Rotational pressure-correction method for the Stokes/Darcy model based on the modular grad-div stabilization

Abstract This paper presents the rotational form of the pressure-correction method with the modular grad-div stabilization for the non-stationary Stokes/Darcy model. The presented methods can improve not only the efficiency of calculation but also the conservation of mass in finite element methods. Moreover, the stability of the presented methods is shown. Finally, some numerical results are given to show that the presented methods are reliable.

[1]  M. Olshanskii.  A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods , 2002 .

[2]  Leo G. Rebholz,et al.  Error analysis and iterative solvers for Navier–Stokes projection methods with standard and sparse grad-div stabilization , 2014 .

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

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

[5]  Jie Shen,et al.  On the error estimates for the rotational pressure-correction projection methods , 2003, Math. Comput..

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

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

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

[9]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[10]  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..

[11]  M. Fortin,et al.  A stable finite element for the stokes equations , 1984 .

[12]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[13]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[14]  Maxim A. Olshanskii,et al.  Grad-div stablilization for Stokes equations , 2003, Math. Comput..

[15]  Leo G. Rebholz,et al.  On the convergence rate of grad-div stabilized Taylor–Hood to Scott–Vogelius solutions for incompressible flow problems , 2011 .

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

[17]  Maxim A. Olshanskii,et al.  On the accuracy of the rotation form in simulations of the Navier-Stokes equations , 2009, J. Comput. Phys..

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

[19]  Jim Magiera,et al.  A multiple-time-step technique for coupled free flow and porous medium systems , 2014, J. Comput. Phys..

[20]  M. Olshanskii,et al.  Stable finite‐element calculation of incompressible flows using the rotation form of convection , 2002 .

[21]  Xiaoli Lu,et al.  A Modular Grad-Div Stabilization for the 2D/3D Nonstationary Incompressible Magnetohydrodynamic Equations , 2020, J. Sci. Comput..

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

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

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

[25]  Yanren Hou,et al.  A second‐order partitioned method with different subdomain time steps for the evolutionary Stokes‐Darcy system , 2018 .

[26]  Maxim A. Olshanskii,et al.  Grad–div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations , 2009 .

[27]  R. E. Showalter,et al.  Nonlinear Degenerate Evolution Equations in Mixed Formulation , 2010, SIAM J. Math. Anal..

[28]  W. Layton,et al.  A decoupling method with different subdomain time steps for the nonstationary stokes–darcy model , 2013 .

[29]  Haibiao Zheng,et al.  The efficient rotational pressure-correction schemes for the coupling Stokes/Darcy problem , 2020, Comput. Math. Appl..

[30]  William Layton,et al.  An efficient and modular grad–div stabilization , 2017, Computer Methods in Applied Mechanics and Engineering.

[31]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[32]  Pengzhan Huang,et al.  A grad-div stabilized projection finite element method for a double-diffusive natural convection model , 2020 .