Free Access
Issue
ESAIM: M2AN
Volume 46, Number 6, November-December 2012
Page(s) 1555 - 1576
DOI https://doi.org/10.1051/m2an/2012016
Published online 01 August 2012
  1. I. Babuška and S.A. Sauter, Is the pollution effect of the FEM avoidable for Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000) 451–484. [CrossRef] [MathSciNet] [Google Scholar]
  2. M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An ‘empirical interpolation’ method : application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Sér. I Math. 339 (2004) 667–672. [CrossRef] [MathSciNet] [Google Scholar]
  3. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 2nd edition. Springer (2002). [Google Scholar]
  4. F. Brezzi, and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Comput. Math. 15 (1991). [Google Scholar]
  5. Y. Chen, J. Hesthaven, Y. Maday and J. Rodriguez, A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations. C. R. Acad. Sci. Paris, Sér. I Math. 346 (2008) 1295–1300. [CrossRef] [Google Scholar]
  6. J.L. Eftang, M.A. Grepl and A.T. Patera, A posteriori error bounds for the empirical interpolation method. C. R. Acad. Sci. Paris, Sér. I Math. 348 (2010) 575–579. [CrossRef] [Google Scholar]
  7. J.L. Eftang, A.T. Patera and E.M. Rønquist, An “hp” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32 (2010) 3170–3200. [CrossRef] [MathSciNet] [Google Scholar]
  8. J.L. Eftang, D.J. Knezevic and A.T. Patera, An “hp” certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dyn. 17 (2011) 395–422. [CrossRef] [MathSciNet] [Google Scholar]
  9. J.L. Eftang, D.B.P. Huynh, D.J. Knezevic and A.T. Patera, A two-step certified reduced basis method. J. Sci. Comput. 51 (2012) 28–58. [CrossRef] [Google Scholar]
  10. A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer-Verlag, New York (2004). [Google Scholar]
  11. L.C. Evans, Partial Differential Equations. Amer. Math. Soc. (1998). [Google Scholar]
  12. A.L. Gerner and K. Veroy, Reduced basis a posteriori error bounds for the stokes equations in parametrized domains : a penalty approach. Math. Mod. Methods Appl. Sci. 21 (2011) 2103–2134. [CrossRef] [Google Scholar]
  13. D. Green and W.G. Unruh, The failure of the Tacoma bridge : a physical model. Am. J. Phys. 74 (2006) 706–716. [CrossRef] [Google Scholar]
  14. M.A. Grepl, Y. Maday, N.C. Nguyen and A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM : M2AN 41 (2007) 575–605. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  15. A. Holt and M. Landahl, Aerodynamics of wings and bodies. Dover New York (1985). [Google Scholar]
  16. D.B.P. Huynh and G. Rozza, Reduced basis method and a posteriori error estimation : application to linear elasticity problems (2011). Submitted. [Google Scholar]
  17. D.B.P Huynh, G. Rozza, S. Sen and A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability costants. C. R. Acad. Sci. Paris, Sér. I Math. 345 (2007) 473–478. [CrossRef] [MathSciNet] [Google Scholar]
  18. D.B.P. Huynh, D. Knezevic, Y. Chen, J. Hesthaven and A.T. Patera, A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199 (2010) 1963–1975. [CrossRef] [MathSciNet] [Google Scholar]
  19. D.B.P. Huynh, N.C. Nguyen, A.T. Patera and G. Rozza, Rapid reliable solution of the parametrized partial differential equations of continuum mechanics and transport. Available on http://augustine.mit.edu. [Google Scholar]
  20. T. Lassila and G. Rozza, Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng. 199 (2010) 1583–1592. [CrossRef] [Google Scholar]
  21. T. Lassila and G. Rozza, Model reduction of semiaffinely parametrized partial differential equations by two-level affine approximation. C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 61–66. [CrossRef] [Google Scholar]
  22. T. Lassila, A. Quarteroni and G. Rozza, A reduced basis model with parametric coupling for fluid-structure interaction problems. SIAM J. Sci. Comput. 34 (2012) A1187–A1213. [CrossRef] [Google Scholar]
  23. Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general multipurpose interpolation procedure : the magic points. Commun. Pure Appl. Anal. 8 (2009) 383–404. [CrossRef] [MathSciNet] [Google Scholar]
  24. A. Manzoni, A. Quarteroni and G. Rozza, Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomed. Eng. (2011). In press, DOI: 10.1002/cnm.1465. [Google Scholar]
  25. A. Manzoni, A. Quarteroni and G. Rozza, Shape optimization of cardiovascular geometries by reduced basis methods and free-form deformation techniques. Int. J. Numer. Methods Fluids (2011). In press, DOI: 10.1002/fld.2712. [Google Scholar]
  26. L.M. Milne-Thomson, Theoretical aerodynamics. Dover (1973). [Google Scholar]
  27. B. Mohammadi and O. Pironneau, Applied shape optimization for fluids. Oxford University Press (2001). [Google Scholar]
  28. N.C. Nguyen, A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations. J. Comput. Phys. 227 (2007) 983–1006. [CrossRef] [Google Scholar]
  29. N.C. Nguyen, G. Rozza and A.T. Patera, Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers equation. Calcolo 46 (2009) 157–185. [CrossRef] [MathSciNet] [Google Scholar]
  30. A.T. Patera and G. Rozza, Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equation. Version 1.0, Copyright MIT (2006), to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering (2009). [Google Scholar]
  31. C. Prud’homme, D.V. Rovas, K. Veroy and A.T. Patera, A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. ESAIM : M2AN 36 (2002) 747–771. [CrossRef] [EDP Sciences] [Google Scholar]
  32. A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations in industrial applications. J. Math. Ind. 1 (2011). [Google Scholar]
  33. G. Rozza, Reduced basis approximation and error bounds for potential flows in parametrized geometries. Commun. Comput. Phys. 9 (2011) 1–48. [Google Scholar]
  34. G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15 (2008) 229–275. [CrossRef] [MathSciNet] [Google Scholar]
  35. G. Rozza, D.B.P. Huynh and A. Manzoni, Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries : roles of the inf-sup stability constants. Technical Report 22.2010, MATHICSE (2010). Online version available at : http://cmcs.epfl.ch/people/manzoni. [Google Scholar]
  36. G. Rozza, T. Lassila and A. Manzoni, Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map, in Spectral and High Order Methods for Partial Differential Equations. Selected papers from the ICOSAHOM’09 Conference, Trondheim, Norway, edited by J.S. Hesthaven and E.M. Rønquist. Lect. Notes Comput. Sci. Eng. 76 (2011) 307–315. [Google Scholar]
  37. S. Sen, K. Veroy, P. Huynh, S. Deparis, N.C. Nguyen and A.T. Patera, “Natural norm” a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217 (2006) 37–62. [CrossRef] [MathSciNet] [Google Scholar]
  38. S. Vallaghe, A. Le-Hyaric, M. Fouquemberg and C. Prud’homme, A successive constraint method with minimal offline constraints for lower bounds of parametric coercivity constant. C. R. Acad. Sci. Paris, Sér. I Math. (2011). Submitted. [Google Scholar]
  39. J. Xu and L. Zikatanov, Some observation on Babuška and Brezzi theories. Numer. Math. 94 (2003) 195–202. [CrossRef] [MathSciNet] [Google Scholar]
  40. S. Zhang, Efficient greedy algorithms for successive constraints methods with high-dimensional parameters. C. R. Acad. Sci. Paris, Sér. I Math. (2011). Submitted. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you