Abstract
Both the variational and heterogeneous multiscale methods are presented for non-linear variational problems. We show that the variational multiscale method can be seen as a subset of the methods which can be defined within the heterogeneous multiscale method framework. Our results extend to the approximate forms of the multiscale methods which are of interest to applications.
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References
I. Babuska and J. E. Osborn, Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods, SIAM J. Numer. Anal., 20(3), 510–536, 1983
D. Braess, Finite elements, Cambridge University Press, Cambridge, 1997
A. Brandt, Barriers to Achieving Textbook Multigrid Efficiency (TME) in CFD, URL: http://hdl.handle.net/2002/14809, 1998
A. Brandt, Multiscale Solvers and Systematic Upscaling in Computational Physics, Comp. Phys. Com., 169, 438–441, 2005
F. Brezzi, L. P. Franca, T. J. R. Hughes and A. Russo, b = ∫ \nolimits \nolimits g, Comput. Meth. Appl. Mech. Eng., 145, 329–339, 1997
Z. Chen, Multiscale Methods for Elliptic Homogenization Problems, Numer. Meth. Part. Diff. Equat., 22, 317–360, 2006
Z. Chen and T. Y. Hou, A Mixed Multiscale Finite Element Method for Elliptic Problems with Oscillating Coefficients, Math. Comp. 72(242), 541–576, 2002
Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, Society of Industrial and Applied Mathematics, Philadelphia, 2006.
L. Durlofsky, Coarse Scale Models of Two Phase Flow in Heterogeneous Reservoirs: Volume Averaged Equations and their Relationship to Existing Upscaling Techniques, Comp. Geosci., 2, 73–92, 1998
W. E and B. Engquist, The heterogeneous multi-scale methods, Comm. Math. Sci., 1, 87–133, 2003
T. J. R. Hughes and G. Sangalli, Variational Multiscale Analysis: The Fine-scale Green’s Function, Projection, Optimization, Localization, and Stabilized Methods SINUM, 45(2), 539–557, 2007
T. J. R. Hughes, G. R. Feijoo, L. Mazzei and J. B. Quincy, The Variational Multiscale Method – A Paradigm for Computational Mechanics, Comp. Meth. Appl. Mech. and Eng., 166(1–2), 3–24, 1998
M. G. Larson, A. Malqvist, Adaptive Variational Multiscale Methods Based on a Posteriori Error Estimation: Energy Norm Estimates for Elliptic Problems, Comp. Meth. Appl. Mech. Eng., 196(21–24), 2313–2324, 2007
P. B. Ming and P.-W. Zhang, Analysis of the Heterogeneous Multiscale Method for Parabolic Homogenization Problems, Math. Comput., 76(257), 153–157, 2007
J. Nolen, G. Papanicolaou and O. Pironneau, A Framework for Adaptive Multiscale Methods for Elliptic Problems, SIAM Multiscale Model. Simul., 7, 171–196, 2008
J. M. Nordbotten, Adaptive Variational Multiscale Methods for Multi-Phase Flow in Porous Media, SIAM Multiscale Model. Simul., 7(3), 1455–1473, 2009
M. Ohlberger, A Posteriori Error Estimates for the Heterogeneous Multiscale Finite Element Method for Elliptic Homogenization Problems, SIAM Multiscale Model. Simul., 4(1), 88–144, 2005
M. Shashkov, Conservative finite-difference methods on general grids, CRC, FL, 1996
B. F. Smith, P. E. Bjørstad and W. D. Gropp, Domain Decomposition, Cambridge University Press, Cambridge, 1996
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Nordbotten, J.M. (2010). Variational and Heterogeneous Multiscale Methods. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds) Numerical Mathematics and Advanced Applications 2009. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11795-4_76
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DOI: https://doi.org/10.1007/978-3-642-11795-4_76
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