vel_cor - SIAM J NUMER ANAL Vol 41 No 1 pp 112134 c 2003...

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VELOCITY-CORRECTION PROJECTION METHODS FOR INCOMPRESSIBLE FLOWS J. L. GUERMOND AND JIE SHEN SIAM J. N UMER. A NAL . c ° 2003 Society for Industrial and Applied Mathematics Vol. 41, No. 1, pp. 112–134 Abstract. We introduce and study a new class of projection methods—namely, the velocity- correction methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1 -norm for the velocity and of the L 2 -norm for the pressure. We also show that the class of fractional-step methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput ., 1 (1986), pp. 75–111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys ., 97 (1991), pp. 414–443] can be interpreted as the rotational form of our velocity-correction methods. Thus, to the best of our knowledge, our results provide the Frst rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L 2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the Fnite element methods. Key words. Navier–Stokes equations, projection methods, fractional-step methods, incom- pressibility, Fnite elements, spectral approximations AMS subject classifcations. 65M12, 35Q30, 35J05, 76D05 PII. S0036142901395400 1. Introduction. We consider in this paper the time discretization of the un- steady incompressible Navier–Stokes equations in primitive variables. For a given body force f ( t ) and an initial solenoidal vector ±eld v 0 ,welookfor u and p such that t u ν 2 u + u ·∇ u + p = f in Ω × [0 ,T ] , ∇· u = 0 in Ω × [0 ] , u | Γ =0 , u | t =0 = v 0 in Ω . (1.1) The boundary condition on the velocity is set to zero for the sake of simplicity. The fluid domain Ω is open and bounded in R d ( d = 2 or 3 in practical situations). The domain boundary Γ is assumed to be smooth; e.g., Γ is Lipschitzian and Ω is locally on one side of its boundary. The goal of this paper is to present a new class of fractional-step projection meth- ods. The original projection method was introduced by Chorin [3] and Temam [15] in the late 60s. An important class of projection methods is the so-called pressure- correction methods introduced in [5, 8]. These schemes consist of two substeps per time step: the pressure is treated explicitly in the ±rst substep and corrected in the sec- ond substep by projecting the intermediate velocity onto the space of divergence-free Received by the editors September 20, 2001; accepted for publication (in revised form) August 12, 2002; published electronically March 13, 2003. http://www.siam.org/journals/sinum/41-1/39540.html LIMSI (CNRS-UPR 3152), BP 133, 91403 Orsay, ±rance ([email protected]). Part of this work was completed while this author was visiting Texas Institute of Computational and Applied Mathematics, Austin, TX, during the period of August, 2001, to July, 2002, under a TICAM Visiting ±aculty ±ellowship.
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vel_cor - SIAM J NUMER ANAL Vol 41 No 1 pp 112134 c 2003...

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