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A multilevel mesh independence principle for the Navier-Stokes equations. (English) Zbl 0844.76053

Multilevel, finite element discretization methods for the Navier-Stokes equations are considered. In contrast to usual multilevel methods, a superlinear scaling of the consecutive meshwidths \(h_{j + 1} = {\mathcal O} (h_j^{\alpha (j)})\) is used. On the coarsest mesh the discretized system is solved, which is small and nonlinear. On each subsequent mesh only one or two Newton correction steps are performed, so that only one or two larger, linear systems are solved. The scalings of the meshwidths that lead to optimal accuracy of the approximate solution in both the \(H^1\)- and \(L^2\)-norm are investigated. An error analysis is also presented for the basic finite element method using the conservation form of the nonlinear term.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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