Abstract
We extend previous work on a parameter multi-element hp certified reduced basis method for elliptic equations to the case of parabolic equations. A POD (in time)/Greedy (in parameter) sampling procedure is invoked both in the partitioning of the parameter domain (h-refinement) and in the construction of individual reduced basis approximation spaces for each parameter subdomain (p-refinement). The critical new issue is proper balance between additional POD modes and additional parameter values in the initial subdivision process. We present numerical results to compare the computational cost of the new approach to the standard (p-type) reduced basis method.
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
D. Amsallem, J. Cortial, and C. Farhat. On-demand CFD-based aeroelastic predictions using a database of reduced-order bases and models. In: 47th AIAA Aerospace Sciences Meeting (2009)
B. O. Almroth, P. Stern, and F. A. Brogan. Automatic choice of global shape functions in structural analysis. AIAA J., 16, 525–528, 1978
J. L. Eftang, A. T. Patera, and E. M. Rønquist. An hp certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput., accepted 2010
M. A. Grepl and A. T. Patera. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN, 39, 157–181, 2005
B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. M2AN Math. Model. Numer. Anal., 42, 277–302, 2008
A. K. Noor and J. M. Peters. Reduced basis technique for nonlinear analysis of structures. AIAA J., 18, 455–462, 1980
D. J. Knezevic and A. T. Patera. A certified reduced basis method for the Fokker-Planck equation of dilute polymeric fluids: FENE dumbbells in extensional flow. SIAM J. Sci. Comput., 32(2):793–817, 2010
N. C. Nguyen, G. Rozza, D. B. P. Huynh, and A. T. Patera. Reduced Basis Approximation and A Posteriori Error Estimation for Paramtrized Parabolic PDEs; Application to Real-Time Bayesian Parameter Estimation. In: Biegler, et al. Computational Methods for Large Scale Inverse Problems and Uncertainty Quantification. Wiley, London (2009)
G. Rozza, D. B. P. Huynh, and A. T. Patera. Reduced Basis Approximation and A Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations. Arch. Comput. Methods Eng., 15, 229–275, 2008
Acknowledgements
We acknowledge helpful discussions with D. Knezevic, N. C. Nguyen, D. B. P. Huynh, and S. Boyaval. The work has been supported by the Norwegian University of Science and Technology, AFOSR Grant FA 9550-07-1-0425, and OSD/AFOSR Grant FA 9550-09-1-0613.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Berlin Heidelberg
About this paper
Cite this paper
Eftang, J.L., Patera, A.T., Rønquist, E.M. (2011). An hp Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations. In: Hesthaven, J., Rønquist, E. (eds) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15337-2_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-15337-2_15
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15336-5
Online ISBN: 978-3-642-15337-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)