    ## 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, doi: 10.3934/dcdss.2020380
##### References:
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##### References:
  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  J. A. Trangenstein, Numerical Solution of Elliptic and Parabolic Partial Differential Equations, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025508.  Google Scholar
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