We have completed the following work, available as a preprint:
G. R. Barrenechea, M. Bosy, V. Dolean, F. Nataf, P.-H. Tournier, "Hybrid discontinuous Galerkin discretization and preconditioning for the Stokes problem with non-standard boundary conditions", (arXiv preprint) submitted to the Journal of Scientific Computing.
Summary: Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this paper we present the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. An analysis of the boundary value problem with non standard condition is provided as well as the numerical evidence showing the advantages of the new method.