A reduced-order LSMFE formulation based on POD method and implementation of algorithm for parabolic equations

Authors:
Zhendong Luo;Hong Li;Yueqiang Shang;Zhichao Fang
Affiliations:
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China;School of Mathematical Sciences, Inner Mongolia University, Huhhot 010021, China;School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China;School of Mathematical Sciences, Inner Mongolia University, Huhhot 010021, China
Venue:
Finite Elements in Analysis and Design
Year:
2012

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Abstract

Proper orthogonal decomposition (POD) technology has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method to a usual least-squares mixed finite element (LSMFE) formulation for two-dimensional parabolic equations with real practical applied background. POD bases here are constructed using the method of snapshots. Karhunen-Loeve projection is used to develop a reduced-order LSMFE formulation obtained by projecting the usual LSMFE formulation onto the most important POD bases. The reduced-order LSMFE formulation has lower dimensions and sufficiently high accuracy, is suitable for finding approximate solutions for two-dimensional parabolic equations, and can circumvent the constraint of the convergence stability what is called Brezzi-Babuska condition. We derive the error estimates between the reduced-order LSMFE solutions and the usual LSMFE solutions for parabolic equations and provide the implementation of algorithm for solving the reduced-order LSMFE formulation so as to supply scientific theoretic basis and computing way for practical applications. Two numerical examples are used to validate that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order LSMFE formulation based on POD method is feasible and efficient for finding numerical solutions of parabolic equations.