Convergence of a nonconforming multiscale finite element method

SIAM Journal on Numerical Analysis

Volumn 37, Issue 3, 2000, Pages 888-910

  Efendiev, Yalchin R ^a   Hou, Thomas Y ^a   Wu, Xiao Hui ^a,b  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b EXXONMOBIL UPSTREAM RESEARCH COMPANY   (United States)

Author keywords

Finite element; Homogenization; Multiscale; Resonant sampling

Indexed keywords

EID: 0000713961     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142997330329     Document Type: Article

References (19)

1. M. AVELLANEDA AND F.-H. LIN, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1989), pp. 803-843.
2. I. BABUŠKA, G. CALOZ, AND E. OSBORN, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal., 31 (1994), pp. 945-981.
3. A. BENSOUSSAN, J. L. LIONS, AND G. PAPANICOLAOU, Asymptotic Analysis for Periodic Structure, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.
4. A. BENSOUSSAN, J. L. LIONS, AND G. PAPANICOLAOU, Boundary layer analysis in homogenization of diffusion equations with Dirichlet conditions in the half space, in Proceedings of the International Symposium on Stochastic Differential Equations, K. Ito, ed., John Wiley, New York, Chichester, Brisbane, 1978, pp. 21-40.
5. F. BREZZI, L. FRANCA, T. HUGHES, AND A. RUSSO, b is equal to integral of g, Comput. Methods Appl. Mech. Engrg., 145 (1997), pp. 329-339.
6. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
7. Y. R. EFENDIEV, The Multiscale Finite Element Method (MsFEM) and Its Applications, Ph.D. thesis, Applied Mathematics, Caltech, Pasadena, CA, 1999.
8. 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.
9. T. Y. HOU AND X. H. WU, A multiscale finite element method for PDEs with oscillatory coefficients, in Numerical Treatment of Multi-Scale Problems, Notes Numer. Fluid Mech. 70, Vieweg-Verlag, Wiesbaden, 1999, pp. 58-69.
10. 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.
11. T. HUGHES, G. FEIJOO, L. MAZZEI, AND J. QUINCY, The variational multiscale method: A paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), pp. 3-24.
12. V. V. JIKOV, S. M. KOZLOV, AND O. A. OLEINIK, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
13. O. A. LADYZHENSKAIA AND N. N. URALTSEVA, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
14. F.-H. LIN, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44 (1991), pp. 287-308.
15. S. MOSKOW AND M. VOGELIUS, First order corrections to the homogenized eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, 127A (1997), pp. 1263-1299.
16. E. ORIORDAN AND M. STYNES, A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in 2 dimensions, Math. Comp., 57 (1991), pp. 47-62.
17. F. SANTOSA AND M. VOGELIUS, First-order corrections to the homogenized eigenvalues of a periodic composite medium, SIAM J. Appl. Math., 53 (1993), pp. 1636-1668.
18. G. STRANG AND G. J. FIX, An Analysis of the Finite Element, Prentice-Hall, Englewood Cliffs, NJ, 1973.
19. Y. ZHANG, T. Y. HOU, AND X. H. WU, Convergence of a Petrov-Galerkin nonconforming multiscale finite element method, SIAM J. Numer. Anal., to appear.