Gradient Schemes for Stokes problem

International audienceThe gradient scheme framework encompasses several conforming and non-conforming numericalschemes for diffusion equations. We develop here this framework for the approximation of the steadystate and transient incompressible Stokes equations with homogeneous Dirichlet boundary conditions.Using this framework, we establish generic convergence results – by error estimates in the case ofthe steady problem, and by compactness arguments in the case of the transient problem – thatare applicable to both old and new schemes for Stokes’ equations. Three classical methods (MAC,Taylor–Hood and Crouzeix–Raviart schemes) are shown to fit into the gradient schemes framework;some of the convergence results obtained for those through the framework are new. We also show thata Hybrid Mixed Mimetic scheme, extension of the Crouzeix–Raviart scheme to any polyhedral mesh,can be designed within the gradient scheme framework; this scheme is new for Stokes’ equations, andour abstract analysis establishes its convergence along with error estimates