Finite Element Approximation of Nonsolenoidal, Viscous Flows around Porous and Solid Obstacles

• Authors: Ross Ingram
• Venue: SIAM Journal on Numerical Analysis
• Year: 2011

We analyze a finite element discretization of the Brinkman equation for modeling non-Darcian fluid flow by allowing the Brinkman viscosity $\tilde{\nu}\rightarrow\infty$ and permeability $K\rightarrow0$ in solid obstacles, and $K\rightarrow\infty$ in the fluid domain. In this context, the Brinkman parameters are generally highly discontinuous. Furthermore, we consider nonhomogeneous Dirichlet boundary conditions $\mathbf{u}|_{\partial\Omega}=\phi\neq0$ and nonsolenoidal velocity $\nabla\cdot\mathbf{u}=g\neq0$ (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 $\tilde{\nu}\rightarrow\infty$ and $K\rightarrow0$ in solid obstacles, as $K\rightarrow\infty$ in the fluid region, and as the mesh width $h\rightarrow0$. For nonsolenoidal Brinkman flows, we include a small data condition on $\phi$ 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).