Free Access
Issue
ESAIM: M2AN
Volume 42, Number 2, March-April 2008
Page(s) 277 - 302
DOI https://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] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  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. [Google Scholar]
  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). [Google Scholar]
  6. J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147 (1999) 269–361. [CrossRef] [MathSciNet] [Google Scholar]
  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). [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  10. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, volume VII, North-Holland, Amsterdam (2000) 713–1020. [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  13. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996). [Google Scholar]
  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). [Google Scholar]
  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] [MathSciNet] [Google Scholar]
  16. P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées 22 [Research in Applied Mathematics]. Masson, Paris (1992). [Google Scholar]
  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. [Google Scholar]
  18. R.L. Higdon, Initial-boundary value problems for linear hyperbolic systems. SIAM Rev. 28 (1986) 177–217. [CrossRef] [MathSciNet] [Google Scholar]
  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] [Google Scholar]
  20. D. Kröner, Numerical Schemes for Conservation Laws. John Wiley & Sons and Teubner (1997). [Google Scholar]
  21. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). [Google Scholar]
  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] [Google Scholar]
  23. M. Mangold and M. Sheng, Nonlinear model reduction of a 2D MCFC model with internal reforming. Fuel Cells 4 (2004) 68–77. [CrossRef] [Google Scholar]
  24. B.C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Automat. Control AC-26 (1981) 17–32. [Google Scholar]
  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. [Google Scholar]
  26. A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455–462. [CrossRef] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  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. [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  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). [Google Scholar]
  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] [Google Scholar]
  36. C.W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcat. Chaos 15 (2005) 997–1013. [CrossRef] [Google Scholar]
  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). [Google Scholar]
  38. B. Schölkopf and A.J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond. MIT Press (2002). [Google Scholar]
  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). [Google Scholar]
  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] [MathSciNet] [Google Scholar]
  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. [Google Scholar]

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