A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations

2011 ◽  
Vol 88 (1) ◽  
pp. 31-46 ◽  
Author(s):  
Zhendong Luo ◽  
Juan Du ◽  
Zhenghui Xie ◽  
Yan Guo
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Element Formulation ◽  
Navier Stokes Equations ◽  
Stationary Navier Stokes Equations ◽  
Proper Orthogonal ◽  
Stabilized Mixed Finite Element

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

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Aiwen Wang ◽  
Jian Li ◽  
Zhenhua Di ◽  
Xiangjun Tian ◽  
Dongxiu Xie
Keyword(s):  
Finite Element ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Proper Orthogonal ◽  
Stabilized Mixed Finite Element

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.

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

2014 ◽  
Vol 6 (5) ◽  
pp. 615-636 ◽  
Author(s):  
Zhendong Luo
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Stokes Equations ◽  
Volume Element ◽  
Navier Stokes ◽  
Element Formulation ◽  
Navier Stokes Equations ◽  
Fully Discrete ◽  
Finite Volume Element ◽  
Stationary Navier Stokes Equations

AbstractA semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.

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