TY - JOUR

T1 - Generalized multiscale approximation of mixed finite elements with velocity elimination for subsurface flow

AU - Chen, Jie

AU - Chung, Eric T.

AU - He, Zhengkang

AU - Sun, Shuyu

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): BAS/1/1351-01
Acknowledgements: The work is supported by the National Natural Science Foundation of China (No.11401467 and No.51874262), Introduction plan of senior foreign experts (G20190027012), and XJTLU Key Programme Special Fund (KSF-P-02). The work is also supported in part by funding from King Abdullah University of Science and Technology (KAUST) through the grant BAS/1/1351-01. The research of Eric Chung is supported by Hong Kong RGC General Research Fund (project 14317516) and CUHK Direct Grant for Research 2017-18.

PY - 2019/11/25

Y1 - 2019/11/25

N2 - 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 fine-grid 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 [E. T. Chung, Y. Efendiev, and C. S. Lee. Mixed generalized multiscale finite element methods and applications. Multiscale Modeling and Simulation, 13(1):338-366, 2015.], 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.

AB - 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 fine-grid 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 [E. T. Chung, Y. Efendiev, and C. S. Lee. Mixed generalized multiscale finite element methods and applications. Multiscale Modeling and Simulation, 13(1):338-366, 2015.], 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.

UR - http://hdl.handle.net/10754/660459

UR - https://linkinghub.elsevier.com/retrieve/pii/S0021999119308381

UR - http://www.scopus.com/inward/record.url?scp=85076475029&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2019.109133

DO - 10.1016/j.jcp.2019.109133

M3 - Article

SN - 0021-9991

VL - 404

SP - 109133

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -