American Institute of Mathematical Sciences

Advanced
March  2021, 14(3): 1001-1015. doi: 10.3934/dcdss.2020380

Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems

Kazunori Matsui

Division of Mathematical and Physical Sciences, Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa 920-1192, Japan

Received  December 2018 Revised  January 2020 Published  March 2021 Early access  June 2020

We consider a boundary value problem for the stationary Stokes problem and the corresponding pressure-Poisson equation. We propose a new formulation for the pressure-Poisson problem with an appropriate additional boundary condition. We establish error estimates between solutions to the Stokes problem and the pressure-Poisson problem in terms of the additional boundary condition. As boundary conditions for the Stokes problem, we use a traction boundary condition and a pressure boundary condition introduced in C. Conca et al (1994).

Keywords: Stokes problem, pressure-Poisson equation.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35A35, 76D07.
Citation: Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380
References:
[1]

A. A. Amsden and F. H. Harlow, A simplified MAC technique for incompressible fluid flow calculations, J. Comput. Phys., 6 (1970), 322-325.  doi: 10.1016/0021-9991(70)90029-X.  Google Scholar

[2]

C. Bernardi, T. Chacón Rebollo and D. Yakoubi, Finite element discretization of the Stokes and Navier–Stokes equations with boundary conditions on the pressure, SIAM J. Numer. Anal., 53 (2015), 1256-1279.  doi: 10.1137/140972299.  Google Scholar

[3]

S. Bertoluzza, V. Chabannes, C. Prud'homme and M. Szopos, Boundary conditions involving pressure for the Stokes problem and applications in computational hemodynamics, Comput. Methods Appl. Mech. Engrg., 322 (2017), 58-80.  doi: 10.1016/j.cma.2017.04.024.  Google Scholar

[4]

A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[5]

C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Jpn. J. Math. (N.S.), 20 (1994), 279-318.  doi: 10.4099/math1924.20.279.  Google Scholar

[6]

C. Conca, C. Parés, O. Pironneau and M. Thiriet, Navier-Stokes equations with imposed pressure and velocity fluxes, Int. J. Numer. Meth. Fluids, 20 (1995), 267-287.  doi: 10.1002/fld.1650200402.  Google Scholar

[7]

S. J. Cummins and M. Rudman, An SPH projection method, J. Comput. Phys., 152 (1999), 584-607.  doi: 10.1006/jcph.1999.6246.  Google Scholar

[8]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[9]

P. M. Gresho and R. L. Sani, On pressure boundary conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 7 (1987), 1111-1145.  doi: 10.1002/fld.1650071008.  Google Scholar

[10]

J.-L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6011-6045.  doi: 10.1016/j.cma.2005.10.010.  Google Scholar

[11]

J.-L. Guermond and L. Quartapelle, On stability and convergence of projection methods based on pressure Poisson equation, Int. J. Numer. Meth. Fluids, 26 (1998), 1039-1053.  doi: 10.1002/(SICI)1097-0363(19980515)26:9<1039::AID-FLD675>3.0.CO;2-U.  Google Scholar

[12]

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface, The Physics of Fluids, 8 (1965), 2182-2189.  doi: 10.1063/1.1761178.  Google Scholar

[13]

J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323.  doi: 10.1016/0021-9991(85)90148-2.  Google Scholar

[14]

J. Liu, Open and traction boundary conditions for the incompressible Navier-Stokes equations, J. Comput. Phys., 228 (2009), 7250-7267.  doi: 10.1016/j.jcp.2009.06.021.  Google Scholar

[15]

S. Marušić, On the Navier-Stokes system with pressure boundary condition, Ann. Univ. Ferrara, 53 (2007), 319-331.  doi: 10.1007/s11565-007-0024-y.  Google Scholar

[16]

S. McKee, M. F. Tomé, J. A. Cuminato, A. Castelo and V. G. Ferreira, Recent advances in the marker and cell method, Arch. Comput. Meth. Engng., 11 (2004), 107-142.  doi: 10.1007/BF02905936.  Google Scholar

[17]

J. B. Perot, An analysis of the fractional step method, J. Comput. Phys., 108 (1993), 51-58.  doi: 10.1006/jcph.1993.1162.  Google Scholar

[18]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[19] J. A. Trangenstein, Numerical Solution of Elliptic and Parabolic Partial Differential Equations, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025508.  Google Scholar

show all references

References:
[1]

A. A. Amsden and F. H. Harlow, A simplified MAC technique for incompressible fluid flow calculations, J. Comput. Phys., 6 (1970), 322-325.  doi: 10.1016/0021-9991(70)90029-X.  Google Scholar

[2]

C. Bernardi, T. Chacón Rebollo and D. Yakoubi, Finite element discretization of the Stokes and Navier–Stokes equations with boundary conditions on the pressure, SIAM J. Numer. Anal., 53 (2015), 1256-1279.  doi: 10.1137/140972299.  Google Scholar

[3]

S. Bertoluzza, V. Chabannes, C. Prud'homme and M. Szopos, Boundary conditions involving pressure for the Stokes problem and applications in computational hemodynamics, Comput. Methods Appl. Mech. Engrg., 322 (2017), 58-80.  doi: 10.1016/j.cma.2017.04.024.  Google Scholar

[4]

A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[5]

C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Jpn. J. Math. (N.S.), 20 (1994), 279-318.  doi: 10.4099/math1924.20.279.  Google Scholar

[6]

C. Conca, C. Parés, O. Pironneau and M. Thiriet, Navier-Stokes equations with imposed pressure and velocity fluxes, Int. J. Numer. Meth. Fluids, 20 (1995), 267-287.  doi: 10.1002/fld.1650200402.  Google Scholar

[7]

S. J. Cummins and M. Rudman, An SPH projection method, J. Comput. Phys., 152 (1999), 584-607.  doi: 10.1006/jcph.1999.6246.  Google Scholar

[8]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[9]

P. M. Gresho and R. L. Sani, On pressure boundary conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 7 (1987), 1111-1145.  doi: 10.1002/fld.1650071008.  Google Scholar

[10]

J.-L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6011-6045.  doi: 10.1016/j.cma.2005.10.010.  Google Scholar

[11]

J.-L. Guermond and L. Quartapelle, On stability and convergence of projection methods based on pressure Poisson equation, Int. J. Numer. Meth. Fluids, 26 (1998), 1039-1053.  doi: 10.1002/(SICI)1097-0363(19980515)26:9<1039::AID-FLD675>3.0.CO;2-U.  Google Scholar

[12]

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface, The Physics of Fluids, 8 (1965), 2182-2189.  doi: 10.1063/1.1761178.  Google Scholar

[13]

J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323.  doi: 10.1016/0021-9991(85)90148-2.  Google Scholar

[14]

J. Liu, Open and traction boundary conditions for the incompressible Navier-Stokes equations, J. Comput. Phys., 228 (2009), 7250-7267.  doi: 10.1016/j.jcp.2009.06.021.  Google Scholar

[15]

S. Marušić, On the Navier-Stokes system with pressure boundary condition, Ann. Univ. Ferrara, 53 (2007), 319-331.  doi: 10.1007/s11565-007-0024-y.  Google Scholar

[16]

S. McKee, M. F. Tomé, J. A. Cuminato, A. Castelo and V. G. Ferreira, Recent advances in the marker and cell method, Arch. Comput. Meth. Engng., 11 (2004), 107-142.  doi: 10.1007/BF02905936.  Google Scholar

[17]

J. B. Perot, An analysis of the fractional step method, J. Comput. Phys., 108 (1993), 51-58.  doi: 10.1006/jcph.1993.1162.  Google Scholar

[18]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[19] J. A. Trangenstein, Numerical Solution of Elliptic and Parabolic Partial Differential Equations, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025508.  Google Scholar
Figure 1.  Image of a flow in a pipe
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[2]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[3]

Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109
[4]

Pedro Gabriel Fernández-Dalgo, Pierre Gilles Lemarié–Rieusset. Characterisation of the pressure term in the incompressible Navier–Stokes equations on the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2917-2931. doi: 10.3934/dcdss.2020408
[5]

C. Davini, F. Jourdan. Approximations of degree zero in the Poisson problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 267-281. doi: 10.3934/cpaa.2005.4.267
[6]

Hanchun Yang, Meimei Zhang, Qin Wang. Global solutions of shock reflection problem for the pressure gradient system. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3387-3428. doi: 10.3934/cpaa.2020150
[7]

Gui-Qiang G. Chen, Qin Wang, Shengguo Zhu. Global solutions of a two-dimensional Riemann problem for the pressure gradient system. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2475-2503. doi: 10.3934/cpaa.2021014
[8]

Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521
[9]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[10]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the quantum Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 273-294. doi: 10.3934/cpaa.2017013
[11]

Wolfgang Arendt, Daniel Daners. Varying domains: Stability of the Dirichlet and the Poisson problem. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 21-39. doi: 10.3934/dcds.2008.21.21
[12]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[13]

Paolo Maremonti. A remark on the Stokes problem in Lorentz spaces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1323-1342. doi: 10.3934/dcdss.2013.6.1323
[14]

Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic & Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063
[15]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939
[16]

Luca Lussardi. On a Poisson's equation arising from magnetism. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 769-772. doi: 10.3934/dcdss.2015.8.769
[17]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292
[18]

Yinzheng Sun, Qin Wang, Kyungwoo Song. Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4899-4920. doi: 10.3934/cpaa.2020217
[19]

Peng Zhang, Jiequan Li, Tong Zhang. On two-dimensional Riemann problem for pressure-gradient equations of the Euler system. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 609-634. doi: 10.3934/dcds.1998.4.609
[20]

Zhendong Fang, Hao Wang. Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021231

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (135)
  • HTML views (276)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences