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**An Oseen two-level stabilized mixed finite-element method for the 2D/3D stationary Navier-Stokes equations.**
*(English)*
Zbl 1237.76075

Summary: We investigate an Oseen two-level stabilized finite-element method based on the local pressure projection for the 2D/3D steady Navier-Stokes equations by the lowest order conforming finite-element pairs (i.e., \(Q_1 - P_0\) and \(P_1 - P_0\)). Firstly, in contrast to other stabilized methods, they are parameter free, no calculation of higher-order derivatives and edge-based data structures, implemented at the element level with minimal cost. In addition, the Oseen two-level stabilized method involves solving one small nonlinear Navier-Stokes problem on the coarse mesh with mesh size \(H\), a large general Stokes equation on the fine mesh with mesh size \(h = O(H)^2\). The Oseen two-level stabilized finite-element method provides an approximate solution \((u^h, p^h)\) with the convergence rate of the same order as the usual stabilized finite-element solutions, which involves solving a large Navier-Stokes problem on a fine mesh with mesh size \(h\). Therefore, the method presented in this paper can save a large amount of computational time. Finally, numerical tests confirm the theoretical results. Conclusion can be drawn that the Oseen two-level stabilized finite-element method is simple and efficient for solving the 2D/3D steady Navier-Stokes equations.

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

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\textit{A. Wang} et al., Abstr. Appl. Anal. 2012, Article ID 520818, 12 p. (2012; Zbl 1237.76075)

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### References:

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