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Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

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Abstract

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard \(L^2\)-norm and broken \(H^1\)-norm are obtained for three discontinuous finite volume element methods (symmetric, non-symmetric and incomplete types). A series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, mass conservation, capability to deal with complicated geometries, and applicability to the problems with realistic parameters.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Rui Li, Yali Gao & Zhangxin Chen

  2. Department of Mathematics, School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an, 710021, People’s Republic of China

    Jian Li

  3. Department of Mathematics, Baoji University of Arts and Sciences, Baoji, 721013, People’s Republic of China

    Jian Li

  4. Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, T2N 1N4, Canada

    Zhangxin Chen

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  1. Rui Li
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  2. Yali Gao
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Correspondence to Zhangxin Chen.

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This work is partially supported by NSF of China (Nos.11371031, 11371288), Foundation CMG in Xi’an Jiaotong University, and the Key Project of Baoji university of Arts and Sciences (No. ZK15040).

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Li, R., Gao, Y., Li, J. et al. Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem. J Sci Comput 74, 693–727 (2018). https://doi.org/10.1007/s10915-017-0454-3

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