In this paper, we develop an iterative scheme to construct multiscale basis functions within the framework of the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for the mixed formulation. The iterative procedure starts with the construction of an energy minimizing snapshot space that can be used for approximating the solution of the model problem. A spectral decomposition is then performed on the snapshot space to form global multiscale space. Under this setting, each global multiscale basis function can be split into a non-decaying and a decaying parts. The non-decaying part of a global basis is localized and it is fixed during the iteration. Then, one can approximate the decaying part via a modified Richardson scheme with an appropriately defined preconditioner. Using this set of iterative-based multiscale basis functions, first-order convergence with respect to the coarse mesh size can be shown if sufficiently many times of iterations with regularization parameter being in an appropriate range are performed. Numerical results are presented to illustrate the effectiveness and efficiency of the proposed computational multiscale method.

在本文中，我们开发了一种迭代方案，以在混合公式的约束能量最小化广义多尺度有限元方法 (CEM-GMsFEM) 的框架内构建多尺度基函数。迭代过程从构建能量最小化快照空间开始，该空间可用于逼近模型问题的解。然后对快照空间进行谱分解以形成全局多尺度空间。在这种设置下，每个全局多尺度基函数可以分为非衰减和衰减部分。全局基的非衰减部分是局部的，并且在迭代过程中是固定的。然后，可以通过修改后的理查森方案和适当定义的预处理器来近似衰减部分。使用这组基于迭代的多尺度基函数，如果执行足够多次的迭代并且正则化参数在适当的范围内，则可以显示相对于粗网格尺寸的一阶收敛。数值结果用于说明所提出的计算多尺度方法的有效性和效率。