A standard numerical method in order to approach the solution of a time
dependent convection-diffusion equation in φ
transported with velocity u, consists to multiply the full equation by a space
dependent test function ψ, to integrate it on the computational domain
Ω and to discretize it in space with a finite element method and in
time with a finite difference scheme. The diffusion term is integrated by
part on Ω but not the advected term u.gradφ.


In the convection dominated regime, a streamline upwind method SUPG is
used in order to stabilize the numerical scheme. In principle, when the flow
is incompressible and confined in Ω, i.e. when divu=0
in Ω and u.n=0 on the boundary ∂Ω,
the integral of φ on the domain Ω
remains constant in time when the source term is vanishing (conservation of
the mass balance). However, on a practical point of view, the velocity
u is often computed with a Navier-Stokes solver which leads to
an approximation u_h which is not exactly divergence
free.

As an unwelcome numerical effect, the mass balance is not conserved
when the time goes up. Especially the mass balance defect can be important
when the equation is integrated on a long time. In this talk, we propose an
original modification of the standard numerical scheme in order to eliminate
this defect and we establish some error estimates produced by this scheme.

[Invited by J-P. Berrut]