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首页> 外文期刊>Journal of Computational Physics >Generalized multiscale approximation of mixed finite elements with velocity elimination for subsurface flow
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Generalized multiscale approximation of mixed finite elements with velocity elimination for subsurface flow

机译：混合有限元的广义多尺度近似，具有速度消除液面流量

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A frame work of the mixed generalized multiscale finite element method (GMsFEM) for solving Darcy's law in heterogeneous media is studied in this paper. Our approach approximates pressure in multiscale function space that is between fine-grid space and coarse-grid space and solves velocity directly in the fine-grid space. To construct multiscale basis functions for each coarse-grid element, three types of snapshot space are raised. The first one is taken as the fine-grid space for pressure and the other two cases need to solve a local problem on each coarse-grid element. We describe a spectral decomposition in the snapshot space motivated by the analysis to further reduce the dimension of the space that is used to approximate the pressure. Since the velocity is directly solved in the finegrid space, in the linear system for the mixed finite elements, the velocity matrix can be approximated by a diagonal matrix without losing any accuracy. Thus it can be inverted easily. This reduces computational cost greatly and makes our scheme simple and easy for application. Comparing to our previous work of mixed generalized multiscale finite element method (Chung et al. (2015) [1.4]), both the pressure and velocity space in this approach are bigger. As a consequence, this method enjoys better accuracy. While the computational cost does not increase because of the good property of velocity matrix. Moreover, the proposed method preserves the local mass conservation property that is important for subsurface problems. Numerical examples are presented to illustrate the good properties of the proposed approach. If offline spaces are appropriately selected, one can achieve good accuracy with only a few basis functions per coarse element according to the numerical results. (C) 2019 Elsevier Inc. All rights reserved.
机译：本文研究了用于解决异构介质中达西法律的混合广义多尺度有限元法（GMSFEM）的帧工作。我们的方法近似于多尺度函数空间的压力，这些功能空间在微电网空间和粗网空间之间，并直接在细网空间中解决速度。要为每个粗略网格元素构造多尺度基函数，提出了三种类型的快照空间。第一个是作为压力的细网空间，另外两个案例需要在每个粗网元件上解决局部问题。我们在快照空间中描述了通过分析的快照空间中的光谱分解，以进一步减小用于近似压力的空间的尺寸。由于速度在FineGrid空间中直接求解，因此在用于混合有限元的线性系统中，速度矩阵可以通过对角矩阵近似而不损失任何精度。因此它可以容易地反转。这极大地降低了计算成本，使我们的方案简单易于应用。比较我们以前的混合通用多尺度有限元方法（Chung等人[1.4]），这种方法中的压力和速度空间都大。因此，这种方法享有更好的准确性。由于速度矩阵的良好性质，计算成本不会增加。此外，所提出的方法保留了对地下问题重要的局部质量保护特性。提出了数值例子以说明所提出的方法的良好性质。 If offline spaces are appropriately selected, one can achieve good accuracy with only a few basis functions per coarse element according to the numerical results. （c）2019 Elsevier Inc.保留所有权利。

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