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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(H 2 ) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|logh| 1/2 H 3 ). The methods we study provide an approximate solution (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:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35Q30Stokes and Navier-Stokes equations
76M10Finite element methods (fluid mechanics)
35L70Nonlinear second-order hyperbolic equations
76D06Statistical solutions of Navier-Stokes and related equations
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