Two-level stabilized finite element method for the transient Navier–Stokes equations

Abstract

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h≪H. A H ¹-optimal velocity approximation and a L ²-optimal pressure approximation are obtained. If we choose h=O(H ²), the two-level method gives the same order of approximation as the standard stabilized finite element method.

Keywords:

• Navier–Stokes equations
• stabilized finite element method
• two-level method
• local Gauss integration

2000 AMS Subject Classifications :

• 35Q10
• 65N30
• 76D06

Acknowledgements

The authors would like to thank the editor and the referees for their valuable comments and suggestions which helped to improve the results of this article.

Supported by the Natural Science Foundation of China (No. 10671154) and the National Basic Research Program (No. 2005CB321703); Sichuan Science and Technology Project (No. 05GG006-006-2) and Science Research Foundation of UESTC. E-mail: [email protected] (K. Wang), [email protected] (Y. He).