Abstract
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.
Highlights
The Marker-And-Cell (MAC) scheme, introduced in [13] is one of the most popular methods [19, 24] for the approximation of the Navier-Stokes equations in the engineering framework, because of its simplicity and of its remarkable mathematical properties
Error estimates may be obtained by viewing the MAC scheme as a mixed finite element method of the vorticity formulation [11], or by a mixed method in primitive variables, with the pressures approximated by Q1 finite elements [12]
It is proven in [14] that a divergence conforming DG scheme based on the lowest order Raviart-Thomas space on rectangular meshes is algebraically equivalent to the MAC scheme
Summary
The Marker-And-Cell (MAC) scheme, introduced in [13] is one of the most popular methods [19, 24] for the approximation of the Navier-Stokes equations in the engineering framework, because of its simplicity and of its remarkable mathematical properties. The aim of this paper is to provide the complete mathematical analysis of an extension of the MAC scheme on possibly non conforming meshes, allowing local refinement. The modification concerns the discretization of the momentum equation, which was performed on dual Voronoı cells in [3], while it is performed by a linear finite element method on a Delaunay triangulation built from the centers of the set of edges (in 2D), or faces (in 3D) where each component of the velocity is defined This modification was found necessary in the mathematical analysis of the scheme for the steady–state and time–dependent nonlinear Navier-Stokes equations, in order to prove the convergence of the scheme. We consider the time–dependent Green–Taylor vortex problem, that we approximate on a circular domain
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