This work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density. One original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation, to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid. It is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step. In particular, a second order accurate BDF projection method is presented and its numerical performance is illustrated by test computations and comparisons.

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 functional setting for projection methods must accommodate two different spaces 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 algorithm. 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

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This work investigates the proper choices of spatial approximations for velocity and pressure in fractional-step projection methods. Numerical results obtained with classical finite element interpolations are presented. These tests confirm the role of the inf-sup LBB condition in non-incremental and incremental versions of the method for computing viscous incompressible flows.

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