A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems

Multiscale Modeling and Simulation

Volumn 4, Issue 1, 2005, Pages 88-114

  Ohlberger, Mario ^a  

^a UNIVERSITY OF FREIBURG   (Germany)

Author keywords

Elliptic equations; Finite elements; Homogenization; Multiscale methods

Indexed keywords

ADAPTIVE ALGORITHMS; COMPUTER SIMULATION; ERROR ANALYSIS; ESTIMATION; FINITE DIFFERENCE METHOD; MATHEMATICAL MODELS;

A POSTERIORI ERROR ESTIMATES; ADAPTIVE MESH; ELLIPTIC EQUATIONS; ELLIPTIC HOMOGENIZATION; MULTISCALE METHODS;

FINITE ELEMENT METHOD;

EID: 33644783757     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/040605229     Document Type: Article

References (37)

1. A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems, J. Comput. Phys., 191 (2003), pp. 18-39.
2. G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), pp. 1482-1518.
3. I. Babuska, Homogenization and its application. Mathematical and computational problems, in Numerical Solution of Partial Differential Equations, III, Academic Press, New York, 1976, pp. 89-116.
4. P. Bastian, M. Droske, C. Engwer, R. Klöfkorn, T. Neubauer, M. Ohlberger, and M. Rumpf, Towards a unified framework for scientific computing, in Proceedings of the 15th International Conference on Domain Decomposition Methods, Berlin, 2003.
5. M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul., 1 (2003), pp. 221-238.
6. P. G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1987.
7. Ph. Clément, Approximation by finite element functions using local regularization, Rev. Prançaise Automat. Informat. Recherche Opérationnelle Sér Rouge Anal. Numér., 9 (1975), pp. 77-84.
8. W. Dörfler and O. Wilderotter, An adaptive finite element method for a linear elliptic equation with variable coefficients, ZAMM Z. Angew. Math. Mech., 80 (2000), pp. 481-491.
9. W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), pp. 87-132.
10. W. E and B. Engquist, Multiscale modeling and computation, Notices Amer. Math. Soc., 50 (2003), pp. 1062-1070.
11. W. E and B. Engquist, The Heterogeneous Multi-scale Method for Homogenization Problems, preprint, Princeton University, Princeton, NJ, 2003.
12. W. E, P. Ming, and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Sec., 18 (2005), pp. 121-156.
13. C. W. Gear and I. G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comput., 24 (2003), pp. 1091-1106.
14. C. W. Gear, J. Li, and I. G. Kevrekids, The gap-tooth method in particle simulations, Phys. Lett. A, 316 (2003), pp. 190-195.
15. T. Gessner, B. Haasdonk, R. Kende, M. Lenz, M. Metscher, R. Neubauer, M. Ohlberger, W. Rosenbaum, M. Rumpf, R. Schwörer, M. Spielberg, and U. Weikard, A Procedural Interface for Multiresolutional Visualization of General Numerical Data, Report 28, SFB 256, Bonn, 1999.
16. V. H. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales, Multiscale Model. Simul., 3 (2005), pp. 195-220.
17. 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. 236-252.
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. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127 (1995), pp. 387-401.
20. T. J. R. Hughes, G. R. Feijóo, L. Mazzei, and J.-B. Quincy, The variational multiscale method - a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), pp. 3-24.
21. T. J. R. Hughes and A. A. Oberai, The variational multiscale formulation of LES with application to turbulent channel flows, in Geometry, Mechanics, and Dynamics, Springer, New York, 2002, pp. 223-239.
22. A.-M. Matache, Sparse two-scale FEM for homogenization problems, J. Sci. Comput., 17 (2002), pp. 659-669.
23. A.-M. Matache and C. Schwab, Two-scale FEM for homogenization problems, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 537-572.
24. P. Ming and X. Yue, Numerical Methods for Multiscale Elliptic Problems, preprint, Princeton University, Princeton, NJ, 2003.
25. P. Ming and P. Zhang, Analysis of the Heterogeneous Multiscale Method for Parabolic Homogenization Problems, preprint, Princeton University, Princeton, NJ, 2003.
26. P. Morin, R. H. Nochetto, and K. G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), pp. 466-488.
27. P. Morin, R. H. Nochetto, and K. G. Seibert, Local problems on stars: A posteriori error estimators, convergence, and performance, Math. Comp., 72 (2003), pp. 1067-1097.
28. N. Neuss, W. Jäger, and G. Wittum, Homogenization and multigrid, Computing, 66 (2001), pp. 1-26.
29. J. T. Oden and S. Prudhomme, Goal-oriented error estimation and adaptivity for the finite element method, Comput. Math. Appl., 41 (2001), pp. 735-756.
30. J. T. Oden and K. S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms, J. Comput. Phys., 164 (2000), pp. 22-47.
31. M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients, Adv. Comput. Math., 16 (2002), pp. 47-75.
32. S. Repin, S. Sauter, and A. Smolianski, A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions, J. Comput. Appl. Math., 164/165 (2004), pp. 601-612.
33. O. Runborg, C. Theodoropoulus, and I. G. Kevrekidis, Effective bifurcation analysis: A time-stepper-based approach, Nonlinearity, 15 (2002), pp. 491-511.
34. C. Schwab and A.-M. Matache, Generalized FEM for homogenization problems, in Multiscale and Multiresolution Methods, Lect. Notes Comput. Sci, Eng. 20, Springer, Berlin, 2002, pp. 197-237.
35. K. S. Vemaganti and J. T. Oden, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. II. A computational environment for adaptive modeling of heterogeneous elastic solids, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 6089-6124.
36. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Advances in Numerical Mathematics, Teubner, Stuttgart, 1996.
37. V. V. Zhikov, S. M. Kozlov, and O. A. Olejnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.