Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants
In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi’s and Babuška’s stability constants. The crucial ingredients of the methodology are a Galerki...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Springer-Verlag
2014
|
| Online Access: | http://hdl.handle.net/1721.1/85654 https://orcid.org/0000-0002-2794-1308 https://orcid.org/0000-0002-0810-8812 |
| _version_ | 1811094965544026112 |
|---|---|
| author | Rozza, Gianluigi Manzoni, Andrea Huynh, Dinh Bao Phuong |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Rozza, Gianluigi Manzoni, Andrea Huynh, Dinh Bao Phuong |
| author_sort | Rozza, Gianluigi |
| collection | MIT |
| description | In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi’s and Babuška’s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi’s saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuška’s inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle. |
| first_indexed | 2024-09-23T16:08:22Z |
| format | Article |
| id | mit-1721.1/85654 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:08:22Z |
| publishDate | 2014 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/856542022-10-02T06:34:48Z Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants Rozza, Gianluigi Manzoni, Andrea Huynh, Dinh Bao Phuong Massachusetts Institute of Technology. Department of Mechanical Engineering Rozza, Gianluigi Rozza, Gianluigi Huynh, Dinh Bao Phuong In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi’s and Babuška’s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi’s saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuška’s inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle. United States. Air Force Office of Scientific Research (Grant FA9550-07-1-0425) United States. Air Force Office of Scientific Research (Office of the Secretary of Defense Grant FA9550-09-1-0613) 2014-03-14T19:47:57Z 2014-03-14T19:47:57Z 2013-03 2012-10 Article http://purl.org/eprint/type/JournalArticle 0029-599X 0945-3245 http://hdl.handle.net/1721.1/85654 Rozza, Gianluigi, D. B. Phuong Huynh, and Andrea Manzoni. “Reduced Basis Approximation and a Posteriori Error Estimation for Stokes Flows in Parametrized Geometries: Roles of the Inf-Sup Stability Constants.” Numerische Mathematik 125, no. 1 (September 2013): 115–152. https://orcid.org/0000-0002-2794-1308 https://orcid.org/0000-0002-0810-8812 en_US http://dx.doi.org/10.1007/s00211-013-0534-8 Numerische Mathematik Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Springer-Verlag Rozza |
| spellingShingle | Rozza, Gianluigi Manzoni, Andrea Huynh, Dinh Bao Phuong Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants |
| title | Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants |
| title_full | Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants |
| title_fullStr | Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants |
| title_full_unstemmed | Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants |
| title_short | Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants |
| title_sort | reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries roles of the inf sup stability constants |
| url | http://hdl.handle.net/1721.1/85654 https://orcid.org/0000-0002-2794-1308 https://orcid.org/0000-0002-0810-8812 |
| work_keys_str_mv | AT rozzagianluigi reducedbasisapproximationandaposteriorierrorestimationforstokesflowsinparametrizedgeometriesrolesoftheinfsupstabilityconstants AT manzoniandrea reducedbasisapproximationandaposteriorierrorestimationforstokesflowsinparametrizedgeometriesrolesoftheinfsupstabilityconstants AT huynhdinhbaophuong reducedbasisapproximationandaposteriorierrorestimationforstokesflowsinparametrizedgeometriesrolesoftheinfsupstabilityconstants |