Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients

Applied Numerical Mathematics

Volumn 59, Issue 12, 2009, Pages 3008-3032

  Dong, Qiao Li ^a   Cao, Li Qun ^a  

^a INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC ENGINEERING COMPUTING   (China)

Author keywords

Finite element method; Homogenization; Multiscale asymptotic expansion; Symplectic geometric scheme; The wave equation with rapidly oscillating coefficients

Indexed keywords

ASYMPTOTIC EXPANSION; CONVERGENCE RATES; CONVERGENCE THEOREM; HIGHER ORDER; HOMOGENIZATION; MULTI SCALE ANALYSIS; MULTISCALE ASYMPTOTIC EXPANSION; MULTISCALES; NUMERICAL ALGORITHMS; NUMERICAL RESULTS; SECOND ORDERS; SYMPLECTIC; SYMPLECTIC GEOMETRIC SCHEME; THE WAVE EQUATION WITH RAPIDLY OSCILLATING COEFFICIENTS; UNSOLVED PROBLEMS;

ASYMPTOTIC ANALYSIS; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; HOMOGENIZATION METHOD; LARGE SCALE SYSTEMS;

WAVE EQUATIONS;

EID: 69249213936     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2009.07.008     Document Type: Article

References (26)

1. Adams A. Sobolev Spaces (1975), Academic Press, New York
2. Arbogast T. Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal. 42 (2004) 576-598
3. Allaire G., and Piatnitski A. Homogenization of the Schrödinger equation and effective mass theorems. Comm. Math. Phys. 258 (2005) 1-22
4. Bensoussan A., Lions J.L., and Papanicolaou G. Asymptotic Analysis of Periodic Structures (1978), North-Holland, Amsterdam
5. J.F. Bourgat, H. Lanchon, Application of the homogenization method to composite materials with periodic structure, IRIA-LABORIA Report No. 208, 1976
6. Brahim-Otsmane S., Francfort G.A., and Murat F. Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 71 (1992) 197-231
7. Cao L.Q. Multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains. Numer. Math. 103 (2006) 11-45
8. Cao L.Q., and Cui J.Z. Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains. Numer. Math. 96 (2004) 525-581
9. Chew W.C. Waves and Fields in Inhomogeneous Media (1990), Van Nostrand-Reinhold, New York
10. Cioranescu D., and Donato P. An Introduction to Homogenization (1999), Oxford University Press, New York
11. Cioranescu D., and Paulin J.S.J. Homogenization of Reticulated Structures (1999), Springer-Verlag, Berlin
12. E W., and Engquist B. The heterogeneous multi-scale methods. Comm. Math. Sci. 1 1 (2003) 87-132
13. Feng K. Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math. 4 3 (1986) 279-289
14. Hairer E., Lubich C., and Wanner G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary. Differential Equations (2002), Springer-Verlag, Berlin
15. Hornung U. Homogenization and Porous Media (1997), Springer-Verlag, New York
16. Hou T.Y., Wu X.H., and Cai Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 68 (1999) 913-943
17. Jiao Y.D., Dai G.D., Feng Q.D., and Tang Y.F. Non-existence of conjugate-symplectic multi-step methods of odd order. J. Comp. Math. 25 6 (2007) 690-696
18. Jikov V.V., Kozlov S.M., and Oleinik O.A. Homogenization of Differential Operators and Integral Functionals (1994), Springer-Verlag, Berlin
19. Lin Q., and Zhu Q.D. The Preprocessing and Postprocessing for the Finite Element Method (1994), Shanghai Scientific & Technical Publishers, Shanghai (in Chinese)
20. Lions J.L., and Magenes E. Non-Homogeneous Boundary Value Problems and Applications, vol. II (1972), Springer-Verlag, Berlin
21. Liu X.S., Qi Y.Y., He J.F., and Ding P.Z. Recent progress in symplectic algorithms for use in quantum systems. Commun. Comput. Phys. 2 1 (2007) 1-53
22. Oleinik O.A., Shamaev A.S., and Yosifian G.A. Mathematical Problems in Elasticity and Homogenization (1992), North-Holland, Amsterdam
23. Qin M.Z. Symplectic schemes for nonautonomous Hamiltonian system. Acta Mathematicae Applicatae Sinica 12 3 (1996) 284-288
24. Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory (1980), Springer-Verlag, Berlin
25. D. Stoffer, On Reversible and Canonical Integration Methods, SAM-Report No. 88-05, ETH-Zürich, 1988
26. Walker J.S. Fourier Analysis (1988), Oxford University Press, New York