Free Access
Volume 48, Number 3, May-June 2014
Page(s) 623 - 663
Published online 20 January 2014
  1. P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Convergence Rates for Greedy Algorithms in Reduced Basis Methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. [CrossRef] [MathSciNet] [Google Scholar]
  2. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, in vol. 15 of Springer Ser. Comput. Math. Springer-Verlag (1991). [Google Scholar]
  3. A. Buffa, Y. Maday, A.T. Patera, C. Prud’homme and G. Turinici, A Priori convergence of the greedy algorithm for the parameterized reduced basis. ESAIM: M2AN 46 (2012) 595–603. [CrossRef] [EDP Sciences] [Google Scholar]
  4. J.M. Cascon, C. Kreuzer, R.H. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. [Google Scholar]
  5. A. Cohen, W. Dahmen and G. Welper, Adaptivity and Variational Stabilization for Convection-Diffusion Equations. ESAIM: M2AN 46 (2012) 1247–1273. [CrossRef] [EDP Sciences] [Google Scholar]
  6. W. Dahmen, Parameter dependent transport equations, in Workshop J.L.L.-SMP: Reduced Basis Methods in High Dimensions. Available at˙ljll˙smp˙rbihd.html [Google Scholar]
  7. W. Dahmen, C. Huang, C. Schwab and G. Welper, Adaptive Petrov−Galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50 (2012) 2420–2445. [CrossRef] [Google Scholar]
  8. L.F. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov−Galerkin Methods I: The transport equation. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1558–1572. [CrossRef] [MathSciNet] [Google Scholar]
  9. L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov−Galerkin methods. Part II: Optimal test functions. Numer. Methods for Partial Differ. Equ. 27 (2011) 70–105. [CrossRef] [Google Scholar]
  10. S. Deparis, Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM J. Numer. Anal. 46 (2008) 2039–2067. [CrossRef] [MathSciNet] [Google Scholar]
  11. R. DeVore, G. Petrova and P. Wojtaszczyk, Greedy algorithms for reduced bases in Banach spaces, Constructive Approximation 37 (2013) 455–466. [Google Scholar]
  12. A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer (2004). [Google Scholar]
  13. A. Gerner and K. Veroy-Grepl, Certified reduced basis methods for parametrized saddle point problems, preprint (2012). To appear in SIAM J. Sci. Comput. [Google Scholar]
  14. A.-L. Gerner and K. Veroy, Reduced basis a posteriori error bounds for the Stokes equations in parameterized domains: A penalty approach. M3AS: Math. Models Methods Appl. Sci. 21 (2011) 2103–2134. [Google Scholar]
  15. M.A. Grepl, Certified Reduced Basis Methods for Nonaffine Linear Time-Varying and Nonlinear Parabolic Partial Differential Equations. M3AS: Math. Models Methods Appl. Sci. 22 (2012) 40. [Google Scholar]
  16. 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]
  17. B. Haasdonk, Convergence rates for the POD-greedy method. ESAIM: M2AN 47 (2013) 859–873. [CrossRef] [EDP Sciences] [Google Scholar]
  18. T. Hughes and G. Sangalli, Variational Multiscale Analysis: the Fine-scale Green’s Function, Projection, Optimization, Localization, and Stabilized Methods. SIAM J. Numer. Anal. 45 (2007) 539–557. [Google Scholar]
  19. G. Kanschat, E. Meinköhn, R. Rannacher and R. Wehrse, Numerical methods in multidimensional radiative transfer, Springer (2009). [Google Scholar]
  20. G.G. Lorentz, M. von Golitschek and Yu. Makovoz, Constructive approximation: Advanced problems, vol. 304. Springer Grundlehren, Berlin (1996). [Google Scholar]
  21. Y. Maday, A.T. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437–446. [CrossRef] [MathSciNet] [Google Scholar]
  22. T. Manteuffel, S. McCormick, J. Ruge and J.G. Schmidt, First-order system ℒℒ (FOSLL) for general scalar elliptic problems in the plane. SIAM J. Numer. Anal. 43 (2005) 2098–2120. [CrossRef] [MathSciNet] [Google Scholar]
  23. N.-C. Nguyen, G. Rozza and A.T. Patera, Reduced basis approximation for the time-dependent viscous Burgers’ equation. Calcolo 46 (2009) 157–185. [CrossRef] [MathSciNet] [Google Scholar]
  24. T. Patera and K. Urban, An improved error bound for reduced basis approximation of linear parabolic problems, submitted to Mathematics of Computation (in press 2013). [Google Scholar]
  25. A.T. Patera and G. Rozza, Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations, Version 1.0, Copyright MIT 2006–2007, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering. [Google Scholar]
  26. H.-J. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, in vol. 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd Edition (2008). [Google Scholar]
  27. G. Rozza and D.B.P. Huynh and A. Manzoni, Reduced basis approximation and a posteriori error estiamtion for Stokes flows in parametrized geometries: roles of the inf-sup stability constants, Numer. Math. DOI: 10.1007/s00211-013-0534-8. [Google Scholar]
  28. 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. [Google Scholar]
  29. G. Rozza and K. Veroy, On the stability of reduced basis techniques for Stokes equations in parametrized domains. Comput. Methods Appl. Mechanics Engrg. 196 (2007) 1244–1260. [Google Scholar]
  30. G. Sangalli, A uniform analysis of non-symmetric and coercive linear operators. SIAM J. Math. Anal. 36 (2005) 2033–2048. [CrossRef] [MathSciNet] [Google Scholar]
  31. M. Schlottbom, On Forward and Inverse Models in Optical Tomography, Ph.D. Thesis. RWTH Aachen (2011). [Google Scholar]
  32. S. Sen, K. Veroy, D.B.P. Huynh, S. Deparis, N.C. Nguyn 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]
  33. R. Verfürth, Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766–1782. [CrossRef] [MathSciNet] [Google Scholar]
  34. G. Welper, Infinite dimensional stabilization of convection-dominated problems, Ph.D. Thesis. RWTH Aachen (2012). [Google Scholar]
  35. J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo and V. Calo, A class of discontinuous Petrov−Galerkin methods. Part IV: Wave propagation. J. Comput. Phys. 230 (2011) 2406–2432. [CrossRef] [Google Scholar]

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