A multiscale finite element method for numerical homogenization

Multiscale Modeling and Simulation

Volumn 4, Issue 3, 2005, Pages 790-812

  Allaire, Grégoire ^a   Brizzi, Robert ^a  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

Multiscale finite element method; Numerical homogenization; Scalar elliptic partial differential equation

Indexed keywords

BOUNDARY VALUE PROBLEMS; ERRORS; ESTIMATION; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS;

MULTISCALE FINITE ELEMENT METHOD; NUMERICAL HOMOGENIZATION; SCALAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION;

FINITE ELEMENT METHOD;

EID: 33748438382     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/040611239     Document Type: Article

References (21)

1. G. Alessandrini and V. Nesi, Univalent σ-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), pp. 155-171.
2. G. Allaire, Shape Optimization by the Homogenization Method, Appl. Math. Sci. 146, Springer, New York, 2002.
3. T. Arbogast, Numerical subgrid upscaling of two-phase flow in porous media, in Numerical Treatment of Multiphase Flows in Porous Media, Lecture Notes in Phys. 552, Z. Chen, R. E. Ewing, and Z. C. Shi, eds., Springer, Berlin, 2000, pp. 35-49.
4. I. Babuška, Solution of interface problems by homogenization. I, SIAM J. Math. Anal., 7 (1976), pp. 603-634.
5. I. Babuška, Solution of interface problems by homogenization. II, SIAM J. Math. Anal., 7 (1976), pp. 635-645.
6. I. Babuška, Solution of interface problems by homogenization. III, SIAM J. Math. Anal., 8 (1977), pp. 923-937.
7. A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.
8. Y. Capdeboscq and M. Vogelius, Wavelet based homogenization of a 2 dimensional elliptic problem, to appear.
9. P. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978.
10. W. E. and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), pp. 87-132.
11. Y. R. Efendiev, T. Y. Hou, and X.-H. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal., 37 (2000), pp. 888-910.
12. Y. Efendiev and X.-H. Wu, Multiscale finite element methods for problems with highly oscillatory coefficients, Numer. Math., 90 (2002), pp. 459-486.
13. T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169-189.
14. T. Y. Hou, X.-H. Wu, and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), pp. 913-943.
15. S. M. Kozlov, Averaging of random operators, Math. USSR-Sb., 37 (1980), pp. 167-180.
16. A.-M. Matache, I. Babuska, and C. Schwab, Generalized p-FEM in homogenization, Numer. Math., 86 (2000), pp. 319-375.
17. A.-M. Matache and C. Schwab, Two-scale FEM for homogenization problems, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 537-572.
18. p-estimate for the gradient of solutions of second-order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), pp. 189-206.
19. F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modeling of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, A. Cherkaev and R. V. Kohn, eds., Birkhaüser, Boston, 1997, pp. 21-43.
20. P. H. Rabinowitz, Théorie du degré topologique et applications à des problèmes aux limites non linéaires, rédigé par H. Berestycki, Laboratoire d'Analyse Numérique, Université de Paris VI, Paris, 1975.
21. G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), pp. 189-258.