This study is mainly dedicated to the development and analysis ofnon-overlapping domain decomposition methods for solving continuous-pressurefinite element formulations of the Stokes problem. These methods have thefollowing special features. By keeping the equations and unknowns unchanged atthe cross points, that is, points shared by more than two subdomains, one caninterpret them as iterative solvers of the actual discrete problem directlyissued from the finite element scheme. In this way, the good stabilityproperties of continuous-pressure mixed finite element approximations of theStokes system are preserved. Estimates ensuring that each iteration can beperformed in a stable way as well as a proof of the convergence of theiterative process provide a theoretical background for the application of therelated solving procedure. Finally some numerical experiments are given todemonstrate the effectiveness of the approach, and particularly to compare itsefficiency with an adaptation to this framework of a standard FETI-DP method.

In this paper we introduce a new class of numerical schemes for the incompressible Navier-Stokes equations, which are inspired by the theory of discrete kinetic schemes for compressible fluids. For these approximations it is possible to give a stability condition, based on a discrete velocities version of the Boltzmann H-theorem. Numerical tests are performed to investigate their convergence and accuracy.