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.
|Number of pages||32|
|Journal||Mathematical Models and Methods in Applied Sciences|
|Publication status||Published - Oct 2011|
Bibliographical noteFunding 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 2011 Elsevier B.V., All rights reserved.
- a posteriori error estimation
- error bounds
- greedy sampling
- offline-online procedure
- real-time computation
- reduced basis approximation
- reduced order model
- saddle point problems
- Stokes equations
- successive constraints method