. Fig, 8 ? From top to bottom, Maximal H 1 error for the RB approximations w N,M (x,·) (left) and L ? error for the RB approximations s N,M (x) (right) obtained with collateral reduced bases of different sizes M using with the usual MP interpolation (top), L 2 projection (middle) and the usual MP interpolation after regularizing the discontinuities with convolutions (bottom)

G. Allaire and R. Brizzi, A Multiscale Finite Element Method for Numerical Homogenization, Multiscale Modeling & Simulation, vol.4, issue.3, pp.790-812, 2005.
DOI : 10.1137/040611239

]. G. All92 and . Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal, vol.23, issue.6, pp.1482-1518, 1992.

J. [. Ainsworth and . Oden, A posteriori error estimation in finite element analysis, Computer Methods in Applied Mechanics and Engineering, vol.142, issue.1-2, 2000.
DOI : 10.1016/S0045-7825(96)01107-3

O. [. Achdou, . N. Pironneauaq92-]-d, J. Arnold, and . Qin, Computational Methods for Option Pricing Quadratic velocity/linear pressure Stokes elements, Society for Industrial and Applied Mathematics Advances in Computer Methods for Partial Differential Equations , volume VII, pp.28-34, 1992.

]. B. Aro04 and . Arouna, Robbins-monroe algorithms and variance reduction in finance, The Journal of Computational Finance, vol.7, issue.2, pp.35-62, 2004.

P. [. Almroth, F. A. Stern, and . Brogan, Automatic choice of global shape functions in structural analysis, AIAA Journal, vol.16, issue.5, pp.525-528, 1978.
DOI : 10.2514/3.7539

]. J. Bb09a, S. Barrett, and . Boyaval, Existence and approximation of a (regularized) FENE-P model, 2009.

]. J. Bb09b, S. Barrett, and . Boyaval, Existence and approximation of a (regularized) Oldroyd-B model. (preprint submitted for publication http, 2009.

]. S. Bbm-+-09, C. Boyaval, Y. Le-bris, N. C. Maday, A. T. Nguyen et al., A reduced basis approach for variational problems with stochastic parameters : Application to heat conduction with variable robin coefficient, Computer Methods in Applied Mechanics and Engineering, vol.198, pp.41-443187, 2009.

]. R. Bcah87a, C. F. Bird, R. C. Curtiss, O. Armstrong, and . Hassager, Dynamics of polymeric liquids, volume 1 : Fluid Mechanics, 1987.

]. R. Bcah87b, C. F. Bird, R. C. Curtiss, O. Armstrong, and . Hassager, Kinetic Theory, Dynamics of Polymeric Liquids, vol.2, 1987.

P. [. Bonito, M. Clément, and . Picasso, Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows, ESAIM: Mathematical Modelling and Numerical Analysis, vol.40, issue.4, pp.785-814, 2006.
DOI : 10.1051/m2an:2006030

B. [. Beris and . Edwards, Thermodynamics of flowing systems with internal microstructure, 1994.

]. W. Bec89 and . Beckner, A generalized poincaré inequality for gaussian measures, Proc. Amer, pp.397-400, 1989.

M. [. Brezzi, M. Fortin, L. P. Behr, T. E. Franca, and . Tezduyar, Mixed and Hybrid Finite Element Methods Stabilized finite element methods for the velocitypressure-stress formulation of incompressible flows, Computer Methods in Applied Mechanics and Engineering, vol.104, pp.31-48, 1991.

M. [. Burkardt, H. C. Gunzburger, and . Lee, POD and CVT-based reduced-order modeling of Navier???Stokes flows, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.1-3, pp.337-355, 2006.
DOI : 10.1016/j.cma.2006.04.004

A. N. Brooks and T. J. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, vol.32, issue.1-3, pp.199-259, 1982.
DOI : 10.1016/0045-7825(82)90071-8

J. [. Barnes, K. F. Hutton, and . Walters, An introduction to rheology, 1989.

J. [. Brezzi, L. D. Douglas, and . Marini, Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik, vol.36, issue.2, pp.217-235, 1985.
DOI : 10.1007/BF01389710

J. [. Brezzi, L. D. Douglas, and . Marini, Recent results on mixed finite element methods for second order elliptic problems, Vistas in Applied Mathematics : Numerical Analysis, Atmospheric Sciences, Immunology, pp.25-43, 1986.

[. Bris and P. Lions, Renormalized solutions of some transport equations with partially W 1,1 velocities and applications, Annali di Matematica pura ed applicata, pp.97-130, 2004.
DOI : 10.1007/s10231-003-0082-4

]. S. Bl09a, T. Boyaval, and . Lelì-evre, A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm, Accepted for publication in Communication in Mathematical Sciences, pp.906-3600, 2009.

]. C. Bl09b, T. Bris, and . Lelì-evre, Multiscale modelling of complex fluids : A mathematical initiation, Multiscale Modeling and Simulation in Science Proceedings of the Summer School in, pp.49-138, 2009.

C. [. Blanc, P. L. Bris, and . Lions, Une variante de la th??orie de l'homog??n??isation stochastique des op??rateurs elliptiques, Comptes Rendus Mathematique, vol.343, issue.11-12, pp.11-12717, 2006.
DOI : 10.1016/j.crma.2006.09.034

T. [. Boyaval, C. Lelì, and . Mangoubi, Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM: Mathematical Modelling and Numerical Analysis, vol.43, issue.3, pp.523-561, 2009.
DOI : 10.1051/m2an/2009008

URL : https://hal.archives-ouvertes.fr/inria-00204620

J. [. Bensoussan, G. Lions, . [. Papanicolaou, A. Baranger, and . Machmoum, Asymptotic analysis for periodic structures, volume 5 of Studies in Mathematics and its applications Existence of approximate solutions and error bounds for viscoelastic fluid flow : Characteristics method, Comput. Methods Appl. Mech. Engrg, vol.148, pp.39-52, 1978.

M. Barrault, N. C. Nguyen, Y. Maday, and A. T. Patera, An ???empirical interpolation??? method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, vol.339, issue.9, pp.667-672, 2004.
DOI : 10.1016/j.crma.2004.08.006

URL : https://hal.archives-ouvertes.fr/hal-00021702

F. [. Babu?ka, R. Nobile, and . Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Journal on Numerical Analysis, vol.45, issue.3, pp.1005-1034, 2007.
DOI : 10.1137/050645142

]. S. Boy08 and . Boyaval, Reduced-basis approach for homogenization beyond the periodic setting, SIAM Multiscale Modeling & Simulation, vol.7, issue.1, pp.466-494, 2008.

]. S. Boy09 and . Boyaval, Mathematical modeling and simulation for material science, 2009.

J. [. Brezzi and . Pitkäranta, On the Stabilization of Finite Element Approximations of the Stokes Equations, Efficient Solution of Elliptic System Notes on Numerical Fluid Mechanics, pp.11-19, 1984.
DOI : 10.1007/978-3-663-14169-3_2

M. [. Bonvin and . Picasso, Variance reduction methods for CONNFFESSIT-like simulations, Journal of Non-Newtonian Fluid Mechanics, vol.84, issue.2-3, pp.191-215, 1999.
DOI : 10.1016/S0377-0257(98)00179-7

A. [. Bourgeat and . Piatnitski, Approximations of effective coefficients in stochastic homogenization, Annales de l?Institut Henri Poincare (B) Probability and Statistics, vol.40, issue.2, pp.153-165, 2004.
DOI : 10.1016/j.anihpb.2003.07.003

M. [. Bonito, M. Picasso, and . Laso, Numerical simulation of 3D viscoelastic flows with free surfaces, Journal of Computational Physics, vol.215, issue.2, pp.691-716, 2006.
DOI : 10.1016/j.jcp.2005.11.013

M. [. Bajaj, J. R. Pasquali, and . Prakash, Coil-stretch transition and the breakdown of computations for viscoelastic fluid flow around a confined cylinder, Journal of Rheology, vol.52, issue.1, pp.197-223, 2008.
DOI : 10.1122/1.2807444

J. Bonvin, M. Picasso, and R. Stenberg, GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.29-30, pp.3893-3914, 2001.
DOI : 10.1016/S0045-7825(00)00307-8

R. [. Becker and . Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica, vol.10, pp.1-225, 2001.
DOI : 10.1017/S0962492901000010

]. M. Bri94 and . Briane, Homogenization of a non periodic material Baranger and D. Sandri. Finite element approximation of viscoelastic fluid flow, J. Math. Pures Appl. Numer. Math, vol.73, issue.63, pp.47-6613, 1992.

]. J. Bs92b, D. Baranger, and . Sandri, A formulation of stokes's problem and the linear elasticity equations suggested by the oldroyd model for viscoelastic flow. RAIRO -Modélisation mathématique et analyse numérique, pp.331-345, 1992.

E. [. Barrett and . Süli, Existence of Global Weak Solutions to Some Regularized Kinetic Models for Dilute Polymers, Multiscale Modeling & Simulation, vol.6, issue.2, pp.506-546, 2007.
DOI : 10.1137/060666810

J. W. Barrett and E. Süli, EXISTENCE OF GLOBAL WEAK SOLUTIONS TO DUMBBELL MODELS FOR DILUTE POLYMERS WITH MICROSCOPIC CUT-OFF, Mathematical Models and Methods in Applied Sciences, vol.18, issue.06, pp.935-971, 2008.
DOI : 10.1142/S0218202508002917

E. [. Barrett and . Süli, Finite element approximation of kinetic dilute polymer models with microscopic cut-off, ESAIM: Mathematical Modelling and Numerical Analysis, vol.45, issue.1, 2009.
DOI : 10.1051/m2an/2010030

C. [. Barrett, E. Schwab, and . Süli, EXISTENCE OF GLOBAL WEAK SOLUTIONS FOR SOME POLYMERIC FLOW MODELS, Mathematical Models and Methods in Applied Sciences, vol.15, issue.06, pp.939-983, 2005.
DOI : 10.1142/S0218202505000625

R. [. Babu?ka, G. Tempone, and . Zouraris, Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.12-16, pp.1251-1294, 2005.
DOI : 10.1016/j.cma.2004.02.026

A. [. Crochet, K. Davies, and . Walters, Numerical Simulation of Non-Newtonian Flow, Journal of Applied Mechanics, vol.52, issue.1, 1984.
DOI : 10.1115/1.3169019

]. D. Cgrdb04, M. A. Chung, J. J. Gutiérrez, R. Remmers, and . De-borst, Stochastic finite element modelling of fibre-metal laminates, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 2004.

J. [. Chen, Y. Hesthaven, J. Maday, and . Rodríguez, A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations, Comptes Rendus Mathematique, vol.346, issue.23-24, 2008.
DOI : 10.1016/j.crma.2008.10.012

URL : https://hal.archives-ouvertes.fr/hal-00873076

. Ph, A. Ciarlet-ere, and . Lozinski, The Finite Element Method for Elliptic Problems. North-Holland Simulation of dilute polymer solutions using a Fokker-Planck equation, Computers and Fluids, vol.33, pp.687-696, 1978.

R. [. Chauvì-ere and . Owens, A new spectral element method for the reliable computation of viscoelastic flow, CMAME, vol.190, issue.31, pp.3999-4018, 2001.

]. R. Cod98 and . Codina, Comparison of some finite element methods for solving the diffusion-convection-reaction equation, Comp. Meth. Appl. Mech. Engrg, vol.156, pp.185-210, 1998.

P. [. Crouzeix and . Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numer, vol.3, pp.33-75, 1973.

I. [. Deb, J. T. Babu?ka, S. F. Oden, and . Edwards, Solution of stochastic partial differential equations using Galerkin finite element techniques, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.48, pp.6359-6372, 1998.
DOI : 10.1016/S0045-7825(01)00237-7

]. S. Dep08 and . Deparis, Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach, SIAM Journal of Numerical Analysis, vol.46, issue.4, pp.2039-2067, 2008.

]. R. Dev93 and . Devore, Constructive approximation, Acta Numerica, vol.7, pp.51-150, 1993.

R. [. Doostan, J. Ghanem, and . Red-horse, Stochastic model reduction for chaos representations, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.37-40, pp.37-403951, 2007.
DOI : 10.1016/j.cma.2006.10.047

]. B. Debusschere, H. N. Najm, P. P. Pebay, O. M. Knio, R. G. Ghanem et al., Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes, SIAM Journal on Scientific Computing, vol.26, issue.2, pp.698-719, 2004.
DOI : 10.1137/S1064827503427741

W. E. , B. Engquist, X. Li, W. Ren, and E. Vanden-eijnden, The heterogeneous multiscale method : A review, Commun. Comput. Phys, vol.2, pp.367-450, 2007.

J. [. Ern and . Guermond, Theory and Practice of Finite Elements, 2004.
DOI : 10.1007/978-1-4757-4355-5

T. [. Li and P. Zhang, Well-posedness for the dumbbell model of polymeric fluids, Comm. Math. Phys, vol.248, pp.409-427, 2004.

]. E. Emm08 and . Emmrich, Convergence of a time discretization for a class of non-newtonian fluid flow, Commun. Math. Sci, vol.6, issue.4, pp.827-843, 2008.

A. [. Eftang, E. M. Patera, and . Rønquist, Personal communication. HP-type reduced basis method for parametrized partial differential equations

E. Fernández-cara, F. Guilì, and R. R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the oldroyd kind Handbook of numerical analysis VIII : Solution of equations in R n (Part 4) Techniques of scientific computing (Part 4) Numerical methods of fluids (Part 2), pp.543-661, 2002.

A. [. Fortin and . Fortin, A new approach for the FEM simulation of viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, vol.32, issue.3, pp.295-310, 1989.
DOI : 10.1016/0377-0257(89)85012-8

R. [. Fortin, R. Guénette, and . Pierre, Numerical analysis of the modified EVSS method, Computer Methods in Applied Mechanics and Engineering, vol.143, issue.1-2, pp.79-95, 1997.
DOI : 10.1016/S0045-7825(96)01145-0

O. [. Fattal, G. Hald, R. Katriel, and . Kupferman, Global stability of equilibrium manifolds, and ???peaking??? behavior in quadratic differential systems related to viscoelastic models, Journal of Non-Newtonian Fluid Mechanics, vol.144, issue.1, pp.30-41, 2007.
DOI : 10.1016/j.jnnfm.2007.03.003

R. [. Fattal and . Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, Journal of Non-Newtonian Fluid Mechanics, vol.123, issue.2-3, pp.2-3281, 2004.
DOI : 10.1016/j.jnnfm.2004.08.008

R. [. Fattal and . Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation, Journal of Non-Newtonian Fluid Mechanics, vol.126, issue.1, pp.23-27, 2005.
DOI : 10.1016/j.jnnfm.2004.12.003

W. [. Fink and . Rheinboldt, On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element Approximations, ZAMM - Zeitschrift f??r Angewandte Mathematik und Mechanik, vol.18, issue.4, pp.21-28, 1983.
DOI : 10.1002/zamm.19830630105

]. A. Fri75 and . Friedman, Stochastic differential equations and applications, 1975.

R. [. Franca and . Stenberg, Error Analysis of Galerkin Least Squares Methods for the Elasticity Equations, SIAM Journal on Numerical Analysis, vol.28, issue.6, pp.1680-1697, 1991.
DOI : 10.1137/0728084

URL : https://hal.archives-ouvertes.fr/inria-00075505

C. [. Frauenfelder, R. A. Schwab, and . Todor, Finite elements for elliptic problems with stochastic coefficients, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.2-5, pp.205-228, 2005.
DOI : 10.1016/j.cma.2004.04.008

S. [. Ghanem and . Dham, Stochastic finite element analysis for multiphase flow in heterogeneous porous media, Tranp. Porous Media, vol.32, issue.239, 1998.

]. A. Glo06a and . Gloria, An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies, Multiscale Modeling and Simulation, vol.5, issue.3, pp.996-1043, 2006.

]. A. Glo06b and . Gloria, A direct approach to numerical homogenization in nonlinear elasticity, NHM, vol.1, issue.1, pp.109-141, 2006.

Y. [. Grepl, N. C. Maday, A. T. Nguyen, and . Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.41, issue.3, pp.575-605, 2007.
DOI : 10.1051/m2an:2007031

URL : https://hal.archives-ouvertes.fr/hal-00112154

A. Gloria and F. Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations. (preprint submitted for publication http, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00383953

A. [. Grepl and . Patera, error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.39, issue.1, pp.157-181, 2005.
DOI : 10.1051/m2an:2005006

]. M. Gre05 and . Grepl, Reduced-Basis Approximations and A Posteriori Error Estimation for Parabolic Partial Differential Equations, 2005.

M. Grmela, Letter to the editor : Comment on " thermodynamics of viscoelastic fluids : The temperature equation " [j. rheol. [bold 42, Journal of Rheology, vol.42, issue.6, pp.999-10191565, 1998.

C. Guillopé and J. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Analysis: Theory, Methods & Applications, vol.15, issue.9, pp.849-869, 1990.
DOI : 10.1016/0362-546X(90)90097-Z

P. [. Ghanem and . Spanos, Stochastic Finite Elements : A Spectral Approach, 1991.
DOI : 10.1007/978-1-4612-3094-6

G. [. Ghanem, A. Saad, and . Doostan, Efficient solution of stochastic systems: Application to the embankment dam problem, Structural Safety, vol.29, issue.3, pp.138-251, 2007.
DOI : 10.1016/j.strusafe.2006.07.015

]. J. Gue99 and . Guermond, Stabilization of galerkin approximations of transport equations by subgrid modeling, Math. Model. Numer. Anal, vol.33, issue.6, pp.1293-1316, 1999.

L. [. Hughes and . Franca, A new finite element formulation for computational fluid dynamics: VII. The stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces, Computer Methods in Applied Mechanics and Engineering, vol.65, issue.1, pp.85-96, 1987.
DOI : 10.1016/0045-7825(87)90184-8

R. [. Hulsen, R. Fattal, and . Kupferman, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: Stabilized simulations using matrix logarithms, Journal of Non-Newtonian Fluid Mechanics, vol.127, issue.1, pp.27-39, 2005.
DOI : 10.1016/j.jnnfm.2005.01.002

D. B. Huynh, D. J. Knezevic, Y. Chen, J. S. Hesthaven, and A. T. Patera, A natural-norm Successive Constraint Method for inf-sup lower bounds, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.29-32, 2009.
DOI : 10.1016/j.cma.2010.02.011

T. [. Hu and . Lelì-evre, New entropy estimates for the Oldroyd-B model and related models, Communications in Mathematical Sciences, vol.5, issue.4, pp.906-916, 2007.
DOI : 10.4310/CMS.2007.v5.n4.a9

URL : https://hal.archives-ouvertes.fr/hal-00135377

M. [. Haasdonk and . Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.42, issue.2, pp.277-302, 2008.
DOI : 10.1051/m2an:2008001

]. J. How09 and . Howell, Computation of viscoelastic fluid flows using continuation methods, J. Comput. Appl. Math, vol.225, issue.1, pp.187-201, 2009.

]. J. Hp07a, T. Hao, and . Pan, Simulation for high Weissenberg number viscoelastic flow by a finite element method, Applied Mathematics Letters, vol.20, pp.988-993, 2007.

]. D. Hp07b, A. T. Huynh, and . Patera, Reduced-basis approximation and a posteriori error estimation for stress intensity factors, Int. J. Num. Meth. Eng, vol.72, issue.10, pp.1219-1259, 2007.

R. [. Heywood and . Rannacher, Finite Element Approximation of the Nonstationary Navier???Stokes Problem. I. Regularity of Solutions and Second-Order Error Estimates for Spatial Discretization, SIAM Journal on Numerical Analysis, vol.19, issue.2, pp.275-311, 1982.
DOI : 10.1137/0719018

]. D. Hrsp07a, G. Huynh, S. Rozza, A. T. Sen, and . Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants, C. R. Acad. Sci. Paris, Analyse Numérique, vol.345, issue.8, pp.473-478, 2007.

]. D. Hrsp07b, G. Huynh, S. Rozza, A. T. Sen, and . Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants, C. R. Math. Acad. Sci. Paris, vol.345, pp.473-478, 2007.

]. M. Hul90 and . Hulsen, A sufficient condition for a positive definite configuration tensor in differential models, J. Non-Newtonian Fluid Mech, vol.38, pp.93-100, 1990.

X. [. Hou and . Wu, A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media, Journal of Computational Physics, vol.134, issue.1, pp.169-189, 1997.
DOI : 10.1006/jcph.1997.5682

J. N. Israelachvili, Intermolecular and Surface Forces, Second Edition : With Applications to Colloidal and Biological Systems (Colloid Science), 1992.

R. [. Jardak and . Ghanem, Spectral stochastic homogenization of divergence-type PDEs, Computer Methods in Applied Mechanics and Engineering, vol.193, issue.6-8, pp.429-447, 2004.
DOI : 10.1016/j.cma.2003.05.001

S. [. Jikov, O. Kozlov, and . Oleinik, Homogenization of differential operators and integral functionals, 1994.
DOI : 10.1007/978-3-642-84659-5

C. [. Jourdain, T. Bris, and . Lelì-evre, On a variance reduction technique for micro???macro simulations of polymeric fluids, Journal of Non-Newtonian Fluid Mechanics, vol.122, issue.1-3, pp.91-106, 2004.
DOI : 10.1016/j.jnnfm.2003.09.006

C. [. Jourdain, T. Le-bris, F. Lelì, and . Otto, Long-time asymptotics of a multiscale model for polymeric fluid flows. Archive for Rational Mechanics and Analysis, pp.97-148, 2006.

B. Jourdain, T. Lelì, and C. L. Bris, Existence of solution for a micro???macro model of polymeric fluid: the FENE model, Journal of Functional Analysis, vol.209, issue.1, pp.162-193, 2004.
DOI : 10.1016/S0022-1236(03)00183-6

]. D. Jos90 and . Joseph, Fluid dynamics of viscoelastic liquids Applied Mat, Jou09] B. Jourdain. to appear, chapter Adaptive variance reduction techniques in finance. Radon Series Comp. Appl. Math 8. De Gruyter, 1990.

[. Jin and J. Zou, Inversion of Robin coefficient by a spectral stochastic finite element approach, Journal of Computational Physics, vol.227, issue.6, pp.3282-3306, 2008.
DOI : 10.1016/j.jcp.2007.11.042

]. K. Kar46 and . Karhunen, Zur spektraltheorie stochastischer prozesse, Annales Academiae Scientiarum Fennicae, vol.37, 1946.

]. R. Kei92 and . Keiller, Numerical instability of time-dependent flows, J. Non-Newtonian Fluid Mech, vol.43, pp.229-246, 1992.

]. R. Keu90 and . Keunings, Fundamentals of computer modeling for polymer processing, Simulation of viscoelastic fluid flow, pp.402-470, 1990.

]. R. Keu00 and . Keunings, A survey of computational rheology, Proc. 13th Int. Congr. on Rheology, pp.7-14, 2000.

A. [. Kwon and . Leonov, Stability constraints in the formulation of viscoelastic constitutive equations, Journal of Non-Newtonian Fluid Mechanics, vol.58, issue.1, pp.25-46, 1995.
DOI : 10.1016/0377-0257(94)01341-E

A. Keese and H. G. Matthies, Hierarchical parallelisation for the solution of stochastic finite element equations, Computers & Structures, vol.83, issue.14, pp.1033-1047, 2005.
DOI : 10.1016/j.compstruc.2004.11.014

C. [. Kupferman, E. Mangoubi, and . Titi, A Beale-Kato-Madja breakdown criterion for an Oldroyd-B fluid in the creeping flow regime, Communications in Mathematical Sciences, vol.6, issue.1, pp.235-256, 2008.
DOI : 10.4310/CMS.2008.v6.n1.a12

E. [. Kloeden, . J. Platen-[-kp09-]-d, A. T. Knezevic, and . Patera, Numerical Solution of Stochastic Differential Equations A certified reduced basis method for the fokker-planck equation of dilute polymeric fluids : Fene dumbbells in extensional flow, 2000.

S. [. Karatzas and . Shreve, Brownian Motion and Stochastic Calculus. SpringerVerlag Analysis of locally stabilized mixed finite element methods for the Stokes problem, Mathematics of Computation, issue.197, pp.581-591, 1991.

]. D. Ks09a, E. Knezevic, and . Süli, A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model. M2AN : Mathematical Modeling and Numerical Analysis, 2009.

]. D. Ks09b, E. Knezevic, and . Süli, Spectral Galerkin approximation of Fokker?Planck equations with unbounded drift, M2AN : Mathematical Modeling and Numerical Analysis, 2009.

S. [. Kunisch and . Volkwein, Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics, SIAM Journal on Numerical Analysis, vol.40, issue.2, pp.492-515, 2002.
DOI : 10.1137/S0036142900382612

C. L. Bris, Systèmes multi-´ echelles : Modélisation & simulation, Mathématiques & applications, vol.47, 2005.
DOI : 10.1007/3-540-37671-2

[. Bris, T. Lelievre, and Y. Maday, Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations. (preprint submitted for publication http, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00336911

C. [. Lozinski and . Chauvì-ere, A fast solver for Fokker???Planck equation applied to viscoelastic flows calculations: 2D FENE model, Journal of Computational Physics, vol.189, issue.2, pp.607-625, 2003.
DOI : 10.1016/S0021-9991(03)00248-1

]. T. Lel04 and . Lelì-evre, Probì emes mathématiques et numériques posés par la simulation d'´ ecoulement de fluides polymériques Available at http ://cermics.enpc.fr/ lelievre/rapports/these.pdf Analysis of simple constitutive equations for viscoelastic liquids. J. non-newton. fluid mech, Ecole Nationale des Ponts et Chaussées, pp.42323-350, 1992.

I. [. Lienhard, J. H. Lienhard, and V. , A Heat Transfer Textbook, 2002.

J. Lions, D. Lukkassen, P. Persson, and . Wall, Reiterated homogenization of monotone operators, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.330, issue.8, pp.675-680, 2000.
DOI : 10.1016/S0764-4442(00)00242-1

[. Lin, C. Liu, and P. W. Zhang, On hydrodynamics of viscoelastic fluids, Communications on Pure and Applied Mathematics, vol.39, issue.11, pp.1437-1471, 2005.
DOI : 10.1002/cpa.20074

C. [. Lei, Y. Liu, and . Zhou, Global solutions for incompressible viscoelastic fluids. Archive for Rational Mechanics and Analysis, pp.371-398, 2008.

H. and L. Meur, Existence locale de solutions deséquationsdeséquations d'un fluide viscoélastique avecfrontì ere libre, C. R. Acad. Sci. Paris Sér. I Math, vol.320, issue.1, pp.125-130, 1995.

[. Lions and N. Masmoudi, GLOBAL SOLUTIONS FOR SOME OLDROYD MODELS OF NON-NEWTONIAN FLOWS, Chinese Annals of Mathematics, vol.21, issue.02, pp.131-146, 2000.
DOI : 10.1142/S0252959900000170

[. Lions and N. Masmoudi, Global existence of weak solutions to some micro-macro models, Comptes Rendus Mathematique, vol.345, issue.1, pp.15-20, 2007.
DOI : 10.1016/j.crma.2007.05.011

URL : https://hal.archives-ouvertes.fr/hal-00667324

R. [. Lozinski and . Owens, An energy estimate for the Oldroyd B model: theory and applications, Journal of Non-Newtonian Fluid Mechanics, vol.112, issue.2-3, pp.161-176, 2003.
DOI : 10.1016/S0377-0257(03)00096-X

]. M. Lò-e78 and . Lò-eve, Probability Theory, volume I-II, 1978.

]. A. Loz03 and . Lozinski, Spectral methods for kinetic theory models of viscoelastic fluids, 2003.

P. [. Lesaint and . Raviart, Mathematical Aspects of Finite Elements in Partial Differential Equations, chapter On a Finite Element Method for Solving the Neutron Transport Equation, pp.89-123, 1974.

J. [. Lee and . Xu, New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.9-12, pp.1180-1206, 2006.
DOI : 10.1016/j.cma.2005.04.008

P. [. Li and . Zhang, Mathematical Analysis of Multi-Scale Models of Complex Fluids, Communications in Mathematical Sciences, vol.5, issue.1, pp.1-51, 2007.
DOI : 10.4310/CMS.2007.v5.n1.a1

H. [. Li, P. Zhang, and . Zhang, Local Existence for the Dumbbell Model of Polymeric Fluids, Communications in Partial Differential Equations, vol.22, issue.2, pp.903-923, 2004.
DOI : 10.1137/0521076

]. Y. Mad06 and . Maday, Reduced?basis method for the rapid and reliable solution of partial differential equations, Proceedings of International Conference of Mathematicians, 2006.

]. N. Mas08 and . Masmoudi, Well posedness of the FENE dumbbell model of polymeric flows, Comm. Pure Appl. Math, vol.61, pp.1685-1714, 2008.

[. Matache, I. Babuska, and C. Schwab, Generalized p-FEM in homogenization, Numerische Mathematik, vol.86, issue.2, pp.319-375, 2000.
DOI : 10.1007/PL00005409

M. [. Marchal and . Crochet, A new mixed finite element for calculating viscoelastic flow, Journal of Non-Newtonian Fluid Mechanics, vol.26, issue.1, pp.77-114, 1987.
DOI : 10.1016/0377-0257(87)85048-6

M. [. Mangoubi, R. Hulsen, and . Kupferman, Numerical stability of the method of Brownian configuration fields, Journal of Non-Newtonian Fluid Mechanics, vol.157, issue.3, pp.188-196, 2009.
DOI : 10.1016/j.jnnfm.2008.11.009

M. [. Mathelin, T. A. Hussaini, and . Zang, Stochastic approaches to uncertainty quantification in CFD simulations, Numerical Algorithms, vol.187, issue.2, pp.209-236, 2005.
DOI : 10.1007/BF02810624

H. G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.12-16, pp.1295-1331, 2005.
DOI : 10.1016/j.cma.2004.05.027

. Mmo-+-00-]-l, Y. Machiels, I. B. Maday, A. T. Oliveira, D. V. Patera et al., Output bounds for reducedbasis approximations of symmetric positive definite eigenvalue problems, C. R. Acad. Sci. Paris, Série I, vol.331, issue.2, pp.153-158, 2000.

L. Machiels, Y. Maday, and A. T. Patera, Output bounds for reduced-order approximations of elliptic partial differential equations Multi-element stochastic reduced basis methods, Comp. Meth. Appl. Mech. Engrg. Computer Methods in Applied Mechanics and Engineering, vol.190, issue.197, pp.26-273413, 2001.

Y. Maday, N. C. Nguyen, A. T. Patera, and G. Pau, A general multipurpose interpolation procedure: the magic points, C. ¨ Ottinger. Variance reduced simulations of stochastic differential equations, pp.383-404, 2009.
DOI : 10.3934/cpaa.2009.8.383

URL : https://hal.archives-ouvertes.fr/hal-00174797

A. [. Morton, E. Priestley, and . Süli, Convergence analysis of the Lagrange-Galerkin method with non-exact integration, J. Chem. Phys, vol.103, issue.224, pp.9506-9509625, 1988.

Y. Maday, A. T. Patera, and G. Turinici, A Priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations, Journal of Scientific Computing, vol.17, pp.1-4437, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00798389

Y. Maday, A. T. Patera, G. Turinicimt86-]-r, F. Mneimne, . N. Testard-[-mt06-]-g et al., Introduction a la théorie des groupes de Lie classiques. Hermann Practical variance reduction via regression for simulating diffusions Variance reduction for simulated diffusions Model reduction via the Karhunen-Loeve expansion part i : an exposition Stochastic reduced basis methods [Nou07] A. Nouy. A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations Generalized spectral decomposition method for solving stochastic finite element equations : Invariant subspace problem and dedicated algorithms Reduced basis technique for nonlinear analysis of structures, Nou08] A. NouyNP80] A. K. Noor and J. M. Peters, pp.289-2941780, 1980.

N. C. Nguyen, G. Rozza, D. B. Huynh, A. Patera-biegler, . Biros et al., Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers??? equation, NTW08] F. Nobile, R. Tempone, and C.G. Webster. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal, pp.157-1852309, 2008.
DOI : 10.1007/s10092-009-0005-x

N. C. Nguyen, K. Veroy, and A. T. Patera, Certified real-time solution of parametrized partial differential equations, Handbook of Materials Modeling, pp.1523-1558, 2005.

]. N. Nvp05b, K. Nguyen, A. T. Veroy, and . Patera, Certified real-time solution of parametrized partial differential equations, pp.1523-1558, 2005.

¨. [. Ottinger, . C. O05-]-h, ]. R. Ottingerop02, T. N. Owens, X. Starzewski et al., Beyond equilibrium thermodynamics Stochastic finite elements as a bridge between random material microstructure and global response Brownian configuration fields and variance reduced CONNFFESSIT, Stochastic Processes in Polymeric FluidsOks03] B. Oksendal. Stochastic Differential Equations. An Introduction with Applications, pp.35-49255, 1996.

M. F. Pellissetti, R. G. Ghanempir82, ]. A. Pironneaupor85-]-t, . T. Porschingpr07a-]-a, E. M. Patera et al., [Pin85] A. Pinkus. n-Widths in Approximation Theory On the transport-diffusion algorithm and its application to the Navier-Stokes equations Calibration of options on a reduced basis Estimation of the error in the reduced basis method solution of nonlinear equations Pagès and J. Printems. Functional quantization for numerics with an application to option pricing Reduced basis approximations and a posteriori error estimation for a Boltzmann model, Iterative solution of systems of linear equations arising in the context of stochastic finite elements, pp.31-39, 1982.

A. Prohl, Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system, ESAIM : M2AN, pp.1065-1087, 2008.
DOI : 10.1051/m2an:2008034

. Prv-+-02-]-c, D. Prud-'homme, K. Rovas, Y. Veroy, A. T. Maday et al., Reliable real-time solution of parametrized partial differential equations : Reduced-basis output bounds methods Numerical Models for Differential Problems, volume 2 of Modeling, Simulation and Applications Mathematical Analysis of Viscoelastic Flows Do we understand the physics in the constitutive equation ? Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations ? application to transport and continuum mechanics Non integrable extra stress tensor solution for a flow in a bounded domain of an Oldroyd fluid, CBMS-NSF Conference Series in Applied Mathematics. SIAMS ¨ 88] E. Süli. Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equationsSB95] R Sureshkumar and A N Beris. Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows. Journal of Non-Newtonian Fluid Mech, pp.70-8037, 1988.

B. Sudret, M. Berveiller, and M. Lemaire, A stochastic finite element procedure for moment and reliability analysis, Revue europ??enne de m??canique num??rique, vol.15, issue.7-8, pp.825-866, 2006.
DOI : 10.3166/remn.15.825-866

]. J. Sch06 and . Schieber, Generalized Brownian configuration field for Fokker?Planck equations including center-of-mass diffusion, J. Non-Newtonian Fluid Mech, vol.135, pp.179-181, 2006.

]. S. Sen08 and . Sen, Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems. Numerical Heat Transfer, Part B : Fundamentals, vol.54, issue.5, 2008.

]. G. Sf73a, G. J. Strang, and . Fix, An Analysis of the Finite Element Method, 1973.

]. W. Sf73b, G. J. Strang, and . Fix, An Analysis of the Finite Element Method, 1973.

R. [. Soize, ]. J. Ghanemsim87, . K. Simon-[-snk06-]-s, P. B. Sachdeva, A. J. Nair et al., Physical systems with random uncertainties : Chaos representations with arbitrary probability measure Compact sets in the space L p (0 Hybridization of stochastic reduced basis methods with polynomial chaos expansions Numerical simulation method for viscoelastic flows with free surfaces -fringe element generation method, SIAM Journal on Scientific Computing Ann. Math. Pura. Appl. Probabilistic Engineering Mechanics International Journal for Numerical Methods in Fluids, vol.26, issue.197, pp.395-41065, 1987.

. Ch, E. Schwab, R. Süli, and . Todor, Sparse finite element approximation of high-dimensional transport-dominated diffusion problems Karhunenlò eve approximation of random fields by generalized fast multipole methods, Math. Models Methods Appl. Sci. Journal of Computational Physics, vol.42, issue.2171, pp.777-820100, 2006.

M. [. Scott and . Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, ESAIM: Mathematical Modelling and Numerical Analysis, vol.19, issue.1, pp.111-143, 1985.
DOI : 10.1051/m2an/1985190101111

]. S. Svh-+-06, K. Sen, D. B. Veroy, S. Huynh, N. C. Deparis et al., Natural norm " a posteriori error estimators for reduced basis approximations, Journal of Computational Physics, vol.217, pp.37-62, 2006.

]. R. Tem66 and . Temam, Sur l'approximation deséquationsdeséquations de Navier-Stokes, C. R. Acad. Sc. Paris, Série A, vol.262, pp.219-221, 1966.

]. R. Tem84 and . Temam, Navier?Stokes Equations. Theory and Numerical Analysis, of Studies in Mathematics and its Applications, 1984.

]. B. Ts07a, M. Thomases, and . Shelley, Emergence of singular structures in Oldroyd-B fluids, Phys. Fluids, vol.1912, issue.12, pp.103103-103104, 2007.

]. R. Ts07b, C. Todor, and . Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA Journal of Numerical Analysis, vol.27, pp.232-261, 2007.

M. [. Venkiteswaran and . Junk, Quasi-Monte Carlo algorithms for diffusion equations in high dimensions, Mathematics and Computers in Simulation, vol.68, issue.1, pp.23-41, 2005.
DOI : 10.1016/j.matcom.2004.09.003

[. V. N08-]-v and . Temlyakov, Nonlinear methods of approximation, Foundations of Computational Mathematics, vol.3, pp.33-107, 2008.

A. [. Veroy and . Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basisa posteriori error bounds, International Journal for Numerical Methods in Fluids, vol.42, issue.8-9, pp.773-788, 2005.
DOI : 10.1002/fld.867

]. K. Vprp03a, C. Veroy, D. V. Prud-'homme, A. T. Rovas, and . Patera, A Posteriori error bounds for reducedbasis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003.

]. K. Vprp03b, C. Veroy, D. V. Prud-'homme, A. T. Rovas, and . Patera, A posteriori error bounds for reducedbasis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, 16th AIAA Computational Fluid Dynamics Conference, 2003.

D. [. Veroy, A. T. Rovas, and . Patera, A Posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : " convex inverse " bound conditioners. ESAIM : Control, Optimization and Calculus of Variations, pp.1007-1028, 2002.

P. Wapperom and M. A. Hulsen, Response to " comment on : 'thermodynamics of viscoelastic fluids : The temperature equation' " [j. rheol. [bold 42, Journal of Rheology, vol.42, issue.6, pp.1565-15671569, 1998.

]. P. Wh98b, M. A. Wapperom, and . Hulsen, Thermodynamics of viscoelastic fluids : the temperature equation, J. Rheol, vol.42, issue.5, pp.999-1019, 1998.

]. N. Wie38 and . Wiener, The homogeneous chaos, Am. J. Math, vol.60, pp.897-936, 1938.

G. [. Wan, . Karniadakis-[-wk09-]-x, G. E. Wan, and . Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations Error control in multi-element generalized polynomial chaos method for elliptic problems with random coefficients, Journal of Computational Physics Communications in Computational Physics, vol.209, issue.5, pp.617-6422, 2005.

R. [. Wapperom, V. Keunings, and . Legat, The backward-tracking Lagrangian particle method for transient viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, vol.91, issue.2-3, pp.273-295, 2000.
DOI : 10.1016/S0377-0257(99)00095-6

J. [. Xiu and . Hesthaven, High-Order Collocation Methods for Differential Equations with Random Inputs, SIAM Journal on Scientific Computing, vol.27, issue.3, pp.1118-1139, 2005.
DOI : 10.1137/040615201

G. [. Xiu and . Karniadakis, The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations, SIAM Journal on Scientific Computing, vol.24, issue.2, pp.619-644, 2002.
DOI : 10.1137/S1064827501387826

]. X. Xu07 and . Xu, A multiscale stochastic finite element method on elliptic problems involving uncertainties The reduced model multiscale method (r3m) for the non-linear homogenization of hyperelastic media at finite strains, Computer Methods in Applied Mechanics and Engineering Journal of Computational Physics, vol.196, issue.223, pp.25-282723, 2007.