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Convergence analysis of a class of massively parallel direction splitting algorithms for the Navier-Stokes equations in simple domains

Authors: Jean-Luc Guermond, Peter D. Minev and Abner J. Salgado
Journal: Math. Comp. 81 (2012), 1951-1977
MSC (2010): Primary 65N12, 65N15, 35Q30
Published electronically: April 13, 2012
MathSciNet review: 2945144
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Abstract: We provide a convergence analysis for a new fractional time-stepping technique for the incompressible Navier-Stokes equations based on direction splitting. This new technique is of linear complexity, unconditionally stable and convergent, and suitable for massive parallelization.

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  • 1. B.F. Armally, F. Durst, J.C.F. Pereira, and B. Schönung, Experimental and theoretical investigation of backward-facing step flow, J. Fluid Mech. 127 (1983), 473-496.
  • 2. J.H. Bramble, J.E. Pasciak, and J. Xu, Parallel multilevel preconditioners., Math. Comp. 55 (1990), no. 191, 1-22 (English). MR 1023042 (90k:65170)
  • 3. J.H. Bramble and X. Zhang, The analysis of multigrid methods, Handbook of numerical analysis, Vol. VII, Handbook Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 173-415. MR 1804746 (2001m:65183)
  • 4. A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968), 745-762. MR 0242392 (39:3723)
  • 5. M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no. 1, 74-97. MR 977489 (90b:35191)
  • 6. J. Douglas, Jr., Alternating direction methods for three space variables, Numer. Math. 4 (1962), 41-63. MR 0136083 (24:B2122)
  • 7. A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138
  • 8. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. theory and algorithms, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, Germany, 1986. MR 88b:65129
  • 9. J.-L. Guermond, Un résultat de convergence d'ordre deux en temps pour l'approximation des équations de Navier-Stokes par une technique de projection incrémentale, M2AN Math. Model. Numer. Anal. 33 (1999), no. 1, 169-189, Also in C. R. Acad. Sci. Paris, Série I, 325:1329-1332, 1997. MR 2000k:65171
  • 10. J.-L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 44-47, 6011-6045. MR 2250931 (2007g:76157)
  • 11. J.-L. Guermond and P.D. Minev, A new class of fractional step techniques for the incompressible Navier-Stokes equations using direction splitting, Comptes Rendus Mathematique 348 (2010), no. 9-10, 581-585. MR 2645177 (2011f:76128)
  • 12. J.-L. Guermond and L. Quartapelle, Calculation of incompressible viscous flows by an unconditionally stable projection FEM, J. Comput. Phys. 132 (1997), no. 1, 12-33. MR 1440332 (97k:76066)
  • 13. J.-L. Guermond and A. Salgado, A splitting method for incompressible flows with variable density based on a pressure Poisson equation, Journal of Computational Physics 228 (2009), no. 8, 2834-2846. MR 2509298 (2010m:65174)
  • 14. J.-L. Guermond and A.J. Salgado, Error analysis of a fractional time-stepping technique for incompressible flows with variable density, SIAM Journal on Numerical Analysis 49 (2011), no. 3, 917-944. MR 2802553
  • 15. J.-L. Guermond and J. Shen, On the error estimates for the rotational pressure-correction projection methods, Math. Comp. 73 (2004), no. 248, 1719-1737 (electronic). MR 2059733
  • 16. J.L. Guermond and P.D. Minev, A new class of massively parallel direction splitting for the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 200 (2011), no. 23-24, 2083-2093. MR 2795163
  • 17. -, Start-up flow in a three-dimensional lid-driven cavity by means of a massively parallel direction splitting algorithm, Int. J. Numer. Meth. Fluids (2011), Available online at
  • 18. J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), no. 2, 275-311. MR 650052 (83d:65260)
  • 19. J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys. 59 (1985), no. 2, 308-323. MR 0796611 (87a:76046)
  • 20. T. Lu, P. Neittaanmäki, and X.-C. Tai, A parallel splitting up method and its application to Navier-Stokes equations, Appl. Math. Lett. 4 (1991), no. 2, 25-29. MR 1095644
  • 21. -, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (1992), no. 6, 673-708. MR 1183413 (93i:65111)
  • 22. D. W. Peaceman and H.H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955), 28-41. MR 0071874 (17,196d)
  • 23. A. Prohl, Projection and quasi-compressibility methods for solving the incompressible Navier-Stokes equations, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1997. MR 1472237 (98k:65058)
  • 24. R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations, The Navier-Stokes Equations II--Theory and Numerical Methods (Oberwolfach, 1991), Lecture Notes in Math., vol. 1530, Springer, Berlin, Germany, 1992, pp. 167-183. MR 95a:65149
  • 25. J. Shen, On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations, Numer. Math. 62 (1992), no. 1, 49-73. MR 1159045 (93a:35122)
  • 26. -, On error estimates of projection methods for the Navier-Stokes equations: second-order schemes, Math. Comp. 65 (1996), no. 215, 1039-1065. MR 1348047 (96j:65091)
  • 27. R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires ii, Arch. Rat. Mech. Anal. 33 (1969), 377-385. MR 0244654 (39:5968)
  • 28. R. Temam, Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001, Theory and numerical analysis, Reprint of the 1984 edition. MR 1846644 (2002j:76001)
  • 29. L.J.P. Timmermans, P.D. Minev, and F.N. van de Vosse, An approximate projection scheme for incompressible flow using spectral elements, Int. J. Numer. Methods Fluids 22 (1996), 673-688.
  • 30. N.N. Yanenko, B.G. Kuznetsov, and Sh. Smagulov, On the approximation of the Navier-Stokes equations for an incompressible fluid by evolutionary-type equations, Numerical methods in fluid dynamics, ``Mir'', Moscow, 1984, pp. 290-314. MR 804995 (87a:65156)

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Additional Information

Jean-Luc Guermond
Affiliation: Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, Texas 77843-3368. On leave from CNRS, France

Peter D. Minev
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta Canada T6G 2G1

Abner J. Salgado
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Keywords: Navier-Stokes, fractional time-stepping, direction splitting
Received by editor(s): January 16, 2011
Received by editor(s) in revised form: June 16, 2011
Published electronically: April 13, 2012
Additional Notes: This material is based upon work supported by the National Science Foundation grants DMS-0713829, by the Air Force Office of Scientific Research, USAF, under grant/contract number FA9550-09-1-0424, and a Discovery grant of the National Science and Engineering Research Council of Canada. This publication is also partially based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST)
The work of P. Minev was also supported by fellowships from the Institute of Applied Mathematics and Computational Science and the Institute of Scientific Computing at Texas A&M University
The work of A.J. Salgado was also been supported by NSF grants CBET-0754983 and DMS-0807811.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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