An extension of the MAC scheme to locally refined meshes: convergence analysis for the full tensor time-dependent Navier–Stokes equations

A variational formulation of the standard marker-and-cell scheme for the approximation of the Navier–Stokes problem yields an extension of the scheme to general 2D and 3D domains and more general meshes. An original discretization of the trilinear form of the nonlinear convection term is proposed; it is designed so as to vanish for discrete divergence free functions. This property allows us to give a mathematical proof of the convergence of the resulting approximate solutions, for the nonlinear Navier–Stokes equations in both steady-state and time-dependent regimes, without any small data condition. Numerical examples (analytical steady and time-dependent ones, inclined driven cavity) confirm the robustness and the accuracy of this method.

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