OPTIMAL ERROR ESTIMATES OF A DECOUPLED SCHEME BASED ON TWO-GRID FINITE ELEMENT FOR MIXED NAVIER-STOKES/DARCY MODEL
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摘要
Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.
Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.
Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.
引文
[1] Arbogast T, Brunson D S. A computational method for approximating a Darcy-Stokes system poverning a vuggy porous medium. Comput Geosci, 2007, 11:207–218
[2] Arbogast T, Gomez M S M. A discretization and multigrid solver for a Darcy-Stokes system of three dimensional vuggy porous media. Comput Geosci, 2009, 13(3):331–348
[3] Arbogast T, Lehr H L. Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput Geosci, 2006, 10(3):291–302
[4] Cao Y, Gunzburger M, Hu X, Hua F, Wang X, Zhao W. Finite element approximations for Stokes-Darcy flow with Beavers-Joseph interface conditions. SIAM J Numer Anal, 2010, 47(6):4239–4256
[5] Cao Y, Gunzburger M, Hua F, Wang X. Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun Math Sci, 2010, 8(1):1-25
[6] Mu M, Xu J C. A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM J Numer Anal, 2007, 45:1801–1813
[7] Ervin V J, Jenkins E W, Sun S. Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM J Numer Anal, 2009, 47(2):929–952
[8] Moraiti M. On the quasistatic approximation in the Stokes-Darcy model of groundwater-surface water flows. J Math Anal Appl, 2012, 394(2):796–808
[9] Cai M, Mu M, Xu J. Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach.SIAM J Numer Anal, 2009, 47(5):3325–3338
[10] Badea L, Discacciati M, Quarteroni A. Numerical analysis of the Navier-Stokes/Darcy coupling. Numerische Mathematik, 2010, 115(2):195–227
[11] C?e?smelioˇglu A, Riviere B. Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J Numer Mathem, 2008, 16(4):249–280
[12] C?e?smelioˇglu A, Riviere B. Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow. J Sci Comput, 2009, 40(1/3):115–140
[13] Chidyagwai P, Riviere B. On the solution of the coupled Navier-Stokes and Darcy equations. Comput Methods Appl Mech Engin, 2009, 198(47/48):3806–3820
[14] Girault V, Riviere B. DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-JosephSaffman interface condition. SIAM J Numer Anal, 2009, 47(3):2052–2089
[15] Hou Y. Optimal Error Estimates of A Decoupled Scheme Based on Two-Grid Finite Element for Mixed Stokes-Darcy Model. Appl Math Lett, 2016, 57:90–96
[16] Beavers G, Joseph D. Boundary conditions at a naturally permeable wall. J Fluid Mech, 1967, 30:197–207
[17] Discacciati M, Miglio E, Quarteroni A. Mathematical and numerical models for coupling surface and groundwater flows. Appl Numer Math, 2002, 43:57–74
[18] Jager W, Mikelic A. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J Appl Math, 2000, 60:1111–1127
[19] Layton W. A two-level discretization method for the Navier-Stokes equations. Comput Math Appl, 1993,26:33–38
[20] Nield D A, Bejan A. Convection in Porous Media. New York:Springer-Verlag, 1999
[21] Saffman P. On the boundary condition at the surface of a porous media. Stud Appl Math, 1971, 50:93–101
[22] Cai M C. Modelling and Numerical Simulation for the Coupling of Surface Flow with Subsurface Flow[D].Hong Kong:The Hong Kong University of Science and Technology, 2008
[23] Discacciati M, Quarteroni A. Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations. Comput Vis Sci, 2004, 6:93–103
[24] Quarteroni A, Valli A. Domain Decomposition Methods for Partial Differential Equations. Oxford:Oxford University Press, 1999
[25] Shan L, Zheng H B, Layton W J. A decoupling method with different subdomain time steps for the nonstationary Stokes-Darcy model. Numer Methods Partial Differ Eqns, 2013, 29:549–583
[26] Shan L, Zheng H B. Partitioned time stepping method for fully evolutionary Stokes-Darcy flow with the Beavers-Joseph interface conditions. SIAM J Numer Anal, 2013, 51:813–839
[27] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York:Springer-Verlag, 1991
[28] Zuo L, Hou Y. A decoupling two-grid algorithm for the mixed Stokes-Darcy model with the Beavers-Joseph interface condition. Numerical Methods for Partial Differential Equations, 2014, 30(3):1066–1082
[29] Zuo L, Hou Y. A two-grid decoupling method for the mixed Stokes-Darcy model. J Comput Appl Math,2015, 275:139-147
[30] Zuo L, Hou Y. Numerical analysis for thr mixed Navier-Stokes and Darcy problem with the Beavers-Joseph interface condition. Numer Methods Partial Differ Equ, 2015, 31(4):1009–1030
[31] Girault V, Raviart P A. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms.Berlin:Springer-Verlag, 1986
[32] Du G, Zuo L. Local and parallel finite element method for the mixed Navier-Stokes/Darcy model with Beavers-Joseph interface conditions. Acta Methematica Scientia, 2017, 37B(5):1331–1347
[2] Arbogast T, Gomez M S M. A discretization and multigrid solver for a Darcy-Stokes system of three dimensional vuggy porous media. Comput Geosci, 2009, 13(3):331–348
[3] Arbogast T, Lehr H L. Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput Geosci, 2006, 10(3):291–302
[4] Cao Y, Gunzburger M, Hu X, Hua F, Wang X, Zhao W. Finite element approximations for Stokes-Darcy flow with Beavers-Joseph interface conditions. SIAM J Numer Anal, 2010, 47(6):4239–4256
[5] Cao Y, Gunzburger M, Hua F, Wang X. Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun Math Sci, 2010, 8(1):1-25
[6] Mu M, Xu J C. A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM J Numer Anal, 2007, 45:1801–1813
[7] Ervin V J, Jenkins E W, Sun S. Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM J Numer Anal, 2009, 47(2):929–952
[8] Moraiti M. On the quasistatic approximation in the Stokes-Darcy model of groundwater-surface water flows. J Math Anal Appl, 2012, 394(2):796–808
[9] Cai M, Mu M, Xu J. Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach.SIAM J Numer Anal, 2009, 47(5):3325–3338
[10] Badea L, Discacciati M, Quarteroni A. Numerical analysis of the Navier-Stokes/Darcy coupling. Numerische Mathematik, 2010, 115(2):195–227
[11] C?e?smelioˇglu A, Riviere B. Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J Numer Mathem, 2008, 16(4):249–280
[12] C?e?smelioˇglu A, Riviere B. Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow. J Sci Comput, 2009, 40(1/3):115–140
[13] Chidyagwai P, Riviere B. On the solution of the coupled Navier-Stokes and Darcy equations. Comput Methods Appl Mech Engin, 2009, 198(47/48):3806–3820
[14] Girault V, Riviere B. DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-JosephSaffman interface condition. SIAM J Numer Anal, 2009, 47(3):2052–2089
[15] Hou Y. Optimal Error Estimates of A Decoupled Scheme Based on Two-Grid Finite Element for Mixed Stokes-Darcy Model. Appl Math Lett, 2016, 57:90–96
[16] Beavers G, Joseph D. Boundary conditions at a naturally permeable wall. J Fluid Mech, 1967, 30:197–207
[17] Discacciati M, Miglio E, Quarteroni A. Mathematical and numerical models for coupling surface and groundwater flows. Appl Numer Math, 2002, 43:57–74
[18] Jager W, Mikelic A. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J Appl Math, 2000, 60:1111–1127
[19] Layton W. A two-level discretization method for the Navier-Stokes equations. Comput Math Appl, 1993,26:33–38
[20] Nield D A, Bejan A. Convection in Porous Media. New York:Springer-Verlag, 1999
[21] Saffman P. On the boundary condition at the surface of a porous media. Stud Appl Math, 1971, 50:93–101
[22] Cai M C. Modelling and Numerical Simulation for the Coupling of Surface Flow with Subsurface Flow[D].Hong Kong:The Hong Kong University of Science and Technology, 2008
[23] Discacciati M, Quarteroni A. Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations. Comput Vis Sci, 2004, 6:93–103
[24] Quarteroni A, Valli A. Domain Decomposition Methods for Partial Differential Equations. Oxford:Oxford University Press, 1999
[25] Shan L, Zheng H B, Layton W J. A decoupling method with different subdomain time steps for the nonstationary Stokes-Darcy model. Numer Methods Partial Differ Eqns, 2013, 29:549–583
[26] Shan L, Zheng H B. Partitioned time stepping method for fully evolutionary Stokes-Darcy flow with the Beavers-Joseph interface conditions. SIAM J Numer Anal, 2013, 51:813–839
[27] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York:Springer-Verlag, 1991
[28] Zuo L, Hou Y. A decoupling two-grid algorithm for the mixed Stokes-Darcy model with the Beavers-Joseph interface condition. Numerical Methods for Partial Differential Equations, 2014, 30(3):1066–1082
[29] Zuo L, Hou Y. A two-grid decoupling method for the mixed Stokes-Darcy model. J Comput Appl Math,2015, 275:139-147
[30] Zuo L, Hou Y. Numerical analysis for thr mixed Navier-Stokes and Darcy problem with the Beavers-Joseph interface condition. Numer Methods Partial Differ Equ, 2015, 31(4):1009–1030
[31] Girault V, Raviart P A. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms.Berlin:Springer-Verlag, 1986
[32] Du G, Zuo L. Local and parallel finite element method for the mixed Navier-Stokes/Darcy model with Beavers-Joseph interface conditions. Acta Methematica Scientia, 2017, 37B(5):1331–1347