numer_math98 - Numer Math(1998 80 207238 Numerische...

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Numer. Math. (1998) 80: 207–238 Numerische Mathematik c ± Springer-Verlag 1998 Electronic Edition On the approximation of the unsteady Navier–Stokes equations by finite element projection methods J.-L. Guermond 1 , L. Quartapelle 2 1 Laboratoire d’Informatique pour la M´ecanique et les Sciences de l’Ing´enieur, CNRS, BP 133, F-91403 Orsay, France; e-mail: [email protected] 2 Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy Received October 2, 1995 / Revised version received July 9, 1997 Summary. This paper provides an analysis of a fractional-step projection method to compute incompressible viscous flows by means of finite element approximations. The analysis is based on the idea that the appropriate func- tionalsettingforprojectionmethodsmustaccommodatetwodifferentspaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method. Such a theoretical distinction leads to a finite element projection method with a Poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algo- rithm. Error estimates in finite time are given. An extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method. Mathematics Subject Classification (1991): 35Q30, 65M12, 65M60 1. Introduction The fractional-step projection method of Chorin [10,11] and Temam [28] (see also Temam [27] and Quartapelle [24]) is the most frequently employed technique for the numerical solution of the primitive variable Navier–Stokes equations. This method is based on a rather peculiar time-discretization of the equations governing viscous incompressible flows, in which the vis- cosity and the incompressibility of the fluid are dealt within two separate steps. The reader is referred to Rannacher [26] for a thorough analysis of the Correspondence to : J.-L. Guermond Numerische Mathematik Electronic Edition page 207 of Numer. Math. (1998) 80: 207–238
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208 J.-L. Guermond, L. Quartapelle convergence in time of the original Chorin–Temam algorithm (i.e. the non- incremental form); it is also shown in [26] that the projection algorithm can be interpreted as a pressure stabilization technique. In practice, the projec- tion method is combined with any kind of spatial discretization technique, viz., finite differences (see e.g. Bell et al. [4]), finite elements (Donea et al. [13], Gresho and Chan [15]), or spectral approximations (Ku et al. [22]). The aim of the present paper to provide a framework and an error analysis for such fully discretized schemes. An important, although almost never analyzed, feature of fractional-step projection methods is the structural difference existing between the equa- tions of the viscous step and those of the incompressible phase of the calcula- tion. In fact the first half-step constitutes an elliptic boundary value problem for an intermediate velocity unknown accounting for the viscous diffusion
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numer_math98 - Numer Math(1998 80 207238 Numerische...

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