A MIXED FINITE ELEMENT AND UPWIND MIXED FINITE ELEMENT MULTI-STEP METHOD FOR THE THREE-DIMENSIONAL POSITIVE SEMI-DEFINITE DARCY-FORCHHEIMER MISCIBLE DISPLACEMENT PROBLEM

A MIXED FINITE ELEMENT AND UPWIND MIXED FINITE ELEMENT MULTI-STEP METHOD FOR THE THREE-DIMENSIONAL POSITIVE SEMI-DEFINITE DARCY-FORCHHEIMER MISCIBLE DISPLACEMENT PROBLEM

A MIXED FINITE ELEMENT AND UPWIND MIXED FINITE ELEMENT MULTI-STEP METHOD FOR THE THREE-DIMENSIONAL POSITIVE SEMI-DEFINITE DARCY-FORCHHEIMER MISCIBLE DISPLACEMENT PROBLEM

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doi:10.1007/s10473-025-0224-5

Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (2): 715-736.doi: 10.1007/s10473-025-0224-5

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Yirang Yuan^1,*, Changfeng Li¹, Huailing Song³, Tongjun Sun¹

1. Institute of Mathematics, Shandong University, Jinan 250100, China;
2. School of Economics, Shandong University, Jinan 250100, China;
3. College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received:2023-11-15 Revised:2024-05-06 Online:2025-03-25 Published:2025-05-08
Contact: *Yirang Yuan, E-mail: yryuan@sdu.edu.cn
About author:Changfeng Li, E-mail: cfli@sdu.edu.cn;Huailing Song, E-mail: shling@hnu.edu.cn;Tongjun Sun, E-mail: tjsun@hnu.edu.cn
Supported by:
This work was supported by the Natural Science Foundation of Shandong Province (ZR2021MA019) and the National Natural Science Foundation of China (11871312).

Abstract: In this paper, a composite numerical scheme is proposed to solve the three-dimensional Darcy-Forchheimer miscible displacement problem with positive semi-definite assumptions. A mixed finite element is used for the flow equation. The velocity and pressure are computed simultaneously. The accuracy of velocity is improved one order. The concentration equation is solved by using mixed finite element, multi-step difference and upwind approximation. A multi-step method is used to approximate time derivative for improving the accuracy. The upwind approximation and an expanded mixed finite element are adopted to solve the convection and diffusion, respectively. The composite method could compute the diffusion flux and its gradient. It possibly becomes an efficient tool for solving convection-dominated diffusion problems. Firstly, the conservation of mass holds. Secondly, the multi-step method has high accuracy. Thirdly, the upwind approximation could avoid numerical dispersion. Using numerical analysis of a priori estimates and special techniques of differential equations, we give an error estimates for a positive definite problem. Numerical experiments illustrate its computational efficiency and feasibility of application.

Key words: Darcy-Forchheimer flow, three-dimensional positive semi-definite problem, upwind mixed finite element multi-step method, conservation of mass, convergence analysis

CLC Number:

65N30

Yirang Yuan, Changfeng Li, Huailing Song, Tongjun Sun. A MIXED FINITE ELEMENT AND UPWIND MIXED FINITE ELEMENT MULTI-STEP METHOD FOR THE THREE-DIMENSIONAL POSITIVE SEMI-DEFINITE DARCY-FORCHHEIMER MISCIBLE DISPLACEMENT PROBLEM[J].Acta mathematica scientia,Series B, 2025, 45(2): 715-736.

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