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Zbl 1099.65111
He, Yinnian; Li, Kaitai
Two-level stabilized finite element methods for the steady Navier-Stokes problem. (English)
[J] Computing 74, No. 4, 337-351 (2005). ISSN 0010-485X; ISSN 1436-5057/e

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 $h = O(\vert\log h\vert^{1/2} H^{3})$. 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 2000:: *65N30 Finite numerical methods (BVP of PDE)
35Q30 Stokes and Navier-Stokes equations
76M10 Finite element methods
35L70 Second order nonlinear hyperbolic equations
76D06 Statistical solutions of Navier-Stokes and related equations

Keywords: Navier-Stokes problem; stabilized finite element; two-level method; error estimate

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Zentralblatt MATH Berlin [Germany]