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

Robert Eymard (1) , Pierre Feron (1) , Cindy Guichard (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 ANGE - Numerical Analysis, Geophysics and Ecology

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 generalized 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 situa- tions. We provide tools for analyzing all the combined methods, and proving their convergence to a weak solution of the problem.

Mots clés: incompressible Navier-Stokes equations; Gradient Discretisation Method; convergence analysis

Domaines: Analyse numérique [math.NA]

Robert Eymard, Pierre Feron, Cindy Guichard. Family of convergent numerical schemes for the incompressible Navier-Stokes equations. Mathematics and Computers in Simulation, 2018, 144, pp.196-218. ⟨10.1016/j.matcom.2017.08.003⟩. ⟨hal-01382924v2⟩