Free access
Issue
ESAIM: M2AN
Volume 42, Number 2, March-April 2008
Page(s) 277 - 302
DOI http://dx.doi.org/10.1051/m2an:2008001
Published online 27 March 2008
  1. B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA J. 16 (1978) 525–528. [CrossRef]
  2. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. [CrossRef] [MathSciNet]
  3. C. Bardos, A.Y. Leroux and J.C. Nedelec, First order quasilinear equations with boundary conditions. Comm. Partial Diff. Eq. 4 (1979) 1017–1034. [CrossRef] [MathSciNet]
  4. 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 Ser. I Math. 339 (2004) 667–672.
  5. T. Barth and M. Ohlberger, Finite volume methods: Foundation and analysis, in Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes Eds., John Wiley & Sons (2004).
  6. J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147 (1999) 269–361. [CrossRef] [MathSciNet]
  7. B. Cockburn, Discontinuous Galerkin methods for computational fluid dynamics, in Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes Eds., John Wiley & Sons (2004).
  8. B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001) 173–261. [CrossRef] [MathSciNet]
  9. Y. Coudiere, J.P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493–516. [CrossRef] [EDP Sciences] [MathSciNet]
  10. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, volume VII, North-Holland, Amsterdam (2000) 713–1020.
  11. R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41–82. [CrossRef] [MathSciNet]
  12. R. Eymard, T. Gallouët and R. Herbin, A cell-centred finite volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal. 26 (2006) 326–353. [CrossRef] [MathSciNet]
  13. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996).
  14. M.A. Grepl, Reduced-basis Approximations and a Posteriori Error Estimation for Parabolic Partial Differential Equations. Ph.D. thesis, Massachusetts Institute of Technology, USA (2005).
  15. M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157–181. [CrossRef] [EDP Sciences]
  16. P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées 22 [Research in Applied Mathematics]. Masson, Paris (1992).
  17. R. Herbin and M. Ohlberger, A posteriori error estimate for finite volume approximations of convection diffusion problems, in Proc. 3rd Int. Symp. on Finite Volumes for Complex Applications - Problems and Perspectives (2002) 753–760.
  18. R.L. Higdon, Initial-boundary value problems for linear hyperbolic systems. SIAM Rev. 28 (1986) 177–217. [CrossRef] [MathSciNet]
  19. M.-J. Jasor and L. Lévi, Singular perturbations for a class of degenerate parabolic equations with mixed Dirichlet-Neumann boundary conditions. Ann. Math. Blaise Pascal 10 (2003) 269–296. [MathSciNet]
  20. D. Kröner, Numerical Schemes for Conservation Laws. John Wiley & Sons and Teubner (1997).
  21. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).
  22. L. Machiels, Y. Maday, I.B. Oliveira, A. Patera and D.V. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Ser. I Math. 331 (2000) 153–158. [CrossRef] [MathSciNet]
  23. M. Mangold and M. Sheng, Nonlinear model reduction of a 2D MCFC model with internal reforming. Fuel Cells 4 (2004) 68–77. [CrossRef]
  24. B.C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Automat. Control AC-26 (1981) 17–32.
  25. N.C. Nguyen, K. Veroy and A.T. Patera, Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling, S. Yip Ed., Springer (2005) 1523–1558.
  26. A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455–462. [CrossRef]
  27. M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. ESAIM: M2AN 35 (2001) 355–387. [CrossRef] [EDP Sciences] [MathSciNet]
  28. M. Ohlberger, A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Numer. Math. 87 (2001) 737–761. [CrossRef] [MathSciNet]
  29. M. Ohlberger and J. Vovelle, Error estimate for the approximation of non-linear conservation laws on bounded domains by the finite volume method. Math. Comp. 75 (2006) 113–150. [CrossRef] [MathSciNet]
  30. A.T. Patera and G. Rozza, Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations. Version 1.0, Copyright MIT 2006, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering.
  31. T.A. Porsching and M.L. Lee, The reduced basis method for initial value problems. SIAM J. Numer. Anal. 24 (1987) 1277–1287. [CrossRef] [MathSciNet]
  32. C. Prud'homme, D. 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]
  33. C. Prud'homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engineering 124 (2002) 70–80. [CrossRef]
  34. A. Quarteroni, G. Rozza, L. Dede and A. Quaini, Numerical approximation of a control problem for advection-diffusion processes, in System Modeling and Optimization, Proceedings of 22nd IFIP TC7 Conference (2006).
  35. D.V. Rovas, L. Machiels and Y. Maday, Reduced basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423–445. [CrossRef] [MathSciNet]
  36. C.W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcat. Chaos 15 (2005) 997–1013. [CrossRef]
  37. G. Rozza, Shape design by optimal flow control and reduced basis techniques: Applications to bypass configurations in haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Switzerland (2005).
  38. B. Schölkopf and A.J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond. MIT Press (2002).
  39. T. Tonn and K. Urban, A reduced-basis method for solving parameter-dependent convection-diffusion problems around rigid bodies. Technical Report 2006-03, Institute for Numerical Mathematics, Ulm University, ECCOMAS CFD (2006).
  40. K. Veroy and A.T. Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Meth. Fluids 47 (2005) 773–788. [CrossRef]
  41. K. Veroy, C. Prud'homme and A.T. Patera, Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. C. R. Acad. Sci. Paris Ser. I Math. 337 (2003) 619–624.

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