Reduced Basis Approximation and a Posteriori Error Estimation for the Parametrized Unsteady Boussinesq Equations
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Patera_Reduced basis app.pdf
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Author(s)
Knezevic, David
Nguyen, Ngoc Cuong
Patera, Anthony T.
Date Issued
July 2011
Journal
Mathematical Models and Methods in Applied Sciences
Publisher
World Scientific
Citation
Knezevic, David J., Ngoc-Cuong Nguyen, and Anthony T. Patera. “REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS.” Mathematical Models and Methods in Applied Sciences 21.07 (2011): 1415–1442.
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Author's final manuscript
Abstract
In this paper we present reduced basis (RB) approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold — to provide dimension reduction; an efficient proper orthogonal decomposition–Greedy sampling method for identification of optimal and numerically stable approximations — to yield rapid convergence; accurate (online) calculation of the solution-dependent stability factor by the successive constraint method — to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the RB approximation and associated outputs — to provide certainty in our predictions; and an offline–online computational decomposition strategy for our RB approximation and associated error bound — to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional "complex" enclosure — a square with a small rectangle cutout — parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the RB approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.
MIT Department
Massachusetts Institute of Technology. Center for Computational Engineering
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Massachusetts Institute of Technology. Department of Mechanical Engineering
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DOI of Published Version
https://doi.org/10.1142/s0218202511005441
