Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra.

*(English)* Zbl 0997.76043
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 ${\mathbf{u}}_{H}$ 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 ${\mathbf{u}}_{H}$ 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:

76M10 | Finite element methods (fluid mechanics) |

76D05 | Navier-Stokes equations (fluid dynamics) |

65N15 | Error bounds (BVP of PDE) |

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

65N55 | Multigrid methods; domain decomposition (BVP of PDE) |