In this paper, we consider a class of multiscale methods for the solution of nonlinear problem in perforated domains. These problems are of multiscale nature and their discretizations lead to large nonlinear systems. To discretize these problems, we construct a fine grid approximation using the finite element method with implicit time approximation and the Newton’s method. In order to solve these large systems efficiently, we will develop a model reduction procedure. To perform the model reduction, we construct a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM). The GMsFEM consists of an offline and online stages. In the offline stage, we construct multiscale basis functions based on the solution of some local spectral problems defined in the snapshot space. Then, we enrich the offline multiscale space by additional multiscale basis that handle non-homogeneous boundary condition. For the accurate solution of the nonlinear problem, we use two techniques on the online stage: (1) residual based multiscale basis functions and (2) residual based local correction. We will present numerical results for two-dimensional Allen–Cahn problems in perforated domains.

在本文中，我们考虑了一类用于解决多孔区域非线性问题的多尺度方法。这些问题具有多尺度性质，其离散化导致大型非线性系统。为了离散化这些问题，我们使用带有隐式时间近似的有限元方法和牛顿法构造了精细的网格近似。为了有效地解决这些大型系统，我们将开发模型简化程序。为了执行模型简化，我们基于广义多尺度有限元方法（GMsFEM）构造了粗网格近似法。GMsFEM由离线阶段和在线阶段组成。在离线阶段，我们基于快照空间中定义的一些局部光谱问题的解决方案构造多尺度基函数。然后，我们通过处理非均匀边界条件的其他多尺度基础来丰富离线多尺度空间。为了精确地解决非线性问题，我们在在线阶段使用了两种技术：（1）基于残差的多尺度基函数和（2）基于残差的局部校正。我们将给出二维多孔区域内Allen-Cahn问题的数值结果。