A space-time certified reduced basis method for Burgers' equation
We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which...
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| Format: | Article |
| Language: | en_US |
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World Scientific
2015
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| Online Access: | http://hdl.handle.net/1721.1/97700 https://orcid.org/0000-0002-8323-9054 https://orcid.org/0000-0002-2631-6463 |
| _version_ | 1811095342824816640 |
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| author | Yano, Masayuki Patera, Anthony T. Urban, Karsten |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Yano, Masayuki Patera, Anthony T. Urban, Karsten |
| author_sort | Yano, Masayuki |
| collection | MIT |
| description | We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which enjoys a favorable inf–sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi–Rappaz–Raviart a posteriori error bounds. We describe computational offline–online decomposition procedures for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf–sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L[superscript 2]-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, in marked contrast to the exponentially growing estimate of the classical formulation for high Peclet number cases. |
| first_indexed | 2024-09-23T16:15:35Z |
| format | Article |
| id | mit-1721.1/97700 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:15:35Z |
| publishDate | 2015 |
| publisher | World Scientific |
| record_format | dspace |
| spelling | mit-1721.1/977002022-09-29T19:11:18Z A space-time certified reduced basis method for Burgers' equation A space-time hp-interpolation-based certified reduced basis method for Burgers' equation Yano, Masayuki Patera, Anthony T. Urban, Karsten Massachusetts Institute of Technology. Department of Mechanical Engineering Yano, Masayuki Patera, Anthony T. We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which enjoys a favorable inf–sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi–Rappaz–Raviart a posteriori error bounds. We describe computational offline–online decomposition procedures for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf–sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L[superscript 2]-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, in marked contrast to the exponentially growing estimate of the classical formulation for high Peclet number cases. United States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (Grant FA9550-09-1-0613) United States. Office of Naval Research (Grant N00014-11-1-0713) Deutsche Forschungsgemeinschaft (Ur-63/9) Deutsche Forschungsgemeinschaft (GrK1100) 2015-07-07T16:33:14Z 2015-07-07T16:33:14Z 2014-02 2013-07 Article http://purl.org/eprint/type/JournalArticle 0218-2025 1793-6314 http://hdl.handle.net/1721.1/97700 Yano, Masayuki, Anthony T. Patera, and Karsten Urban. “A Space-Time Hp-Interpolation-Based Certified Reduced Basis Method for Burgers’ Equation.” Math. Models Methods Appl. Sci. 24, no. 09 (August 2014): 1903–1935. https://orcid.org/0000-0002-8323-9054 https://orcid.org/0000-0002-2631-6463 en_US http://dx.doi.org/10.1142/S0218202514500110 Mathematical Models and Methods in Applied Sciences Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf World Scientific Other univ. web domain |
| spellingShingle | Yano, Masayuki Patera, Anthony T. Urban, Karsten A space-time certified reduced basis method for Burgers' equation |
| title | A space-time certified reduced basis method for Burgers' equation |
| title_full | A space-time certified reduced basis method for Burgers' equation |
| title_fullStr | A space-time certified reduced basis method for Burgers' equation |
| title_full_unstemmed | A space-time certified reduced basis method for Burgers' equation |
| title_short | A space-time certified reduced basis method for Burgers' equation |
| title_sort | space time certified reduced basis method for burgers equation |
| url | http://hdl.handle.net/1721.1/97700 https://orcid.org/0000-0002-8323-9054 https://orcid.org/0000-0002-2631-6463 |
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