Gradient Schemes for Stokes problem

Jerome Droniou (1) , Robert Eymard (2) , Pierre Feron (2)

1 School of Mathematical Sciences [Clayton]
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

The gradient scheme framework encompasses several conforming and non-conforming numerical schemes for diffusion equations. We develop here this framework for the approximation of the steady state and transient incompressible Stokes equations with homogeneous Dirichlet boundary conditions. Using this framework, we establish generic convergence results – by error estimates in the case of the steady problem, and by compactness arguments in the case of the transient problem – that are 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 that a 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, and our abstract analysis establishes its convergence along with error estimates.

Domaines: Analyse numérique [math.NA]

Jerome Droniou, Robert Eymard, Pierre Feron. Gradient Schemes for Stokes problem. IMA Journal of Numerical Analysis, 2015, ⟨10.1093/imanum/drv061⟩. ⟨hal-01070703v2⟩