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

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- by Jean-Luc Guermond, Peter D. Minev and Abner J. Salgado PDF
- Math. Comp.
**81**(2012), 1951-1977 Request permission

## 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.## References

- 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. - James H. Bramble, Joseph E. Pasciak, and Jinchao Xu,
*Parallel multilevel preconditioners*, Math. Comp.**55**(1990), no. 191, 1–22. MR**1023042**, DOI 10.1090/S0025-5718-1990-1023042-6 - James H. Bramble and Xuejun Zhang,
*The analysis of multigrid methods*, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 173–415. MR**1804746** - Alexandre Joel Chorin,
*Numerical solution of the Navier-Stokes equations*, Math. Comp.**22**(1968), 745–762. MR**242392**, DOI 10.1090/S0025-5718-1968-0242392-2 - Monique 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**, DOI 10.1137/0520006 - Jim Douglas Jr.,
*Alternating direction methods for three space variables*, Numer. Math.**4**(1962), 41–63. MR**136083**, DOI 10.1007/BF01386295 - Alexandre Ern and Jean-Luc Guermond,
*Theory and practice of finite elements*, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR**2050138**, DOI 10.1007/978-1-4757-4355-5 - S. Minakshi Sundaram,
*On non-linear partial differential equations of the parabolic type*, Proc. Indian Acad. Sci., Sect. A.**9**(1939), 479–494. MR**0000088**, DOI 10.1007/BF03046993 - P. Erdös,
*On the distribution of normal point groups*, Proc. Nat. Acad. Sci. U.S.A.**26**(1940), 294–297. MR**2000**, DOI 10.1073/pnas.26.4.294 - J. L. Guermond, P. Minev, and Jie Shen,
*An overview of projection methods for incompressible flows*, Comput. Methods Appl. Mech. Engrg.**195**(2006), no. 44-47, 6011–6045. MR**2250931**, DOI 10.1016/j.cma.2005.10.010 - Jean-Luc Guermond and Peter D. Minev,
*A new class of fractional step techniques for the incompressible Navier-Stokes equations using direction splitting*, C. R. Math. Acad. Sci. Paris**348**(2010), no. 9-10, 581–585 (English, with English and French summaries). MR**2645177**, DOI 10.1016/j.crma.2010.03.009 - 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**, DOI 10.1006/jcph.1996.5587 - J.-L. Guermond and Abner Salgado,
*A splitting method for incompressible flows with variable density based on a pressure Poisson equation*, J. Comput. Phys.**228**(2009), no. 8, 2834–2846. MR**2509298**, DOI 10.1016/j.jcp.2008.12.036 - J.-L. Guermond and Abner J. Salgado,
*Error analysis of a fractional time-stepping technique for incompressible flows with variable density*, SIAM J. Numer. Anal.**49**(2011), no. 3, 917–944. MR**2802553**, DOI 10.1137/090768758 - J. L. Guermond and Jie Shen,
*On the error estimates for the rotational pressure-correction projection methods*, Math. Comp.**73**(2004), no. 248, 1719–1737. MR**2059733**, DOI 10.1090/S0025-5718-03-01621-1 - J. L. Guermond and P. D. Minev,
*A new class of massively parallel direction splitting for the incompressible Navier-Stokes equations*, Comput. Methods Appl. Mech. Engrg.**200**(2011), no. 23-24, 2083–2093. MR**2795163**, DOI 10.1016/j.cma.2011.02.007 - —,
*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 http://onlinelibrary.wiley.com/doi/10.1002/fld.2583. - John G. Heywood and Rolf 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**, DOI 10.1137/0719018 - 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**796611**, DOI 10.1016/0021-9991(85)90148-2 - 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**, DOI 10.1016/0893-9659(91)90161-N - T. Lu, P. Neittaanmäki, and X.-C. Tai,
*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 (English, with English and French summaries). MR**1183413**, DOI 10.1051/m2an/1992260606731 - 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**71874**, DOI 10.1137/0103003 - Andreas Prohl,
*Projection and quasi-compressibility methods for solving the incompressible Navier-Stokes equations*, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1997. MR**1472237**, DOI 10.1007/978-3-663-11171-9 - L. Kantorovitch,
*The method of successive approximations for functional equations*, Acta Math.**71**(1939), 63–97. MR**95**, DOI 10.1007/BF02547750 - Jie 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**, DOI 10.1007/BF01396220 - Jie Shen,
*On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes*, Math. Comp.**65**(1996), no. 215, 1039–1065. MR**1348047**, DOI 10.1090/S0025-5718-96-00750-8 - R. Témam,
*Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II*, Arch. Rational Mech. Anal.**33**(1969), 377–385 (French). MR**244654**, DOI 10.1007/BF00247696 - Roger Temam,
*Navier-Stokes equations*, AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis; Reprint of the 1984 edition. MR**1846644**, DOI 10.1090/chel/343 - 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. - 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**

## 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
- Email: guermond@math.tamu.edu
**Peter D. Minev**- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta Canada T6G 2G1
- Email: minev@ualberta.ca
**Abner J. Salgado**- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Email: abnersg@math.umd.edu
- 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. - © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**81**(2012), 1951-1977 - MSC (2010): Primary 65N12, 65N15, 35Q30
- DOI: https://doi.org/10.1090/S0025-5718-2012-02588-9
- MathSciNet review: 2945144