A REDUCED-ORDER MFE FORMULATION BASED ON POD METHOD FOR PARABOLIC EQUATIONS

doi:10.1016/S0252-9602(13)60097-1

Abstract

Abstract:

In this paper, we extend the applications of proper orthogonal decomposition (POD) method, i.e., apply POD method to a mixed finite element (MFE) formulation naturally satisfied Brezz-Babu?ska for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consis-tent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.

Key words: proper orthogonal decomposition method, mixed finite element formulation, parabolic equation, error estimate

LUO Zhen-Dong, LI Lei, SUN Ping. A REDUCED-ORDER MFE FORMULATION BASED ON POD METHOD FOR PARABOLIC EQUATIONS[J].Acta mathematica scientia,Series B, 2013, 33(5): 1471-1484.

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References

[1] Thom´ee V. Galerkin Finite Element Methods for Parabolic Problems. Berlin: Springer, 1997

[2] Luo Z D. Mixed Finite Element Methods and Applications. Beijing: Science Press, 2006

[3] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York: Springer-Verlag, 1991

[4] Li L, Sun P, Luo Z D. A new mixed finite element formulation and error estimates for parabolic equations. Acta Mathematica Scientia, 2012, 32A(6): 1158–1165

[5] Fukunaga K. Introduction to Statistical Recognition. New York: Academic Press, 1990

[6] Jolliffe I T. Principal Component Analysis. Berlin: Springer-Verlag, 2002

[7] Holmes P, Lumley J L, Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge, UK: Cambridge University Press, 1996

[8] Sirovich L. Turbulence and the dynamics of coherent structures: Part I-III. Quarterly of Applied Mathe-matics, 1987, 45(3): 561–590

[9] Lumley J L. Coherent structures in turbulence//Meyer R E, ed. Transition and Turbulence. Academic Press, 1981

[10] Aubry N, Holmes P, Lumley J L, Stone E. The dynamics of coherent structures in the wall region of a turbulent boundary layer. J Fluid Dynamics, 1988, 192: 115–173

[11] Moin P, Moser R D. Characteristic-eddy decomposition of turbulence in channel. J Fluid Mech, 1989, 200: 417–509

[12] Rajaee M, Karlsson S K F, Sirovich L. Low dimensional description of free shear flow coherent structures and their dynamical behavior. J Fluid Mech, 1994, 258: 1–29

[13] Joslin R D, Gunzburger M D, Nicolaides R A, Erlebacher G, Hussaini M Y. A self-contained automated methodology for optimal flow control validated for transition delay. AIAA J, 1997, 35(5): 816–824

[14] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems. Nu-merische Mathematik, 2001, 90(1): 117–148

[15] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for a general equation in fluid
dynamics. SIAM J Num Anal, 2002, 40(2): 492–515

[16] Sun P, Luo Z D, Zhou Y J. Some reduced finite difference schemes based on a proper orthogonal decom-position technique for parabolic equations. Appl Num Math, 2010, 60(1/2): 15–164

[17] Luo Z D, Chen J, Sun P, et al. Finite element formulation based on proper orthogonal decomposition for parabolic equations. Science in China Series A: Mathematics, 2009, 52(3): 587–596

[18] Luo Z D, Chen J, Xie Z H, et al. A reduced second-order time accurate finite element formulation based on POD for parabolic equations (in Chinese). Science in China Series A: Mathematics, 2011, 41(5): 447–460

[19] Cao Y H, Zhu J, Navon I M, et al. A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition. Int J Num Methods Fluids, 2007, 53(10): 1571–1583

[20] Tian X J, Xie Z H, Dai A G. An ensemble-based explicit four-dimensional variational assimilation method. J Geophys Res, 2008, 113(D21124), doi: 10.1029/2008JD010358

[21] Tian X J, Xie Z H, Sun Q. A POD-based ensemble four-dimensional variational assimilation method. Tellus, 2011, 63A: 805–816

[22] Adams R A. Sobolev Spaces. New York: Academic Press, 1975

[23] Girault V, Raviart P A. Finite Element Approximations of the Navier-Stokes Equations//Theorem and Algorithms. New York: Springer-Verlag, 1986

[24] Ciarlet P G. The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland, 1978

[25] Rudin W. Functional and Analysis. 2nd ed. McGraw-Hill Companies, Inc, 1973