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New analysis and recovery technique of mixed FEMs for compressible miscible displacement in porous media

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Abstract

Numerical methods and analysis for compressible miscible flow in porous media have been investigated extensively in the last several decades. Amongst those methods, the lowest-order mixed method is the most popular one in practical applications. The method is based on the linear Lagrange approximation for the concentration and the lowest order (zero-order) Raviart–Thomas mixed approximation for the Darcy velocity/pressure. However, the existing error analysis only provides the first-order accuracy in \(L^2\)-norm for all three physical components in spatial direction, which was proved under certain extra restrictions on both time step and spatial meshes. The analysis is not optimal for the concentration mainly due to the strong coupling of the system and the drawback of the traditional approach which leads to serious pollution to the numerical concentration in analysis. The main task of this paper is to present a new analysis and establish the optimal error estimate of the commonly-used linearized lowest-order mixed FEM. In particular, the second-order accuracy for the concentration in spatial direction is proved unconditionally. Moreover, we propose a simple recovery technique to obtain a new numerical Darcy velocity/pressure of second-order accuracy by re-solving an elliptic pressure equation. Also we extend our analysis to a second-order time discrete scheme to obtain optimal error estimates in both spatial and temporal directions. Numerical results are provided to confirm our theoretical analysis and show the efficiency of the method.

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Acknowledgements

The author would like to thank the anonymous referees for their valuable suggestions and comments and also, Professor R. An (Wenzhou University) and Dr. J. Wang (BCSRC, Beijing) for providing a FreeFEM code and Fig. 2.

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  1. Advanced Institute of Natural Science, Beijing Normal University at Zhuhai, Zhuhai, People’s Republic of China

    Weiwei Sun

  2. Division of Science and Technology, United International College (BNU-HKBU), Zhuhai, 519087, People’s Republic of China

    Weiwei Sun

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Correspondence to Weiwei Sun.

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The research is partially supported by a Grant from National Natural Science Foundation of China under Grant Number 12071040 and internal funds (R5202009, R72021111) from United International College (BNU-HKBU)

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Sun, W. New analysis and recovery technique of mixed FEMs for compressible miscible displacement in porous media. Numer. Math. 150, 179–215 (2022). https://doi.org/10.1007/s00211-021-01249-w

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