Mass conservation of finite element methods for coupled flow-transport problems

Advection-diffusion equations arise in a number of important applications. Their robust and accurate numerical solution is – in case of advection dominated flows – still a challenge. Often it is neglected that the physical processes are governed by a velocity field u which itself is the solution of a hydrodynamical model like the incompressible Navier-Stokes equations. In this paper we address the issue of mass conservation when the underlying velocity field u in the transport equation is replaced by an approximation uh. We consider the simplest case of a coupled flow-transport problem in a bounded domain Ω ⊂ Rd, d = 2, 3. The system is described by the instationary, incompressible Navier–Stokes equations ut − ν4u+ (u · ∇)u+∇p = f in Ω× (0, T ], div u = 0 in Ω× (0, T ], u = ub on ∂Ω× (0, T ], u(0) = u in Ω, (1)