Approximating fast, viscous fluid flow in complicated domains

Typical industrial and biological flows often occur in complicated domains that are either infeasible or impossible to resolve. Alternatives to solving the Navier-Stokes equations (NSE) for the fluid velocity in the pores of these problems must be considered. We propose and analyze a finite element discretization of the Brinkman equation for modeling non-Darcian fluid flow by allowing the Brinkman viscosity tends to infinity and permeability K tends to 0 in solid obstacles, and K tends to infinity in fluid domain. In this context, the Brinkman parameters are generally highly discontinuous. Furthermore, we consider inhomogeneous Dirichlet boundary conditions and non-solenoidal velocity (to model sources/sinks). Coupling between these two conditions makes even existence of solutions subtle. We establish conditions for the well-posedness of the continuous and discrete problem. We also establish convergence as Brinkman viscosity tends to infinity and K tends to 0 in solid obstacles, as K tends to infinity in fluid region, and as the mesh width vanishes. We prove similar results for time-dependent Brinkman equations for backward-Euler (BE) time-stepping. We provide numerical examples confirming theory including convergence of velocity, pressure, and drag/lift.We also investigate the stability and convergence of the fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various contexts since then, all known convergence estimates of CNLE require a time-step restriction. We show herein that no such restriction is required. Moreover, we propose a new linear extrapolation of the convecting velocity for CNLE so that the approximating velocities converge without without time-step restriction in l^{infty}(H^1) along with the discrete time derivative of the velocity in l^2(L^2). The new extrapolation ensures energetic stability of CNLE in the case of inhomogeneous boundary data. Such a result is unknown for conventional CNLE (usual techniques fail!). Numerical illustrations are provided showing that our new extrapolation clearly improves upon stability and accuracy from conventional CNLE