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Finite element formulation based on proper orthogonal decomposition for parabolic equations

Science China Mathematics 2009年第3期52卷 585-596页

作者： LUO ZhenDong CHEN Jing SUN Ping YANG XiaoZhong School of Mathematics and Physics North China Electric Power UniversityBeijing 102206China School of Mathematics and Computer Science Guizhou Normal UniversityGuiyang 550001China College of Science China Agricultural UniversityBeijing 100083China

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method.

关键词： proper orthogonal decomposition finite element formulation parabolic equations error analysis 65N30 35q10

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Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

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Science China Mathematics 2007年第8期50卷 1186-1196页

作者： Zhen-dong LUO Rui-wen WANG Jiang ZHU School of Science Beijing Jiaotong UniversityBeijing 100044China Institute of Atmospheric Physics Chinese Academy of SciencesBeijing 100029China

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes *** optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and *** the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes *** errors between POD approximate solutions and FDS solutions are *** is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical ***,it is also shown that this validates the feasibility and efficiency of POD method.

关键词： proper orthogonal decomposition singular value decomposition finite difference scheme the nonstationary Navier-Stokes equations 65N30 35q10

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