A domain decomposition method for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition

Abstract In this article a domain decomposition method is proposed to solve a time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition. Robin boundary conditions between the Navier-Stokes domain and Darcy domain are constructed by directly re-organizing the terms in the three interface conditions, including the Beavers-Joseph condition. In order to avoid the traditional iteration for the domain decomposition method at each time step, the interface information, which is needed for the Robin type transmission conditions at the current time step, is directly predicted based on the numerical solution of the previous time steps. Backward Euler scheme is first utilized for the temporal discretization while finite elements are used for the spatial discretization. The convergences of this domain decomposition method are rigorously analyzed for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition. The major difficulties in the analysis arise from nonlinear terms and Beavers-Joseph interface condition, including a series of technical treatments and the final special norm used in the discrete Gronwall's inequality for the analysis of full discretization. Based on the above preparation, we further develop a Lagrange multiplier method under the framework of the domain decomposition method to overcome the difficulty of non-unique solutions arising from the defective boundary condition. One interesting finding of this paper is that the Lagrange multipliers are time dependent functions instead of constants. In order to improve the accuracy order for the temporal discretization, a three-step backward differentiation scheme is used to replace the backward Euler scheme. Compared with the first scheme, the second one allows us to use the relative larger time step to reduce the computational cost while keeping the same accuracy. Numerical examples are provided to illustrate the features of the proposed method.

[1]  Xiaoming He,et al.  An artificial compressibility ensemble algorithm for a stochastic Stokes‐Darcy model with random hydraulic conductivity and interface conditions , 2019, International Journal for Numerical Methods in Engineering.

[2]  Christian Vergara,et al.  Prescription of General Defective Boundary Conditions in Fluid-Dynamics , 2012 .

[3]  Trygve K. Karper,et al.  Unified finite element discretizations of coupled Darcy–Stokes flow , 2009 .

[4]  J. Douglas,et al.  Galerkin Methods for Parabolic Equations , 1970 .

[5]  Xiaoming He,et al.  A Dual-Porosity-Stokes Model and Finite Element Method for Coupling Dual-Porosity Flow and Free Flow , 2016, SIAM J. Sci. Comput..

[6]  Xiaoming He,et al.  A Domain Decomposition Method for the Steady-State Navier-Stokes-Darcy Model with Beavers-Joseph Interface Condition , 2015, SIAM J. Sci. Comput..

[7]  G. Gatica,et al.  A conforming mixed finite-element method for the coupling of fluid flow with porous media flow , 2008 .

[8]  Vincent J. Ervin,et al.  Numerical Approximation of a Quasi-Newtonian Stokes Flow Problem with Defective Boundary Conditions , 2007, SIAM J. Numer. Anal..

[9]  Xiaoming He,et al.  Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model , 2018, SIAM J. Sci. Comput..

[10]  T. Arbogast,et al.  A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium , 2007 .

[11]  Xiaoming He,et al.  Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes-Darcy systems , 2014, Math. Comput..

[12]  Wenbin Chen,et al.  Efficient and Long-Time Accurate Second-Order Methods for Stokes-Darcy System , 2012, 1211.0567.

[13]  Xiaoming He,et al.  Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model , 2019, International Journal for Numerical Methods in Engineering.

[14]  Alfio Quarteroni,et al.  Robin-Robin Domain Decomposition Methods for the Stokes-Darcy Coupling , 2007, SIAM J. Numer. Anal..

[15]  Barbara I. Wohlmuth,et al.  Large Scale Lattice Boltzmann Simulation for the Coupling of Free and Porous Media Flow , 2015, HPCSE.

[16]  Mingchao Cai,et al.  A Mixed and Nonconforming FEM with Nonmatching Meshes for a Coupled Stokes-Darcy Model , 2012, J. Sci. Comput..

[17]  Max Gunzburger,et al.  Asymptotic analysis of the differences between the Stokes–Darcy system with different interface conditions and the Stokes–Brinkman system☆ , 2010 .

[18]  Shuyu Sun,et al.  Coupling nonlinear Stokes and Darcy flow using mortar finite elements , 2011 .

[19]  Ivan Yotov,et al.  Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes–Darcy flows on polygonal and polyhedral grids , 2013, Numerische Mathematik.

[20]  Alfio Quarteroni,et al.  Numerical Treatment of Defective Boundary Conditions for the Navier-Stokes Equations , 2002, SIAM J. Numer. Anal..

[21]  G. Gatica,et al.  A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes–Darcy coupled problem , 2011 .

[22]  Shuyu Sun,et al.  Coupled Generalized Nonlinear Stokes Flow with Flow through a Porous Medium , 2009, SIAM J. Numer. Anal..

[23]  V. Nassehi,et al.  Numerical Analysis of Coupled Stokes/Darcy Flows in Industrial Filtrations , 2006 .

[24]  Willi Jäger,et al.  On The Interface Boundary Condition of Beavers, Joseph, and Saffman , 2000, SIAM J. Appl. Math..

[25]  Svetlana Tlupova,et al.  Boundary integral solutions of coupled Stokes and Darcy flows , 2009, J. Comput. Phys..

[26]  Gerhard Starke,et al.  First-Order System Least Squares for Coupled Stokes-Darcy Flow , 2011, SIAM J. Numer. Anal..

[27]  Béatrice Rivière,et al.  Primal Discontinuous Galerkin Methods for Time-Dependent Coupled Surface and Subsurface Flow , 2009, J. Sci. Comput..

[28]  Xiaoming He,et al.  Robin–Robin domain decomposition methods for the steady-state Stokes–Darcy system with the Beavers–Joseph interface condition , 2011, Numerische Mathematik.

[29]  Béatrice Rivière,et al.  Analysis of a Discontinuous Finite Element Method for the Coupled Stokes and Darcy Problems , 2005, J. Sci. Comput..

[30]  Alfio Quarteroni,et al.  Numerical analysis of the Navier–Stokes/Darcy coupling , 2010, Numerische Mathematik.

[31]  J. Galvis,et al.  NON-MATCHING MORTAR DISCRETIZATION ANALYSIS FOR THE COUPLING STOKES-DARCY EQUATIONS , 2007 .

[32]  Béatrice Rivière,et al.  A strongly conservative finite element method for the coupling of Stokes and Darcy flow , 2010, J. Comput. Phys..

[33]  Ricardo Ruiz-Baier,et al.  New fully-mixed finite element methods for the Stokes–Darcy coupling☆ , 2015 .

[34]  M. Wheeler A Priori L_2 Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations , 1973 .

[35]  Chris Lenn,et al.  Measurement of Oil and Water Flow Rates in a Horizontal Well With Chemical Markers and a Pulsed-Neutron Tool , 1997 .

[36]  Vincent J. Ervin,et al.  Approximation of the Stokes–Darcy System by Optimization , 2014, J. Sci. Comput..

[37]  Xiaoming He,et al.  Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface Condition , 2013 .

[38]  B. Rivière,et al.  On the solution of the coupled Navier–Stokes and Darcy equations , 2009 .

[39]  Xiaoming He,et al.  Fabrication and verification of a glass-silicon-glass micro-/nanofluidic model for investigating multi-phase flow in shale-like unconventional dual-porosity tight porous media. , 2019, Lab on a chip.

[40]  Jinchao Xu,et al.  A Two-Grid Method of a Mixed Stokes-Darcy Model for Coupling Fluid Flow with Porous Media Flow , 2007, SIAM J. Numer. Anal..

[41]  D. Joseph,et al.  Boundary conditions at a naturally permeable wall , 1967, Journal of Fluid Mechanics.

[42]  Svetlana Tlupova,et al.  Stokes-Darcy boundary integral solutions using preconditioners , 2009, J. Comput. Phys..

[43]  Béatrice Rivière,et al.  Analysis of time-dependent Navier–Stokes flow coupled with Darcy flow , 2008, J. Num. Math..

[44]  Ivan Yotov,et al.  Coupling Fluid Flow with Porous Media Flow , 2002, SIAM J. Numer. Anal..

[45]  Li Shan,et al.  Partitioned Time Stepping Method for Fully Evolutionary Stokes-Darcy Flow with Beavers-Joseph Interface Conditions , 2013, SIAM J. Numer. Anal..

[46]  Vivette Girault,et al.  Mortar multiscale finite element methods for Stokes–Darcy flows , 2014, Numerische Mathematik.

[47]  M. Gunzburger,et al.  Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition , 2010 .

[48]  Todd Arbogast,et al.  A discretization and multigrid solver for a Darcy–Stokes system of three dimensional vuggy porous media , 2009 .

[49]  Béatrice Rivière,et al.  Locally Conservative Coupling of Stokes and Darcy Flows , 2005 .

[50]  A. Quarteroni,et al.  Navier-Stokes/Darcy Coupling: Modeling, Analysis, and Numerical Approximation , 2009 .

[51]  Wenqiang Feng,et al.  Non-iterative domain decomposition methods for a non-stationary Stokes-Darcy model with Beavers-Joseph interface condition , 2012, Appl. Math. Comput..

[52]  Marco Discacciati,et al.  Domain decomposition methods for the coupling of surface and groundwater flows , 2004 .

[53]  Gabriel N. Gatica,et al.  A Residual-Based A Posteriori Error Estimator for the Stokes-Darcy Coupled Problem , 2010, SIAM J. Numer. Anal..

[54]  A. Quarteroni,et al.  Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations , 2004 .

[55]  F. Z. Nouri,et al.  A posteriori error analysis for Navier–Stokes equations coupled with Darcy problem , 2015 .

[56]  Svetlana Tlupova,et al.  Domain Decomposition Methods for Solving Stokes-Darcy Problems with Boundary Integrals , 2013, SIAM J. Sci. Comput..

[57]  Xiaoming He,et al.  On Stokes-Ritz Projection and Multistep Backward Differentiation Schemes in Decoupling the Stokes-Darcy Model , 2016, SIAM J. Numer. Anal..

[58]  Wenbin Chen,et al.  A Parallel Robin-Robin Domain Decomposition Method for the Stokes-Darcy System , 2011, SIAM J. Numer. Anal..

[59]  Yuri V. Vassilevski,et al.  Computational issues related to iterative coupling of subsurface and channel flows , 2007 .

[60]  Xiaohong Zhu,et al.  Decoupled schemes for a non-stationary mixed Stokes-Darcy model , 2009, Math. Comput..

[61]  Daozhi Han,et al.  Two‐phase flows in karstic geometry , 2014 .

[62]  Béatrice Rivière,et al.  Time-dependent coupling of Navier–Stokes and Darcy flows , 2013 .

[63]  I. Yotov,et al.  Domain decomposition for coupled Stokes and Darcy flows , 2013 .

[64]  E. Miglio,et al.  Mathematical and numerical models for coupling surface and groundwater flows , 2002 .

[65]  T. Arbogast,et al.  Homogenization of a Darcy–Stokes system modeling vuggy porous media , 2006 .

[66]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[67]  Xiaoming He,et al.  A stabilized finite volume element method for a coupled Stokes–Darcy problem , 2017, Applied Numerical Mathematics.

[68]  Weidong Zhao,et al.  Finite Element Approximations for Stokes–darcy Flow with Beavers–joseph Interface Conditions * , 2022 .

[69]  Jinchao Xu,et al.  Numerical Solution to a Mixed Navier-Stokes/Darcy Model by the Two-Grid Approach , 2009, SIAM J. Numer. Anal..

[70]  Wenbin Chen,et al.  An efficient and long-time accurate third-order algorithm for the Stokes–Darcy system , 2016, Numerische Mathematik.

[71]  I. P. Jones,et al.  Low Reynolds number flow past a porous spherical shell , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.

[72]  VIVETTE GIRAULT,et al.  DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition , 2009, SIAM J. Numer. Anal..

[73]  S. Meddahi,et al.  Strong coupling of finite element methods for the Stokes–Darcy problem , 2012, 1203.4717.