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Two-level stabilized finite element methods for the steady Navier-Stokes problem. (English) Zbl 1099.65111
Summary: Two-level stabilized finite element formulations of the two-dimensional steady Navier-Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier-Stokes problem. By satisfying this condition the stability of the ${Q}_{1}-{P}_{0}$ quadrilateral element and the ${P}_{1}-{P}_{0}$ triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size $H$, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size $h=O\left({H}^{2}\right)$ or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size $h=O\left(|logh{|}^{1/2}{H}^{3}\right)$. The methods we study provide an approximate solution $\left({u}^{h}$, ${p}^{h}$) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier-Stokes problem on a fine mesh with mesh size $h$. Hence, our methods can save a large amount of computational time.
MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 35Q30 Stokes and Navier-Stokes equations 76M10 Finite element methods (fluid mechanics) 35L70 Nonlinear second-order hyperbolic equations 76D06 Statistical solutions of Navier-Stokes and related equations
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