The partial homogenization: Continuous and semi-discretized versions

Mathematical Models and Methods in Applied Sciences

Volumn 17, Issue 8, 2007, Pages 1183-1209

  Panasenko, Grigory ^a  

^a UNIVERSITÉ JEAN MONNET   (France)

Author keywords

Homogenization; Partial asymptotic decomposition of the domain; Semi discrete model

Indexed keywords

EID: 34547412908     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202507002248     Document Type: Article

References (55)

1. G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, SIAM J. Multiscale Model. Simul. 4 (2005) 790-812.
2. D. N. Arnold and A. L. Madureira, Asymptotic estimates of hierarchical modeling, Math. Mod. Meth. Appl. Sci. 13 (2003) 1329-1350.
3. J. P. Aubin, Approximation of Elliptic Boundary-Value Problems (John Wiley & Sons, 1972).
4. I. Babuska, Solutions of interface problems by homogenization, SIAM J. Math. Anal. Part 1, 7 (1976) 603-634;
5. Part 2, 7 (1976) 635-645;
6. Part 3, 8 (1976) 923-937.
7. I. Babuska and L. Li, Hierarchic modelling of plates, Comput. Struct. 40 (1991) 419-430.
8. I. Babuska and C. Schwab, A posteriori error estimation for the hierarchic models of elliptic boundary value problems on thin domains, SIAM J. Numer. Anal. 33 (1996) 221-246.
9. N. S. Bakhvalov, On optimization of the methods of solution of boundary value problems with boundary layers, Zh. Vyc. Mat. i Mat. Fis. (ZVMMF) 9 (1969) 841-859.
10. N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging Processes in Periodic Media (Nauka, 1984) [English translation, Kluwer, 1989].
11. A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).
12. P. J. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, 1978).
13. R. Z. Dautov, On a version of the finite element method in the domains with periodic structure, Diff. Uravneniia Diff. Eqns. 21 (1985) 1155-1164.
14. D. Dupuy, G. Panasenko and R. Stavre, Asymptotic analysis for micro-polar fluids in a tube structure, Math. Mod. Meth. Appl. Sci. 14 (2004) 735-758.
15. E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell'energia par operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. 8 (1973) 391-411.
16. T. Y. Hou, X. H. Wu and Z. Q. Cai, Convergence of a multi-scale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comput. 68 (1999) 913-943.
17. A. M. Ilyin, Difference scheme for a differential equation with small parameter in the higher derivative, Mat. Zam. 6 (1969) 237-248.
18. L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis (Noordhoff, 1958).
19. E. M. Landis and G. Panasenko, A theorem on the asymptotic of solutions of elliptic equations with coefficients periodic in all variables except one, Dokl. AN SSSR. 235 (1977) (in Russian)
20. [English transi. Sov. Math. Dokl. 18 (1977) 1140-1143].
21. J. L. Lions, Some Methods in Mathematical Analysis of Systems and Their Control (Science Press Gordon and Breach, 1981).
22. S. A. Nazarov, Two-term asymptotic of solutions of singularly perturbated spectral problems, Math. Sbornik 181 (1990) 291-320.
23. S. A. Nazarov, Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in rectangle, Mat. Sbornik 182 (1991) 692-722
24. [English translation Math. USSR Sbornik 73 (1992) 79-110].
25. S. A. Nazarov and M. Specovius-Neugebauer, Approximation of unbounded domains by bounded domains. Boundary value problems for Lamé operator, St. Petersburg Math. J. 8 (1997) 879-912.
26. S. A. Nazarov and A. S. Zorin, Two-term asymptotics of the problem on longitudinal deformation of a plate with clamped edge, Comput. Mech. Solids 2 (1991) 10-21 (Russian).
27. M. Neuss-Radu, Boundary layers in the homogenization of elliptic problems, PhD thesis, Universität Heidelberg, 1999.
28. n, in Proc. of the Conf. Young Scientists of the Dept. of Numerical Analysis and Informatics of Moscow State University (Moscow Univ. Publ., 1976), pp. 53-61 (in Russian).
29. G. P. Panasenko, Asymptotic of higher order of solutions of equations with rapidly oscillating coefficients, Dokl. AN USSR 240 (1978) 1293-1296 (in Russian)
30. [English transi. Sov. Math. Dokl., 1979].
31. G. P. Panasenko, Higher order asymptotic of solutions of problems on the contact of periodic structures, Math. Sb. 110/152 (1979) 505-538 (in Russian)
32. [English transl. Math. USSR Sb. 38 (1981) 465-494].
33. G. P. Panasenko, Asymptotic solutions of the elasticity theory system of equations for lattice and skeletal structures, Math. Sb. 183 (1992) 89-113 (in Russian).
34. English transl. by AMS in Russian Acad. Sci. Sbornik Math. 75 (1993) 85-110.
35. G. P. Panasenko, Asymptotic analysis of bar systems, I. Russ. J. Math. Phys. 2 (1994) 325-352;
36. II Russ. J. Math. Phys. 4 (1996) 87-116.
37. G. P. Panasenko, Method of asymptotic partial decomposition of domain, Math. Mod. Meth. Appl. Sci. 8 (1998) 139-156.
38. G. P. Panasenko, Method of asymptotic partial decomposition of rod structures, Int. J. Comput. Civil Structural Eng. Begel House Publ. 1 (2000) 57-70.
39. G. P. Panasenko, Asymptotic expansion of the solution of Navier-Stokes equation in tube structure and partial asymptotic decomposition of the domain, Appl. Anal. Int. J. 76 (2000) 363-381.
40. G. P. Panasenko, Multi-Scale Modelling for Structures and Composites (Springer, 2005).
41. A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Modelling of Living Systems, Handbook of Numerical Analysis Series, ed. N. Ayache (2002).
42. E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory (Springer-Verlag, 1980).
43. C. Schwab, A posteriori modelling error estimation for hierarchic plate models, Numer. Math. 74 (1996) 221-259.
44. G. I. Shishkin, Network Approximations for Singularly Perturbed Elliptic and Parabolic Problems (Russian Academy of Sciences, 1992).
45. S. M. Shugrin, Coupling of one-dimensional and two-dimensional models of water flow, Water Res. 5 (1987) 5-15 (in Russian).
46. V. P. Smyshlyaev and K. D. Cherednichenko, On derivation of "strain gradient" effects in the overall behaviour of periodic heterogeneous media, J. Mech. Phys. Solids 48 (2000) 1325-1357.
47. M. Specovius-Neugebauer, Approximation of Stokes problem in unbounded domain, preprint, University of Paderborn, 1997.
48. L. G. Strakhovskaya and R. P. Fedorenko, On one version of the finite element method, Zh. Vyc. Mat. i Mat. Fis. ZVMMF 4 (1979) 950-960.
49. L. Tartar, Probleèmes d' homogénéisation dans les équations aux dérivées partielles Cours Pecot, Collège de France, 1977.
50. M. Vogelius and I. Babuska, On a dimensional reduction method, I. The optimal selection of basis functions, Math. Comput. 37 (1981) 31-46;
51. II Some approximation-theoretic results, Math. Comput. 37 (1981) 47-68;
52. III A posteriori error estimation and an adaptive approach, Math. Comput. 37 (1981) 361-384.
53. M. Vogelius, Papanicolaou, A projection method applied to diffusion in a periodic structure, SIAM J. Appl. Math. 42 (1982) 1302-1322.
54. V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators (Nauka, 1993)
55. [English version: V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Partial Differential Operators and Integral Functionals (Springer-Verlag, 1994)].