Finite difference heterogeneous multi-scale method for homogenization problems

Journal of Computational Physics

Volumn 191, Issue 1, 2003, Pages 18-39

  Abdulle, Assyr ^a   Weinan, E ^b  

^a ETH ZURICH   (Switzerland)

^b PRINCETON UNIVERSITY   (United States)

Author keywords

Finite difference; Heterogeneous multi scale method; Homogenization; Multi scale problem

Indexed keywords

HOMOGENIZATION METHOD;

FINITE DIFFERENCE; GRID LEVELS; HETEROGENEOUS MULTI-SCALE METHOD; HOMOGENIZATION; HOMOGENIZATION PROBLEMS; MULTI-SCALES; MULTISCALE METHOD; MULTISCALE PROBLEM; NUMERICAL RESULTS; PARABOLIC PROBLEMS;

NUMERICAL METHODS;

EID: 0141682262     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9991(03)00303-6     Document Type: Article

References (20)

1. A. Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput. 23 (6) (2002) 2041-2054.
2. A. Abdulle, A.A. Medovikov, Second order Chebyshev methods based on orthogonal polynomials, Numer. Math. 90 (1) (2001) 1-18.
3. G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (6) (1992) 1482-1518.
4. I. Babuska, Homogenization and its applications, in: B. Hubbard (Ed.), SYNSPADE, 1975, pp. 89-116.
5. A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
6. D. Cioranescu, P. Donato, An Introduction to Homogenization, Oxford University Press, Oxford, 1999.
7. W. E, Homogenization of linear and nonlinear transport equations, Commun. Pure Appl. XLV (1992) 301-326.
8. W. E, B. Engquist, The heterogeneous multi-scale methods, Comm. Math. Sci. 1 (1) (2003) 87-132.
9. B. Engquist, Computation of oscillatory solutions to hyperbolic differential equations, Springer Lecture Notes Math. 1270 (1987) 10-22.
10. M. Dorobantu, B. Engquist, Wavelets-based numerical homogenization, SIAM J. Numer. Anal. 35 (2) (1998) 540-559.
11. B. Engquist, O. Runborg, Wavelets-based numerical homogenization with applications, in: Multiscale and Multiresolution Methods, Lecture Notes in Computational Science and Engineering, vol. 20, Springer, Berlin, 2002, pp. 97-148.
12. E. Hairer, S.P. Nørsett, G. Wanner, Solving ordinary differential equations I. Nonstiff problems, Springer Series in Computational Mathematics, vol. 8, second ed., Springer, Berlin, 1993.
13. E. Hairer, G. Wanner, Solving ordinary differential equations II. Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, vol. 14, second ed., Springer, Berlin, 1996.
14. T.-Y. Hou, X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997) 169-189.
15. T.-Y. Hou, X.-H. Wu, Z. Cai, Convergence of a multi-scale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comput. 68 (227) (1999) 913-943.
16. J.L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, vols. 1 and 2, Paris, Dunod, 1968 (English Transl., Springer, 1970).
17. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (3) (1989) 608-623.
18. A-M. Matache, I. Babuska, C. Schwab, Generalized p-FEM in homogenization, Numer. Math. 86 (2) (2000) 319-375.
19. C. Schwab, A.-M. Matache, Generalized FEM for homogenization problems, in: Multiscale and Multiresolution Methods, Lecture Notes in Computational Science and Engineering, vol. 20, Springer, Berlin, 2002, pp. 197-238.
20. N. Neuss, W. Jäger, G. Wittum, Homogenization and multigrid, Computing 66 (1) (2001) 1-26.