实现；然而，它缺少我们在第二种方法中介绍的非线性插值。在这种方法中，局部解被用作来自过采样区域中解的局部平均值（约束）的非线性前向映射。该局部细网格解决方案进一步用于制定粗网格问题。