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Two-grid finite element schemes for the transient Navier-Stokes problem. (English) Zbl 1032.76032

Summary: We semi-discretize in space a time-dependent 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 semi-discretized on a coarse grid, with mesh-size \(H\). In the second step, the problem is linearized by substituting into the nonlinear term the velocity \(u _H\) computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size \(h\). This approach is motivated by the fact that, on a convex polyhedron and under adequate assumptions on the data, the contribution of \(u _H\) to the error analysis is measured in the \(L^2\) norm in space and time, and thus, for the lowest-degree elements, is of the order of \(H^2\). Hence, an error of the order of \(h\) can be recovered at the second step, provided \(h = H^2\).

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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