REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

By David J. Knezevic, Ngoc Cuong Nguyen, Anthony T. Patera. 2011

88 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c9b026a80c50d018f99e2b4559f6400a8082f8d9

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Citations in the corpus

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations (2013, methodology): “[17] proposed a rapidly convergent reduced basis method with associated a posteriori error bounds for the parametrized unsteady Boussinesq equations.”
CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS (2012, background): “In the recent past, reduced basis approximations and associated a posteriori error estimation for linear and at most quadratically nonlinear elliptic and parabolic PDEs honoring this requirement have been successfully developed.(19,20,23,32,30,36,46,48,49) In Ref.”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, background): “However, the most important challenge deals with the reliability and/or the certification of the methodology in the unsteady – parabolic – problems [23,62]: in these cases exponential instability seriously compromises a priori and a posteriori error estimates, yielding to bounds which are limited to”
A new error bound for reduced basis approximation of parabolic partial differential equations (2012, background): “In contrast, a standard energy approach [3] gives effective inf-sup constants on the order of e−μ1 T (here about 10−8).”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, methodology): “Certified RB methods have been applied to parametrized (moderate Reynolds) unsteady viscous flows in [70], where a nonisothermal viscous flow is modelled by Boussinesq equations describing natural convection.”