TY - JOUR
T1 - Reduced basis a posteriori error bounds for the stokes equations in parametrized domains
T2 - A penalty approach
AU - Gerner, Anna Lena
AU - Veroy, Karen
N1 - Funding Information:
The authors would like to thank Prof. Anthony T. Patera of MIT for many helpful discussions and comments, and for his invaluable support and encouragement. The authors would also like to thank Prof. Arnold Reusken of RWTH Aachen University, and Lorenzo Zanon of Politecnico Torino for helpful comments, and also Dr. Gian-luigi Rozza of EPFL for previous contributions. Financial support from the Deutsche Forschungsgemeinschaft (German Research Foundation) through Grant GSC 111 is gratefully acknowledged.
Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2011/10
Y1 - 2011/10
N2 - 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.
AB - 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.
KW - a posteriori error estimation
KW - error bounds
KW - greedy sampling
KW - offline-online procedure
KW - penalty
KW - real-time computation
KW - reduced basis approximation
KW - reduced order model
KW - saddle point problems
KW - Stokes equations
KW - successive constraints method
UR - http://www.scopus.com/inward/record.url?scp=80054958204&partnerID=8YFLogxK
U2 - 10.1142/S0218202511005672
DO - 10.1142/S0218202511005672
M3 - Article
AN - SCOPUS:80054958204
VL - 21
SP - 2103
EP - 2134
JO - Mathematical Models & Methods in Applied Sciences
JF - Mathematical Models & Methods in Applied Sciences
SN - 0218-2025
IS - 10
ER -