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

Received  December 2018 Revised  January 2020 Published  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).

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. BernardiT. 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. BertoluzzaV. ChabannesC. 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. ConcaF. 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. ConcaC. ParésO. 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. GuermondP. 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. McKeeM. F. ToméJ. A. CuminatoA. 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. BernardiT. 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. BertoluzzaV. ChabannesC. 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. ConcaF. 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. ConcaC. ParésO. 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. GuermondP. 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. McKeeM. F. ToméJ. A. CuminatoA. 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
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