Reduced Basis Approximation and a Posteriori Error Estimation for the Parametrized Unsteady Boussinesq Equations
Author(s)
Knezevic, David; Nguyen, Ngoc Cuong; Patera, Anthony T.
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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.
Date issued
2011-07Department
Massachusetts Institute of Technology. Center for Computational Engineering; Massachusetts Institute of Technology. Department of Aeronautics and Astronautics; Massachusetts Institute of Technology. Department of Mechanical EngineeringJournal
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.
Version: Author's final manuscript
ISSN
0218-2025
1793-6314