Articles

A REDUCED MFE FORMULATION BASED ON POD FOR THE NON-STATIONARY CONDUCTION-CONVECTION PROBLEMS

  • LUO Zhen-Dong ,
  • XIE Zheng-Hui ,
  • CHEN Jing
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  • School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China|LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China|College of Sciences, China Agricultural University, Beijing 100083, China

Received date: 2010-06-09

  Revised date: 2010-10-13

  Online published: 2011-09-20

Supported by

Research of this work was supported by the National Science Foundation of China (10871022; 11061009; 40821092); the National Basic Research Program (2010CB428403; 2009CB421407; 2010CB951001); Natural Science Foundation of Hebei Province (A2010001663); and Chinese Universities Scientific Fund (2009-2-05).

Abstract

In this article, a reduced mixed finite element (MFE) formulation based on proper orthogonal decomposition (POD) for the non-stationary conduction-convection problems is presented. Also the error estimates between the reduced MFE solutions based
on POD and usual MFE solutions are derived. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced MFE formulation based on POD is feasible and efficient in finding numerical solutions for the non-stationary conduction-convection problems.

Cite this article

LUO Zhen-Dong , XIE Zheng-Hui , CHEN Jing . A REDUCED MFE FORMULATION BASED ON POD FOR THE NON-STATIONARY CONDUCTION-CONVECTION PROBLEMS[J]. Acta mathematica scientia, Series B, 2011 , 31(5) : 1765 -1785 . DOI: 10.1016/S0252-9602(11)60360-3

References

[1] Tang X J. On the existence and uniqueness of the solution to the Navier-Stokes equation. Acta Mathematics Scientia, 1995, 15: 342–351

[2] Christon M A, Gresho P M, Sutton S B. Computational predictability of time-dependent natural convection flows in enclosures (including a benchmark solution). Special Issue: MIT Special Issue on Thermal Convection. International Journal for Numerical Methods in Fluids, 2002, 40(8): 953-980

[3] Gresho P M, Lee R L, Chen S T. Solution of the time-dependent incompressible Navier–Stokes and Boussinesq equation using the Galerkin finite element method//Approximation methods for the Navier–Stokes problems. Lecture Notes in Mathematics, 771. Berlin: Springer, 1980: 203–222

[4] Fukunaga K. Introduction to Statistical Recognition. Academic Press, 1990

[5] Jolliffe I T. Principal Component Analysis. Springer-Verlag, 2002

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

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

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

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

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

[11] Rajaee M, Karlsson S K F, Sirovich L. Low dimensional description of free shear flow coherent structures and their dynamical behavior. Journal of Fluid Mechanics, 1994, 258: 1–19

[12] 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 Journal, 1997, 35: 816–824

[13] Ly H V, Tran H T. Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Quarterly of Applied Mathematics, 2002, 60: 631–656

[14] Rediniotis O K, Ko J, Yue X, Kurdila A J. Synthetic jets, their reduced order modeling and applications to flow control. AIAA Paper number 99-1000, 37 Aerospace Sciences Meeting & Exhibit, Reno, 1999

[15] Cao Y H, Zhu J, Luo Z D, Navon I M. Reduced order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition. Computers and Mathematics with Applications, 2006, 52: 1373–1386

[16] Cao Y H, Zhu J, Navon I M, Luo Z D. A reduced order approach to four-dimensional variational data assimilation using proper orthogonal decomposition. International Journal for Numerical Methods in Fluids, 2007, 53: 1571–1583

[17] Luo Z D, Zhu J, Wang R W, Navon I M. Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model. Computer Methods in Applied Mechanics and Engineering, 2007, 196(41-44): 4184–4195

[18] Luo Z D, Chen J, Zhu J, Wang R W, Navon I M. An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model. International Journal for Numerical Methods in Fluids, 2007, 55(2): 143–161

[19] Luo Z D, Wang R W, Zhu J. Finite difference scheme based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations. Science in China Series A: Mathematics, 2007, 50(8): 1186–1196

[20] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik, 2001, 90: 117–148

[21] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM Journal on Numerical Analysis, 2002, 40: 492–515

[22] Kunisch K, Volkwein S. Control of Burgers’ equation by a reduced order approach using proper orthogonal decomposition. Journal of Optimization Theory and Applications, 1999, 102: 345–371

[23] Ahlman D, S¨odelund F, Jackson J, Kurdila A, Shyy W. Proper orthogonal decomposition for timedependent lid-driven cavity flows. Numerical Heal Transfer Part B–Fundamentals, 2002, 42: 285–306

[24] Patera A T, Rønquist E M. Reduced basis approximation and a posteriori error estimation for a Boltzmann model. Computer Methods in Applied Mechanics and Engineering, 2007, 196: 2925–2942

[25] Rovas D V, Machiels L, Maday Y. Reduced-basic output bound methods for parabolic problems. IMA Journal of Numerical Analysis, 2006, 26: 423–445

[26] Adams R A. Sobolev Space. New York: Academic Press, 1975

[27] Luo Z D. The bases and Applications of Mixed Finite Element Methods. Beijing: Science Press, 2006

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

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

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