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New Analysis of Galerkin FEMs for Miscible Displacement in Porous Media

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Abstract

The paper is concerned with optimal error estimates of classical Galerkin FEMs for the equations of incompressible miscible flows in porous media. The analysis done in the last several decades shows that classical Galerkin FEMs provide the numerical concentration of the accuracy \(O(\tau ^k+h^{r+1} + h^s)\) in \(L^2\)-norm. This analysis leads to the use of higher order finite element approximation to the pressure than that to the concentration in various numerical simulations to achieve the best rate of convergence. However, this error estimate is not optimal. The purpose of this paper is to establish the optimal \(L^2\) error estimate \(O(\tau ^k + h^{r+1} + h^{s+1})\), from which one can see that the best convergence rate can be achieved by taking the same order \((r=s\)) approximation to the concentration and pressure. Clearly Galerkin FEMs with \(r=s\) are less expensive in computation and easier for implementation. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis.

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  1. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Chengda Wu & Weiwei Sun

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The work of the authors was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302718).

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Wu, C., Sun, W. New Analysis of Galerkin FEMs for Miscible Displacement in Porous Media. J Sci Comput 80, 903–923 (2019). https://doi.org/10.1007/s10915-019-00963-w

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