Reduced Basis A Posteriori Error Bounds for the Stokes Equations in Parametrized Domains: A Penalty Approach

2010 ◽  
Author(s):  
Anna-Lena Gerner ◽  
Karen Veroy ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Error Bounds ◽  
Stokes Equations ◽  
Posteriori Error ◽  
A Posteriori ◽  
Reduced Basis ◽  
A Posteriori Error ◽  
Penalty Approach ◽  
A Posteriori Error Bounds

REDUCED BASISA POSTERIORIERROR BOUNDS FOR THE STOKES EQUATIONS IN PARAMETRIZED DOMAINS: A PENALTY APPROACH

2011 ◽  
Vol 21 (10) ◽  
pp. 2103-2134 ◽  
Author(s):  
ANNA-LENA GERNER ◽  
KAREN VEROY
Keyword(s):  
Error Bounds ◽  
Stokes Equations ◽  
Posteriori Error ◽  
Galerkin Projection ◽  
Penalty Parameter ◽  
A Posteriori ◽  
Reduced Basis ◽  
A Posteriori Error ◽  
A Posteriori Error Bounds ◽  
Low Dimensional

We present reduced basis approximations and associated rigorous a posteriori error bounds for the Stokes equations in parametrized domains. The method, built upon the penalty formulation for saddle point problems, provides error bounds not only for the velocity but also for the pressure approximation, while simultaneously admitting affine geometric variations with relative ease. The essential ingredients are: (i) dimension reduction through Galerkin projection onto a low-dimensional reduced basis space; (ii) stable, good approximation of the pressure through supremizer-enrichment of the velocity reduced basis space; (iii) optimal and numerically stable approximations identified through an efficient greedy sampling method; (iv) certainty, through rigorous a posteriori bounds for the errors in the reduced basis approximation; and (v) efficiency, through an offline-online computational strategy. The method is applied to a flow problem in a two-dimensional channel with a (parametrized) rectangular obstacle. Numerical results show that the reduced basis approximation converges rapidly, the effectivities associated with the (inexpensive) rigorous a posteriori error bounds remain good even for reasonably small values of the penalty parameter, and that the effects of the penalty parameter are relatively benign.

CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (03) ◽  
pp. 1150015 ◽  
Author(s):  
MARTIN A. GREPL
Keyword(s):  
Error Bounds ◽  
Linear Time ◽  
Posteriori Error ◽  
Parabolic Partial Differential Equations ◽  
Coefficient Function ◽  
Time Varying ◽  
A Posteriori ◽  
Reduced Basis ◽  
A Posteriori Error ◽  
A Posteriori Error Bounds

We present reduced basis approximations and associated a posteriori error bounds for parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii ) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct "affine" coefficient-function approximations of the "nonaffine" (or nonlinear) parametrized functions. We consider linear time-invariant as well as linear time-varying nonaffine functions and introduce a new sampling approach to generate the function approximation space for the latter case. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. Numerical results are presented to confirm and test our approach.

scholarly journals Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation

2018 ◽  
Vol 19 (3) ◽  
pp. 663-695 ◽  
Author(s):  
Mark Kärcher ◽  
Sébastien Boyaval ◽  
Martin A. Grepl ◽  
Karen Veroy
Keyword(s):  
Data Assimilation ◽  
Error Bounds ◽  
Posteriori Error ◽  
A Posteriori ◽  
Reduced Basis ◽  
A Posteriori Error ◽  
A Posteriori Error Bounds

scholarly journals REDUCED BASIS APPROXIMATION ANDA POSTERIORIERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

2011 ◽  
Vol 21 (07) ◽  
pp. 1415-1442 ◽  
Author(s):  
DAVID J. KNEZEVIC ◽  
NGOC-CUONG NGUYEN ◽  
ANTHONY T. PATERA
Keyword(s):  
Error Bounds ◽  
Dimensional Space ◽  
Boussinesq Equations ◽  
Posteriori Error ◽  
Galerkin Projection ◽  
Stability Factor ◽  
A Posteriori ◽  
Reduced Basis ◽  
A Posteriori Error ◽  
A Posteriori Error Bounds

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.

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