Family of convergent numerical schemes for the incompressible Navier-Stokes equations

This paper presents the common mathematical features which are leading to convergence properties for a family of numerical schemes applied to the discretisation of the steady and transient incompressible Navier-Stokes equations with homogeneous Dirichlet's boundary conditions. This family includes the Taylor-Hood scheme, the MAC scheme, the Crouzeix-Raviart scheme generalised into the Hybrid Mixed Mimetic scheme, which can be combined with a variety of discretisations for the nonlinear convection term, each of them being more efficient than the others in particular situations. We provide tools for analysing all the combined methods, and proving their convergence to a weak solution of the problem.