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New analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media


Authors: Weiwei Sun and Chengda Wu
Journal: Math. Comp. 90 (2021), 81-102
MSC (2010): Primary 35K61, 65N12, 65N30
DOI: https://doi.org/10.1090/mcom/3561
Published electronically: September 8, 2020
MathSciNet review: 4166454
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Abstract: Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure. The previous works only showed the first-order accuracy of the method in $ L^2$-norm in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial mesh. In this paper, we provide new and optimal $ L^2$-norm error estimates of Galerkin-mixed FEMs for all three components in a general case. In particular, for the lowest-order Galerkin-mixed FEM, we show unconditionally the second-order accuracy in $ L^2$-norm for the concentration. Numerical results for both two- and three-dimensional models are presented to confirm our theoretical analysis. More important is that our approach can be extended to the analysis of mixed FEMs for many strongly coupled systems to obtain optimal error estimates for all components.


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Additional Information

Weiwei Sun
Affiliation: Advanced Institute of Natural Sciences, Beijing Normal University at Zhuhai, 519087, People’s Republic of China; and Division of Science and Technology, United International College (BNU-HKBU), Zhuhai, 519087, People’s Republic of China
Email: maweiw@uic.edu.cn

Chengda Wu
Affiliation: Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, People’s Repubilic of China
MR Author ID: 1269306
Email: chengda.wu@my.cityu.edu.hk

DOI: https://doi.org/10.1090/mcom/3561
Received by editor(s): September 12, 2018
Received by editor(s) in revised form: September 15, 2019, and February 12, 2020
Published electronically: September 8, 2020
Additional Notes: The research was supported in part by the Zhujiang Scholar program and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, People’s Republic of China (Project No. CityU 11302915).
Article copyright: © Copyright 2020 American Mathematical Society