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Summary: CONVERGENCE ANALYSIS OF A FINITE ELEMENT
PROJECTION/LAGRANGE-GALERKIN METHOD FOR THE INCOMPRESSIBLE
NAVIER{STOKES EQUATIONS
Y. ACHDOU AND J.-L. GUERMOND y
Abstract. This paper provides a convergence analysis of a fractional-step methods to compute incompressible
viscous
ows by means of nite element approximations. In the proposed algorithm, the convection, the di usion and the
incompressibility are treated in three di erent substeps. The convection is treated rst by means of a Lagrange/Galerkin
technique whereas the di usion and the incompressibility are treated separatedly in two subsequent substeps by means
of a projection method. It is shown that provided the time step is of O(h d=4 ), where h is the meshsize and d is the
space dimension (2 d 3), the proposed method yields for nite time T an error of O(h l+1 + Æt) in the L 2 norm for
the velocity and an error of O(h l + Æt) in the H 1 norm (or the L 2 norm for the pressure).
Key words. Incompressible Navier{Stokes equations, Projection method, Lagrange/Galerkin method, Fractional-
step method, Finite elements.
35A40, 35Q30, 65M12, 65N30
1. Introduction. The Lagrange-Galerkin method is a numerical technique for solving convection{
dominated convection{di usion problems. It consists of combining a Galerkin nite element procedure
with a discretization of the Lagrangian material derivative along the characteristics. It combines the
advantages of the methods which stabilize the convection (eg. upwinding, Petrov{Galerkin, etc.) with
the advantages of the methods which treat the convection in an explicit manner, that is to say, the
linear systems to be solved at each time step involve only di usion, are symmetric, and are time{
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