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.
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
.
DOI: 10.1007/s10473-025-0224-5
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