Finite element approximation of nonsolenoidal, viscous flows around porous and solid obstacles

摘要

We analyze a finite element discretization of the Brinkman equation for modeling non-Darcian fluid flow by allowing the Brinkman viscosity ?? →∞ and permeability K → 0 in solid obstacles, and K →∞ in the fluid domain. In this context, the Brinkman parameters are generally highly discontinuous. Furthermore, we consider nonhomogeneous Dirichlet boundary conditions u|?Ω = φ ≠ 0 and nonsolenoidal velocity ? ·u = g ≠ 0 (to model sources/sinks). Coupling between these two conditions makes even existence of solutions subtle. We establish well-posedness of the continuous and discrete problem, a priori stability estimates, and convergence as c? →∞and K → 0 in solid obstacles, as K →∞ in the fluid region, and as the mesh width h → 0. For nonsolenoidal Brinkman flows, we include a small data condition on ? and g to ensure existence of solutions (a similar conclusion is attainable for existence of solutions to the steady Navier-Stokes equations with nonhomogeneous data).