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March
2021, 14(3): 1001-1015.
doi: 10.3934/dcdss.2020380
Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems
Division of Mathematical and Physical Sciences, Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa 920-1192, Japan |
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 and Continuous Dynamical Systems - S,
2021, 14
(3)
: 1001-1015.
doi: 10.3934/dcdss.2020380
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
S. J. Cummins and M. Rudman,
An SPH projection method, J. Comput. Phys., 152 (1999), 584-607.
doi: 10.1006/jcph.1999.6246. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
J. B. Perot,
An analysis of the fractional step method, J. Comput. Phys., 108 (1993), 51-58.
doi: 10.1006/jcph.1993.1162. |
| [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. |
| [19] |
J. A. Trangenstein, Numerical Solution of Elliptic and Parabolic Partial Differential Equations, Cambridge University Press, Cambridge, 2013.
doi: 10.1017/CBO9781139025508.
|
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
S. J. Cummins and M. Rudman,
An SPH projection method, J. Comput. Phys., 152 (1999), 584-607.
doi: 10.1006/jcph.1999.6246. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
J. B. Perot,
An analysis of the fractional step method, J. Comput. Phys., 108 (1993), 51-58.
doi: 10.1006/jcph.1993.1162. |
| [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. |
| [19] |
J. A. Trangenstein, Numerical Solution of Elliptic and Parabolic Partial Differential Equations, Cambridge University Press, Cambridge, 2013.
doi: 10.1017/CBO9781139025508.
|
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