Finite element heterogeneous multiscale methods with near optimal computational complexity

Multiscale Modeling and Simulation

Volumn 6, Issue 4, 2007, Pages 1059-1084

  Abdulle, Assyr ^a   Engquist, Bjorn ^b  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Elliptic homogenization; Heterogeneous finite element method; Multiscale method; Spectral method

Indexed keywords

ELLIPTIC HOMOGENIZATION; ELLIPTIC PROBLEMS; FINITE ELEMENTS; HETEROGENEOUS FINITE ELEMENT METHOD; HETEROGENEOUS MULTISCALE METHODS; MULTISCALE; MULTISCALE METHOD; OPTIMAL COMPUTATIONAL COMPLEXITIES; PERIODIC HOMOGENIZATIONS; PSEUDOSPECTRAL METHODS; SPECTRAL METHOD;

COMPUTATIONAL COMPLEXITY; HIDDEN MARKOV MODELS; HOMOGENIZATION METHOD; NUMERICAL METHODS;

FINITE ELEMENT METHOD;

EID: 55249110253     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/060676118     Document Type: Article

References (30)

1. A. ABDULLE AND W. E, Finite difference HMM for homogenization problems, J. Comput. Phys., 191 (2003), pp. 18-39.
2. A. ABDULLE AND C. SCHWAB, Heterogeneous multiscale FEM for diffusion problems on rough surfaces, Multiscale Model. Simul., 3 (2005), pp. 195-220.
3. A. ABDULLE, On a priori error analysis of fully discrete heterogeneous multiscale FEM, Multiscale Model. Simul., 4 (2005), pp. 447-459.
4. A. ABDULLE, Multiscale methods for advection-diffusion problems, Discrete Contin. Dyn. Syst., Suppl. Vol. (2005), pp. 11-21.
5. A. ABDULLE, Analysis of a heterogeneous multiscale FEM for problems in elasticity, Math. Models Methods Appl. Sci., 16 (2006), pp. 615-635.
6. A. BENSOUSSAN, J.-L. LIONS, AND G. P APANICOLAOU, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
7. C. BERNARDI, Y. MADAY, AND F. RAPETTI, Discrétisations variationnelles de problèmes aux limites elliptiques, Math. Appl. (Berlin) 45, Springer, Berlin, 2004.
8. S. N. BERNSTEIN, Sur l'ordre de la meilleure approximation des fonctions continues par les polynômes de degre ́donné , Mémoire de l'Acad. Royale de Belgique, 4 (1912), pp. 1-104.
9. L. BERS, F. JOHN, AND M. SCHECHTER, Partial Differential Equations, Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, CO, 1957.
10. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer, New York, 1987.
11. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
12. D. CIORANESCU and P. DONATO, An Introduction to Homogenization, Oxford University Press, New York, 1999.
13. W. E and B. ENGQUIST, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), pp. 87-132.
14. W. E, P. MING, and P. ZHANG, Analysis of the heterogeneous multi-scale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2004), pp. 121-156.
15. B. ENGQUIST and Y.-H. TSAI, Heterogeneous multiscale methods for stiff ordinary differential equations, Math. Comp., 74 (2005), pp. 1707-1742.
16. D. GOTTLIEB and S. A. ORSZAG, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math. 26, SIAM, Philadelphia, 1977.
17. V. H. HOANG AND S. SCHWAB, High-dimensional finite elements for elliptic problems with multiple scales, Multiscale Model. Simul., 3 (2005), pp. 168-194.
18. 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.
19. T. 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.
20. D. JACKSON, The Theory of Approximation, Amer. Math. Soc. Colloq. Publ. 11, AMS, New York, 1930.
21. V. V. JIKOV, S. M. KOZLOV, and O. A. OLEINIK, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, Heidelberg, 1994.
22. H.-O. KREISS and J. OLIGER, Stability of the Fourier method, SIAM J. Numer. Anal., 16 (1979), pp. 421-433.
23. O. A. LADYZHENSKAYA and N. N. URAL'TSEVA, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
24. P. MLNG and Y. YUE, Numerical methods for multiscale elliptic problems, J. Comput. Phys., 214 (2006), pp. 421-445.
25. P. MLNG and Y. YUE, Numerical methods for multiscale elliptic problems, J. Comput. Phys., 214 (2006), pp. 421-445.
26. S. MOSKOW and M. VOGELIUS, First order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy Soc. Edinburgh Sect. A, 127 (1997), pp. 1263-1299.
27. L. NIRENBERG and C. B. MORREY, JR., On the analycity of the solution of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math., 10 (1957), pp. 271-290.
28. J. T. ODEN and K. S. VEMAGANTI, Estimation of local modeling error and global-oriented adaptive modeling of heterogeneous materials: Error estimates and adaptive algorithms, J. Comput. Phys., 164 (2000), pp. 22-47.
29. E. TADMOR, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal., 23 (1986), pp. 1-10.
30. T. C. WALLSTROM. S. HOU, M. A. CHRISTIE, L. J. DURLOFSKY, AND D. H. SHARP, Accurate scale up of two phase flow using renormalization and nonuniform coarsening, Comput. Geosci., 3 (1999), pp. 69-87.