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]
  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]
  3. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 2nd edition. Springer (2002).
  4. F. Brezzi, and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Comput. Math. 15 (1991).
  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]
  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]
  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]
  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]
  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]
  10. A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer-Verlag, New York (2004).
  11. L.C. Evans, Partial Differential Equations. Amer. Math. Soc. (1998).
  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]
  13. D. Green and W.G. Unruh, The failure of the Tacoma bridge : a physical model. Am. J. Phys. 74 (2006) 706–716. [CrossRef]
  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]
  15. A. Holt and M. Landahl, Aerodynamics of wings and bodies. Dover New York (1985).
  16. D.B.P. Huynh and G. Rozza, Reduced basis method and a posteriori error estimation : application to linear elasticity problems (2011). Submitted.
  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]
  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]
  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.
  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]
  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]
  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]
  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]
  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.
  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.
  26. L.M. Milne-Thomson, Theoretical aerodynamics. Dover (1973).
  27. B. Mohammadi and O. Pironneau, Applied shape optimization for fluids. Oxford University Press (2001).
  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]
  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]
  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).
  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]
  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).
  33. G. Rozza, Reduced basis approximation and error bounds for potential flows in parametrized geometries. Commun. Comput. Phys. 9 (2011) 1–48.
  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]
  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.
  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.
  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]
  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.
  39. J. Xu and L. Zikatanov, Some observation on Babuška and Brezzi theories. Numer. Math. 94 (2003) 195–202. [CrossRef] [MathSciNet]
  40. S. Zhang, Efficient greedy algorithms for successive constraints methods with high-dimensional parameters. C. R. Acad. Sci. Paris, Sér. I Math. (2011). Submitted.

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