A. Abdulle, W. E. , and T. Li, Effectiveness of implicit methods for stiff stochastic differential equations, Commun. Comput. Phys, vol.3, issue.2, pp.295-307, 2008.

L. Arnold, Random Dynamical Systems, 1998.

A. N. Borodin and P. Salminen, Handbook of Brownian motion: Facts and formulae, 2002.

C. Brehier, Approximation of the invariant measure via a Euler scheme for stochastic PDEs driven by space-time white noise

C. Brehier, Analysis of a HMM discretization scheme for SPDEs

P. Brémaud, Markov chains Gibbs fields, Monte Carlo simulation, and queues, Texts in Applied Mathematics, 1999.

H. Brézis, Functional analysisAnalyse fonctionnelle, Théorie et applications.). Collection Mathématiques Appliquées pour la Maîtrise. Paris: Masson. 248 p, 1994.

S. Cerrai, Second order PDE's in finite and infinite dimension, Lecture Notes in Mathematics, vol.1762, 2001.
DOI : 10.1007/b80743

S. Cerrai, A Khasminskii type averaging principle for stochastic reaction???diffusion equations, The Annals of Applied Probability, vol.19, issue.3, pp.899-948, 2009.
DOI : 10.1214/08-AAP560

S. Cerrai, Normal deviations from the averaged motion for some reaction???diffusion equations with fast oscillating perturbation, Journal de Math??matiques Pures et Appliqu??es, vol.91, issue.6, pp.614-647, 2009.
DOI : 10.1016/j.matpur.2009.04.007

S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction???diffusion equations, Probability Theory and Related Fields, vol.17, issue.1-2, pp.137-177, 2009.
DOI : 10.1007/s00440-008-0144-z

M. Crouzeix and V. Thomée, On the discretization in time of semilinear parabolic equations with nonsmooth initial data, Mathematics of Computation, vol.49, issue.180, pp.359-377, 1987.
DOI : 10.1090/S0025-5718-1987-0906176-3

G. Da-prato and J. Zabczyk, Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and Its Applications . 44. Cambridge etc, 1992.

G. Da-prato and J. Zabczyk, Ergodicity for infinite dimensional systems, London Mathematical Society Lecture Note Series, vol.229, issue.339, 1996.
DOI : 10.1017/CBO9780511662829

A. M. Davie and J. G. Gaines, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Mathematics of Computation, vol.70, issue.233, pp.121-134, 2001.
DOI : 10.1090/S0025-5718-00-01224-2

A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Mathematics of Computation, vol.80, issue.273, pp.89-117, 2011.
DOI : 10.1090/S0025-5718-2010-02395-6

A. Debussche, Y. Hu, and G. Tessitore, Ergodic BSDEs under weak dissipative assumptions, Stochastic Processes and their Applications, pp.407-426, 2011.
DOI : 10.1016/j.spa.2010.11.009

URL : https://hal.archives-ouvertes.fr/hal-00472146

A. Debussche and J. Printems, Weak order for the discretization of the stochastic heat equation, Mathematics of Computation, vol.78, issue.266, pp.845-863, 2009.
DOI : 10.1090/S0025-5718-08-02184-4

URL : https://hal.archives-ouvertes.fr/hal-00183249

W. Doeblin, Expose de la théorie des chaînes simples constantes de Markoff à un nombre fini d'etats, Rev. Math. Union Interbalkan, vol.2, pp.77-105, 1938.
DOI : 10.24033/asens.883

W. E. and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci, vol.1, issue.1, pp.87-132, 2003.

W. E. , B. Engquist, X. Li, W. Ren, and E. Vanden-eijnden, Heterogeneous multiscale methods: a review, Commun. Comput. Phys, vol.2, issue.3, pp.367-450, 2007.

W. E. , D. Liu, and E. Vanden-eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math, vol.58, issue.11, pp.1544-1585, 2005.

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol.19, 2010.

E. Faou, Analysis of splitting methods for reaction-diffusion problems using stochastic calculus, Mathematics of Computation, vol.78, issue.267, pp.1467-1483, 2009.
DOI : 10.1090/S0025-5718-08-02185-6

URL : https://hal.archives-ouvertes.fr/hal-00777649

R. Ferretti, A Technique for High-Order Treatment of Diffusion Terms in Semi-Lagrangian Schemes, Communications in Computational Physics, 2000.
DOI : 10.4208/cicp.070709.011209a

J. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave propagation and time reversal in randomly layered media. Stochastic Modelling and Applied Probability 56, p.612, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00172124

M. I. Freidlin and A. D. , Random perturbations of dynamical systems. Transl. from the Russian by Joseph Szuecs, 1998.

H. Fu and J. Liu, Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, Journal of Mathematical Analysis and Applications, vol.384, issue.1, pp.70-86, 2011.
DOI : 10.1016/j.jmaa.2011.02.076

E. Gobet, Euler schemes and half-space approximation for the simulation of diffusion in a domain, ESAIM: Probability and Statistics, vol.5, pp.261-297, 2001.
DOI : 10.1051/ps:2001112

J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.44-47, pp.44-47, 2006.
DOI : 10.1016/j.cma.2005.10.010

I. Gyongy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I. Potential Anal, pp.1-25, 1998.

I. Gyongy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise, Potential Analysis, vol.11, issue.1, pp.1-37, 1999.
DOI : 10.1023/A:1008699504438

I. Gyongy and D. Nualart, Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise. Potential Anal, pp.725-757, 1997.

R. Z. Khasminskii, On an averaging principle for Itô stochastic differential equations, Kibernetica, issue.4, pp.260-279, 1968.

R. Z. Khasminskii and G. Yin, Limit behavior of two-time-scale diffusions revisited, Journal of Differential Equations, vol.212, issue.1, pp.85-113, 2005.
DOI : 10.1016/j.jde.2004.08.013

E. Hausenblas, Approximation for semilinear stochastic evolution equations. Potential Anal, pp.141-186, 2003.

J. C. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Processes and their Applications, vol.101, issue.2, pp.185-232, 2002.
DOI : 10.1016/S0304-4149(02)00150-3

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

I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg et al., Equation-free, coarsegrained multiscale computation: enabling microscopic simulators to perform system-level tasks, Communications in Mathematical Sciences, vol.1, issue.4, pp.715-762, 2003.

P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics, vol.23, 1992.

S. Kuksin and A. Shirikyan, A Coupling Approach??to Randomly Forced Nonlinear PDE's. I, Communications in Mathematical Physics, vol.221, issue.2, pp.351-366, 2001.
DOI : 10.1007/s002200100479

M. Roux, Semidiscretization in Time for Parabolic Problems, Mathematics of Computation, vol.33, issue.147, pp.919-931, 1979.
DOI : 10.2307/2006068

F. Legoll, T. Lelievre, and G. Samaey, A Micro-Macro Parareal Algorithm: Application to Singularly Perturbed Ordinary Differential Equations, SIAM Journal on Scientific Computing, vol.35, issue.4
DOI : 10.1137/120872681

URL : https://hal.archives-ouvertes.fr/hal-00691939

D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Communications in Mathematical Sciences, vol.8, issue.4, pp.999-1020, 2010.
DOI : 10.4310/CMS.2010.v8.n4.a11

R. Mannella, Absorbing boundaries and optimal stopping in a stochastic differential equation, Physics Letters A, vol.254, issue.5, pp.257-262, 1999.
DOI : 10.1016/S0375-9601(99)00117-6

J. C. Mattingly, Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics, Communications in Mathematical Physics, vol.230, issue.3, pp.421-462, 2002.
DOI : 10.1007/s00220-002-0688-1

J. C. Mattingly, A. M. Stuart, and M. V. Tretyakov, Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations, SIAM Journal on Numerical Analysis, vol.48, issue.2, pp.552-577, 2010.
DOI : 10.1137/090770527

S. Meyn and R. L. Tweedie, Markov chains and stochastic stability, 2009.

G. N. Milstein, Numerical integration of stochastic differential equations. Transl. from the Russian Mathematics and its Applications (Dordrecht). 313, 1994.

G. N. Milstein and M. V. Tretyakov, Stochastic numerics for mathematical physics. Scientific Computation, 2004.
DOI : 10.1007/978-3-662-10063-9

D. Nualart, The Malliavin calculus and related topics, 2006.
DOI : 10.1007/978-1-4757-2437-0

G. A. Pavliotis and A. M. Stuart, Multiscale methods. Averaging and homogenization, Texts in Applied Mathematics 53, 2008.

J. Printems, On the discretization in time of parabolic stochastic partial differential equations, Monte Carlo Methods Appl, vol.7, issue.3-4, pp.359-368, 2001.

M. Sanz-solé, Malliavin calculus with applications to stochastic partial differential equations, Fundamental Sciences: Mathematics . Boca Raton, vol.162, 2005.
DOI : 10.1201/9781439818947

J. Shen, Hopf bifurcation of the unsteady regularized driven cavity flow, Journal of Computational Physics, vol.95, issue.1, pp.228-245, 1991.
DOI : 10.1016/0021-9991(91)90261-I

D. Talay, Discrétisation d'une équation différentielle stochastique et calcul approché d'espérances de fonctionnelles de la solution. (Discretization of a stochastic differential equation and computation of expectations of functions of the solution), 1986.
DOI : 10.1051/m2an/1986200101411

URL : http://archive.numdam.org/article/M2AN_1986__20_1_141_0.pdf

D. Talay, Second-order discretization schemes of stochastic differential systems for the computation of the invariant law, Stochastics and Stochastic Reports, vol.20, issue.1, pp.13-36, 1990.
DOI : 10.1080/17442509008833606

URL : https://hal.archives-ouvertes.fr/inria-00075799

D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Analysis and Applications, vol.20, issue.4, pp.483-509, 1990.
DOI : 10.1080/07362999008809220

URL : https://hal.archives-ouvertes.fr/inria-00075490

R. Temam, Navier Stokes Equations: Theory and Numerical Analysis, Journal of Applied Mechanics, vol.45, issue.2, 2001.
DOI : 10.1115/1.3424338

E. Vanden-eijnden, Numerical Techniques for Multi-Scale Dynamical Systems with Stochastic Effects, Communications in Mathematical Sciences, vol.1, issue.2, pp.385-391, 2003.
DOI : 10.4310/CMS.2003.v1.n2.a11

J. B. Walsh, Finite element methods for parabolic stochastic PDE's. Potential Anal, pp.1-43, 2005.
DOI : 10.1007/s11118-004-2950-y