Incompressible Finite Elements via Hybridization. Part II: The Stokes System in Three Space Dimensions

We introduce a method that gives exactly incompressible velocity approximations to Stokes flow in three space dimensions. The method is designed by extending the ideas in Part I [B. Cockburn and J. Gopalakrishnan, SIAM J. Numer. Anal., 43 (2005), pp. 1627--1650] of this series, where the Stokes system in two space dimensions was considered. Thus we hybridize a vorticity-velocity formulation to obtain a new mixed method coupling approximations of tangential velocity and pressure on mesh faces. Once this relatively small tangential velocity-pressure system is solved, it is possible to recover a globally divergence-free numerical approximation of the fluid velocity, an approximation of the vorticity whose tangential component is continuous across interelement boundaries, and a discontinuous numerical approximation of the pressure. The main difference between our method here and that of the two-dimensional case treated in Part I is in the use of Nédélec elements, which necessitates development of new hybridization techniques. We also generalize the method to allow for varying polynomial degrees on different mesh elements and to incorporate certain nonstandard but physically relevant boundary conditions.