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A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the non‐stationary Navier–Stokes equations
Author(s) -
Luo Zhendong,
Du Juan,
Xie Zhenghui,
Guo Yan
Publication year - 2011
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.3161
Subject(s) - proper orthogonal decomposition , finite element method , mathematics , point of delivery , linear subspace , constraint (computer aided design) , mathematical analysis , mixed finite element method , geometry , engineering , structural engineering , agronomy , biology
Abstract In this paper, a proper orthogonal decomposition (POD) method is applied to reducing a classical stabilized mixed finite element (SMFE) formulation for the non‐stationary Navier–Stokes equations. Error estimates between the classical SMFE solutions and the reduced SMFE solutions based on the POD method are provided. The reduced SMFE formulation based on the POD method could not only greatly reduce its degrees of freedom but also circumvent the constraint of Brezzi–Babuka (BB) condition so that the combination of finite element subspaces could be chosen freely and allow optimal‐order error estimates to be obtained. Numerical experiments show that the errors between the reduced SMFE solutions and the classical SMFE solutions are consistent with theoretical results. Moreover, it is shown that the reduced SMFE formulation based on the POD method is feasible and efficient for solving the non‐stationary Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.