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Convergence of the MAC scheme for the steady-state incompressible Navier-Stokes equations on non-uniform grids. (English) Zbl 1304.76038

Fuhrmann, Jürgen (ed.) et al., Finite volumes for complex applications VII – methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, Germany, June 15–20, 2014. Vol. I. Cham: Springer (ISBN 978-3-319-05683-8/hbk; 978-3-319-05684-5/ebook; 978-3-319-06402-4/set). Springer Proceedings in Mathematics & Statistics 77, 343-351 (2014).
Summary: We prove in this paper the convergence of the Marker and cell (MAC) scheme for the discretization of the steady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step of which tends to zero. We then establish that the limit is a weak solution to the continuous problem.
For the entire collection see [Zbl 1291.65004].

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65N08 Finite volume methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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