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

Kazunori Matsui

doi:10.3934/dcdss.2020380

Article Contents

2021, Volume 14, Issue 3: 1001-1015. Doi: 10.3934/dcdss.2020380

This issue Previous Article Computational optimization in solving the geodetic boundary value problems Next Article An automated segmentation of NATURA 2000 habitats from Sentinel-2 optical data

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: November 30, 2018

Revised: December 31, 2019

Early access: June 2020

Published: March 2021

Abstract Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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:

Full Text(HTML)

Figure 1. Image of a flow in a pipe

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.