Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 895386, 19 pages
http://dx.doi.org/10.1155/2011/895386
Research Article

A New Reduced Stabilized Mixed Finite-Element Method Based on Proper Orthogonal Decomposition for the Transient Navier-Stokes Equations

Aiwen Wang,1 Jian Li,2 Zhenhua Di,3 Xiangjun Tian,4 and Dongxiu Xie1

1School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
2Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721007, China
3College of Global Change and Earth System Science, Beijing Normal University, Beijing 100009, China
4Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

Received 31 May 2011; Accepted 15 August 2011

Academic Editor: Jan Sladek

Copyright © 2011 Aiwen Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A reduced stabilized mixed finite-element (RSMFE) formulation based on proper orthogonal decomposition (POD) for the transient Navier-Stokes equations is presented. An ensemble of snapshots is compiled from the transient solutions derived from a stabilized mixed finite-element (SMFE) method based on two local Gauss integrations for the two-dimensional transient Navier-Stokes equations by using the lowest equal-order pair of finite elements. Then, the optimal orthogonal bases are reconstructed by implementing POD techniques for the ensemble snapshots. Combining POD with the SMFE formulation, a new low-dimensional and highly accurate SMFE method for the transient Navier-Stokes equations is obtained. The RSMFE formulation could not only greatly reduce its degrees of freedom but also circumvent the constraint of inf-sup stability condition. Error estimates between the SMFE solutions and the RSMFE solutions are derived. Numerical tests confirm that the errors between the RSMFE solutions and the SMFE solutions are consistent with the the theoretical results. Conclusion can be drawn that RSMFE method is feasible and efficient for solving the transient Navier-Stokes equations.