Volume 17, Issue 3
Efficient Galerkin-Mixed FEMs for Incompressible Miscible Flow in Porous Media

Int. J. Numer. Anal. Mod., 17 (2020), pp. 350-367.

Published online: 2020-05

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• Abstract

The paper focuses on numerical study of the incompressible miscible flow in porous media. The proposed algorithm is based on a fully decoupled and linearized scheme in the temporal direction, classical Galerkin-mixed approximations in the FE space ($V^r_h$, $S^{r-1}_h$ × $H^{r-1}_h$) ($r$ ≥ 1) in the spatial direction and a post-processing technique for the velocity/pressure, where $V^r_h$ and $S^{r-1}_h$ × $H^{r-1}_h$ denotes the standard $C^0$ Lagrange FE and the Raviart-Thomas FE spaces, respectively. The decoupled and linearized Galerkin-mixed FEM enjoys many advantages over existing methods. At each time step, the method only requires solving two linear systems for the concentration and velocity/pressure. Analysis in our recent work [37] shows that the classical Galerkin-mixed method provides the optimal accuracy $O$($h^{r+1}$) for the numerical concentration in $L^2$-norm, instead of $O$($h^r$) as shown in previous analysis. A new numerical velocity/pressure of the same order accuracy as the concentration can be obtained by the post-processing in the proposed algorithm. Extensive numerical experiments in both two- and three-dimensional spaces, including smooth and non-smooth problems, are presented to illustrate the accuracy and stability of the algorithm. Our numerical results show that the one-order lower approximation to the velocity/pressure does not influence the accuracy of the numerical concentration, which is more important in applications.

35R35, 49J40, 60G40

maweiw@uic.edu.hk (Weiwei Sun)

chengda.wu@my.cityu.edu.hk (Chengda Wu)

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@Article{IJNAM-17-350, author = {Sun , Weiwei and Wu , Chengda}, title = {Efficient Galerkin-Mixed FEMs for Incompressible Miscible Flow in Porous Media}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2020}, volume = {17}, number = {3}, pages = {350--367}, abstract = {

The paper focuses on numerical study of the incompressible miscible flow in porous media. The proposed algorithm is based on a fully decoupled and linearized scheme in the temporal direction, classical Galerkin-mixed approximations in the FE space ($V^r_h$, $S^{r-1}_h$ × $H^{r-1}_h$) ($r$ ≥ 1) in the spatial direction and a post-processing technique for the velocity/pressure, where $V^r_h$ and $S^{r-1}_h$ × $H^{r-1}_h$ denotes the standard $C^0$ Lagrange FE and the Raviart-Thomas FE spaces, respectively. The decoupled and linearized Galerkin-mixed FEM enjoys many advantages over existing methods. At each time step, the method only requires solving two linear systems for the concentration and velocity/pressure. Analysis in our recent work [37] shows that the classical Galerkin-mixed method provides the optimal accuracy $O$($h^{r+1}$) for the numerical concentration in $L^2$-norm, instead of $O$($h^r$) as shown in previous analysis. A new numerical velocity/pressure of the same order accuracy as the concentration can be obtained by the post-processing in the proposed algorithm. Extensive numerical experiments in both two- and three-dimensional spaces, including smooth and non-smooth problems, are presented to illustrate the accuracy and stability of the algorithm. Our numerical results show that the one-order lower approximation to the velocity/pressure does not influence the accuracy of the numerical concentration, which is more important in applications.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/16863.html} }
TY - JOUR T1 - Efficient Galerkin-Mixed FEMs for Incompressible Miscible Flow in Porous Media AU - Sun , Weiwei AU - Wu , Chengda JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 350 EP - 367 PY - 2020 DA - 2020/05 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/16863.html KW - Galerkin-mixed FEM, incompressible miscible flow in porous media, fully linearized scheme. AB -

The paper focuses on numerical study of the incompressible miscible flow in porous media. The proposed algorithm is based on a fully decoupled and linearized scheme in the temporal direction, classical Galerkin-mixed approximations in the FE space ($V^r_h$, $S^{r-1}_h$ × $H^{r-1}_h$) ($r$ ≥ 1) in the spatial direction and a post-processing technique for the velocity/pressure, where $V^r_h$ and $S^{r-1}_h$ × $H^{r-1}_h$ denotes the standard $C^0$ Lagrange FE and the Raviart-Thomas FE spaces, respectively. The decoupled and linearized Galerkin-mixed FEM enjoys many advantages over existing methods. At each time step, the method only requires solving two linear systems for the concentration and velocity/pressure. Analysis in our recent work [37] shows that the classical Galerkin-mixed method provides the optimal accuracy $O$($h^{r+1}$) for the numerical concentration in $L^2$-norm, instead of $O$($h^r$) as shown in previous analysis. A new numerical velocity/pressure of the same order accuracy as the concentration can be obtained by the post-processing in the proposed algorithm. Extensive numerical experiments in both two- and three-dimensional spaces, including smooth and non-smooth problems, are presented to illustrate the accuracy and stability of the algorithm. Our numerical results show that the one-order lower approximation to the velocity/pressure does not influence the accuracy of the numerical concentration, which is more important in applications.

Weiwei Sun & Chengda Wu. (2020). Efficient Galerkin-Mixed FEMs for Incompressible Miscible Flow in Porous Media. International Journal of Numerical Analysis and Modeling. 17 (3). 350-367. doi:
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