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Zbl 0997.76043
Girault, V.; Lions, J.-L.
Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. (English)
[J] Port. Math. (N.S.) 58, No.1, 25-57 (2001). ISSN 0032-5155; ISSN 1662-2758/e

Summary: We discretize the steady Navier-Stokes system on a three-dimensional polyhedron by finite element schemes defined on two grids. At the first step, the fully nonlinear problem is solved on a coarse grid, with mesh size $H$. At the second step, the problem is linearized by substituting into the nonlinear term the velocity ${\bold u}_H$ computed at step one, and the linearized problem is solved on a fine grid with mesh size $h$. This approach is motivated by the fact that the contribution of ${\bold u}_H$ to the error analysis is measured in the $L^3$ norm, and thus, for the lowest-degree elements on a Lipschitz polyhedron, is of the order of $H^{3/2}$. Hence, an error of the order of $h$ can be recovered at the second step, provided $h=H^{3/2}$. When the domain is convex, a similar result can be obtained with $h=H^2$. Both results are valid in two dimensions.

MSC 2000:: *76M10 Finite element methods
76D05 Navier-Stokes equations (fluid dynamics)
65N15 Error bounds (BVP of PDE)
65N30 Finite numerical methods (BVP of PDE)
65N55 Multigrid methods; domain decomposition (BVP of PDE)

Keywords: two-grid scheme; linearization; steady Navier-Stokes system; three-dimensional polyhedron; finite element schemes; coarse grid; fine grid; error analysis

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