Summary
The variational multiscale method (VMM) provides a general framework for construction of multiscale finite element methods. In this paper we propose a method for parallel solution of the fine scale problem based on localized Dirichlet problems which are solved numerically. Next we present a posteriori error representation formulas for VMM which relates the error in linear functionals to the discretization errors, resolution and size of patches in the localized problems, in the fine scale approximation. These formulas are derived by using duality techniques. Based on the a posteriori error representation formula we propose an adaptive VMM with automatic tuning of the critical parameters. We primarily study elliptic second order partial differential equations with highly oscillating coefficients or localized singularities.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Arbogast and S. L. Bryant, A two-scale numerical subgrid technique for waterflood simulations, SPE J., Dec. 2002, pp 446–457.
B. Aksoylu, S. Bond, and M. Holst An odyssey into local refinement and multilevel preconditioning III: Implementation of numerical experiments, SIAM J. Sci. Comput., 25(2003), 478–498.
B. Aksoylu and M. Holst An odyssey into local refinement and multilevel preconditioning I: Optimality of the BPX preconditioner, SIAM J. Numer. Anal. (2002) in review
B. Aksoylu and M. Holst An odyssey into local refinement and multilevel preconditioning II: stabilizing hierarchical basis methods, SIAM J. Numer. Anal. in review
S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Springer Verlag, 1994.
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) 169–189.
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) 387–401.
T. J. R. Hughes, L Mazzei, A. A. Oberai, and M. G. Larson, The continuous Galerkin method is locally conservative, J. Comput. Phys., Vol. 163(2) (2000) 467–488.
T. J. R. Hughes, G. R. Feijóo, L. Mazzei and Jean-Baptiste Quincy, The variational multiscale method-a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998) 3–24.
M. G. Larson and A. Målqvist, Adaptive Variational Multiscale Methods Based on A Posteriori Error Estimates: Energy Norm Estimates for Elliptic Problems, preprint
A. A. Oberai and P. M. Pinsky, A multiscale finite element method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 154 (1998) 281–297.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Larson, M.G., Målqvist, A. (2005). Adaptive Variational Multiscale Methods Based on A Posteriori Error Estimation: Duality Techniques for Elliptic Problems. In: Engquist, B., Runborg, O., Lötstedt, P. (eds) Multiscale Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26444-2_9
Download citation
DOI: https://doi.org/10.1007/3-540-26444-2_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25335-8
Online ISBN: 978-3-540-26444-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)