Two-Grid Finite-Element Schemes for The Steady Navier--Stokes Problem in Polyhedra

V. Girault and J.-L. Lions

Abstract: We discretize a steady Navier--Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully nonlinear problem is solved on a coarse grid, with mesh-size $H$. In the second step, the problem is linearized by substituting into the nonlinear term, the velocity ${\bf u}_H$ 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 ${\bf u}_H$ 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.

Keywords: Two grids; Nonsingular solutions; Duality.

Classification (MSC2000): 76D05, 65N15, 65N30, 65N55.

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