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Attractors for the threedimensional incompressible NavierStokes equations with damping
Macroscopic discrete modelling of stochastic reactiondiffusion equations on a periodic domain
1.  School of Mathematical Sciences, University of Adelaide, South Australia 5005, Australia, Australia 
References:
[1] 
H. Attouch, "Variational Convergence for Functions and Operators," Pitman Publishing Limited, London, 1984. 
[2] 
W. E, X. Li and E. VandenEijnden, Some recent progress in multiscale modeling, in "Multiscale Modelling and Simulation," Lect. Notes Comput. Sci. Eng., 39, 321, Springer, Berlin, 2004. 
[3] 
P. Imkeller and A. Monahan, editors, "Stochastic Climate Dynamics," a Special Issue in the journal Stoch. and Dyna., 2, 2002. 
[4] 
D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: Model problems and algorithms, Nonlinearity, 17 (2004), R55R127. doi: 10.1088/09517715/17/6/R01. 
[5] 
H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys., 65 (1979), 97128. doi: 10.1007/BF01225144. 
[6] 
G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. 
[7] 
A. J. Roberts, A holistic finite difference approach models linear dynamics consistently, Mathematics of Computation, 72 (2003), 247262. Available from: http://www.ams.org/mcom/200372241/S0025571802014485. doi: 10.1090/S0025571802014485. 
[8] 
A. J. Roberts, A step towards holistic discretisation of stochastic partial differential equations, ANZIAM J., 45 (2003/04), C1C15. Available from: http://anziamj.austms.org.au/V45/CTAC2003/Robe. 
[9] 
A. J. Roberts, Resolving the multitude of microscale interactions accurately models stochastic partial differential equations, LMS J. Computation and Math., 9 (2006), 193221. Available from: http://www.lms.ac.uk/jcm/9/lms2005032. 
[10] 
A. J. Roberts, Subgrid and interelement interactions affect discretisations of stochastically forced diffusion, ANZIAM J., 48 (2006/07), C169C188. Available from: http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/36. 
[11] 
A. J. Roberts, Choose interelement coupling to preserve selfadjoint dynamics in multiscale modelling and computation, Applied Numerical Modelling, 60 (2010), 949973. Available from: http://www.sciencedirect.com/science/article/pii/S0168927410001145 
[12] 
Tony MacKenzie and A. J. Roberts, Holistic discretisation ensures fidelity to dynamics in two spatial dimensions, preprint, 2009. 
[13] 
J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 6596. doi: 10.1007/BF01762360. 
[14] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003. 
[15] 
S. Engblom, L. Ferm, A. Hellander and P. Lötstedt, Simulation of stochastic reaction diffusion processes on unstructured meshes, preprint, 2008. Available from: http://arXiv.org/abs/0804.3288. 
[16] 
V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," SpringerVerlag, Berlin, 1997. 
[17] 
W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007). 
[18] 
W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions, Comm. Math. Phys., 275 (2007), 163186. doi: 10.1007/s0022000703018. 
[19] 
W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reactiondiffusion equations, preprint, 2008. Available from: http://arxiv.org/abs/0812.1837. 
[20] 
W. Wang and A. J. Roberts, Average and deviation for slowfast stochastic partial differential equations, preprint, 2008. Available from: http://arxiv.org/abs/0904.1462. 
[21] 
W. Wang and A. J. Roberts, Macroscopic discrete modelling of stochastic reactiondiffusion equations, preprint, 2009. 
[22] 
H. Watanabe, Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients, Probab. Th. Rel. Fields, 77 (1988), 359378. doi: 10.1007/BF00319294. 
[23] 
E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics," IMA, 140, SpringerVerlag, New York, 2005. 
[24] 
Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 13631384. doi: 10.1137/040605278. 
[25] 
Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations, SIAM J. Numer. Anal., 40 (2002), 14211445. doi: 10.1137/S0036142901387956. 
show all references
References:
[1] 
H. Attouch, "Variational Convergence for Functions and Operators," Pitman Publishing Limited, London, 1984. 
[2] 
W. E, X. Li and E. VandenEijnden, Some recent progress in multiscale modeling, in "Multiscale Modelling and Simulation," Lect. Notes Comput. Sci. Eng., 39, 321, Springer, Berlin, 2004. 
[3] 
P. Imkeller and A. Monahan, editors, "Stochastic Climate Dynamics," a Special Issue in the journal Stoch. and Dyna., 2, 2002. 
[4] 
D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: Model problems and algorithms, Nonlinearity, 17 (2004), R55R127. doi: 10.1088/09517715/17/6/R01. 
[5] 
H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys., 65 (1979), 97128. doi: 10.1007/BF01225144. 
[6] 
G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. 
[7] 
A. J. Roberts, A holistic finite difference approach models linear dynamics consistently, Mathematics of Computation, 72 (2003), 247262. Available from: http://www.ams.org/mcom/200372241/S0025571802014485. doi: 10.1090/S0025571802014485. 
[8] 
A. J. Roberts, A step towards holistic discretisation of stochastic partial differential equations, ANZIAM J., 45 (2003/04), C1C15. Available from: http://anziamj.austms.org.au/V45/CTAC2003/Robe. 
[9] 
A. J. Roberts, Resolving the multitude of microscale interactions accurately models stochastic partial differential equations, LMS J. Computation and Math., 9 (2006), 193221. Available from: http://www.lms.ac.uk/jcm/9/lms2005032. 
[10] 
A. J. Roberts, Subgrid and interelement interactions affect discretisations of stochastically forced diffusion, ANZIAM J., 48 (2006/07), C169C188. Available from: http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/36. 
[11] 
A. J. Roberts, Choose interelement coupling to preserve selfadjoint dynamics in multiscale modelling and computation, Applied Numerical Modelling, 60 (2010), 949973. Available from: http://www.sciencedirect.com/science/article/pii/S0168927410001145 
[12] 
Tony MacKenzie and A. J. Roberts, Holistic discretisation ensures fidelity to dynamics in two spatial dimensions, preprint, 2009. 
[13] 
J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 6596. doi: 10.1007/BF01762360. 
[14] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003. 
[15] 
S. Engblom, L. Ferm, A. Hellander and P. Lötstedt, Simulation of stochastic reaction diffusion processes on unstructured meshes, preprint, 2008. Available from: http://arXiv.org/abs/0804.3288. 
[16] 
V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," SpringerVerlag, Berlin, 1997. 
[17] 
W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007). 
[18] 
W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions, Comm. Math. Phys., 275 (2007), 163186. doi: 10.1007/s0022000703018. 
[19] 
W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reactiondiffusion equations, preprint, 2008. Available from: http://arxiv.org/abs/0812.1837. 
[20] 
W. Wang and A. J. Roberts, Average and deviation for slowfast stochastic partial differential equations, preprint, 2008. Available from: http://arxiv.org/abs/0904.1462. 
[21] 
W. Wang and A. J. Roberts, Macroscopic discrete modelling of stochastic reactiondiffusion equations, preprint, 2009. 
[22] 
H. Watanabe, Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients, Probab. Th. Rel. Fields, 77 (1988), 359378. doi: 10.1007/BF00319294. 
[23] 
E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics," IMA, 140, SpringerVerlag, New York, 2005. 
[24] 
Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 13631384. doi: 10.1137/040605278. 
[25] 
Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations, SIAM J. Numer. Anal., 40 (2002), 14211445. doi: 10.1137/S0036142901387956. 
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