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The nonlinear galerkin method: A multiscale method applied to the simulation of homogeneous turbulent flows

A. Debussche, T. Dubois and R. Temam
Theoretical and Computational Fluid Dynamics 7 (4) 279 (1995)
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Long Time Behavior of Solutions to the 2D Boussinesq Equations with Zero Diffusivity

Igor Kukavica and Weinan Wang
Journal of Dynamics and Differential Equations 32 (4) 2061 (2020)
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Incremental unknowns, multilevel methods and the numerical simulation of turbulence

T. Dubois, F. Jauberteau and R. Temam
Computer Methods in Applied Mechanics and Engineering 159 (1-2) 123 (1998)
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Uniform stability of spectral nonlinear Galerkin methods

Yinnian He, Kaitai Li and Chunshan Zhao
Numerical Methods for Partial Differential Equations 20 (5) 723 (2004)
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A priori analysis of reduced description of dynamical systems using approximate inertial manifolds

Maryam Akram, Malik Hassanaly and Venkat Raman
Journal of Computational Physics 409 109344 (2020)
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A two-level correction method in space and time based on Crank–Nicolson scheme for Navier–Stokes equations

Qingfang Liu and Yanren Hou
International Journal of Computer Mathematics 87 (11) 2520 (2010)
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Error Estimates on a New Nonlinear Galerkin Method Based on Two-Grid Finite Elements

Martine Marion and Jinchao Xu
SIAM Journal on Numerical Analysis 32 (4) 1170 (1995)
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The Additive Turbulent Decomposition for the Two-Dimensional Incompressible Navier--Stokes Equations: Convergence Theorems and Error Estimates

Peter Perry, Russell M. Brown and Zhongwei Shen
SIAM Journal on Applied Mathematics 59 (1) 139 (1998)
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Computational efficiency and approximate inertial manifolds for a Bénard convection system

M. D. Graham, P. H. Steen and E. S. Titi
Journal of Nonlinear Science 3 (1) 153 (1993)
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Approximation of exponential order of the attractor of a turbulent flow

A. Debussche and T. Dubois
Physica D: Nonlinear Phenomena 72 (4) 372 (1994)
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Flattening, squeezing and the existence of random attractors

Peter E Kloeden and José A Langa
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463 (2077) 163 (2007)
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Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

Ramon Codina
Computer Methods in Applied Mechanics and Engineering 191 (39-40) 4295 (2002)
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The Postprocessing Galerkin and Nonlinear Galerkin Methods---A Truncation Analysis Point of View

Len G. Margolin, Edriss S. Titi and Shannon Wynne
SIAM Journal on Numerical Analysis 41 (2) 695 (2003)
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Nonlinear galerkin method and subgrid-scale model for two-dimensional turbulent flows

Fr�d�ric Pascal and Claude Basdevant
Theoretical and Computational Fluid Dynamics 3 (5) 267 (1992)
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Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method

T. Dubois, F. Jauberteau and R. Temam
Journal of Scientific Computing 8 (2) 167 (1993)
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On computing the long-time solution of the two-dimensional Navier-Stokes equations

M. S. Jolly and C. Xiong
Theoretical and Computational Fluid Dynamics 7 (4) 261 (1995)
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Multilevel Methods in Space and Time for the Navier--Stokes Equations

J. B. Burie and M. Marion
SIAM Journal on Numerical Analysis 34 (4) 1574 (1997)
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A nonlinear galerkin mixed element method and a posteriori error estimator for the stationary navier-stokes equations

Luo Zhen-dong and Zhu Jiang
Applied Mathematics and Mechanics 23 (10) 1194 (2002)
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Attractors for nonautonomous two-dimensional space periodic Navier–Stokes equations

Dongshan Gong, Haitao Song and Chengkui Zhong
Journal of Mathematical Physics 50 (10) 102706 (2009)
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A Class of Numerical Algorithms for Large Time Integration: The Nonlinear Galerkin Methods

Christophe Devulder and Martine Marion
SIAM Journal on Numerical Analysis 29 (2) 462 (1992)
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A nonlinear Galerkin method for the Navier-Stokes equations

F. Jauberteau, C. Rosier and R. Temam
Computer Methods in Applied Mechanics and Engineering 80 (1-3) 245 (1990)
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A Remark on Quasi-Stationary Approximate Inertial Manifolds for the Navier–Stokes Equations

D. A. Jones and E. S. Titi
SIAM Journal on Mathematical Analysis 25 (3) 894 (1994)
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Maryam Akram, Malik Hassanaly and Venkatramanan Raman
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A ladder inequality for the Navier-Stokes equation

Igor Kukavica
Nonlinearity 13 (3) 639 (2000)
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MULTILEVEL SCHEMES FOR SOLVING UNSTEADY EQUATIONS

O. GOYON
International Journal for Numerical Methods in Fluids 22 (10) 937 (1996)
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Gilead Tadmor, Oliver Lehmann, Bernd R. Noack and Marek Morzyński
528 151 (2011)
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Stability and convergence for the reform postprocessing Galerkin method

He Yinnian and R.M.M. Mattheij
Nonlinear Analysis: Real World Applications 1 (4) 517 (2000)
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The reality of artificial viscosity

L. G. Margolin
Shock Waves 29 (1) 27 (2019)
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A Nonlinear Galerkin Method for the Shallow-Water Equations on Periodic Domains

Saulo R.M. Barros and José W. Cárdenas
Journal of Computational Physics 172 (2) 592 (2001)
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Order reduction and nonlinear behaviors of a continuous rotor system

Qian Ding and Kunpeng Zhang
Nonlinear Dynamics 67 (1) 251 (2012)
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A fully discrete nonlinear Galerkin method for the 3D Navier-Stokes equations

J.-L. Guermond and Serge Prudhomme
Numerical Methods for Partial Differential Equations 24 (3) 759 (2008)
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Lipschitz deviation and embeddings of global attractors

Eleonora Pinto de Moura and James C Robinson
Nonlinearity 23 (7) 1695 (2010)
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Nonlinear Galerkin methods based on the concept of determining modes for the magnetohydrodynamic equations

Olaf Schmidtmann, Fred Feudel and Norbert Seehafer
Journal of Physics A: Mathematical and General 31 (34) 7141 (1998)
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INERTIAL ALGORITHMS FOR THE STATIONARY NAVIER-STOKES EQUATIONS

Yanren Hou and R.M.M Mattheij
Acta Mathematica Scientia 23 (2) 219 (2003)
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Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation

M. S. Jolly, I. G. Kevrekidis and E. S. Titi
Journal of Dynamics and Differential Equations 3 (2) 179 (1991)
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Explicit construction of attracting integral manifolds for a dissipative hyperbolic equation

A. Yu. Goritsky
Journal of Mathematical Sciences 143 (4) 3239 (2007)
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IMD based nonlinear Galerkin method

Hou Yan-ren and Li Kai-tai
Applied Mathematics and Mechanics 24 (3) 326 (2003)
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Construction of Approximate Inertial Manifolds Using Wavelets

Olivier Goubet
SIAM Journal on Mathematical Analysis 23 (6) 1455 (1992)
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Amir Averbuch, Ludimila Ioffe, Moshe Israeli and Lev Vozovoi
120 123 (2000)
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Galerkin projections and the proper orthogonal decomposition for equivariant equations

Gal Berkooz and Edriss S. Titi
Physics Letters A 174 (1-2) 94 (1993)
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Regularity of solutions and approximate inertial manifolds for von Karman evolution equations

I. D. Chueshov
Mathematical Methods in the Applied Sciences 17 (9) 667 (1994)
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Localization and approximation of attractors for the Ginzburg-Landau equation

Keith Promislow and Roger Temam
Journal of Dynamics and Differential Equations 3 (4) 491 (1991)
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Thierry Dubois, Francois Jauberteau and Roger Temam
490 388 (1997)
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A two-level method in space and time for the Navier-Stokes equations

Qingfang Liu, Yanren Hou and Qingchang Liu
Numerical Methods for Partial Differential Equations 29 (5) 1504 (2013)
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Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing

Julia García-Luengo, Pedro Marín-Rubio, José Real and James C. Robinson
Discrete & Continuous Dynamical Systems - A 34 (1) 203 (2014)
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DPIV-driven flow simulation: a new computational paradigm

X. Ma, G. E. Karniadakis, H. Park and M. Gharib
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 459 (2031) 547 (2003)
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Scale matters

L. G. Margolin
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376 (2118) 20170235 (2018)
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Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis

Michael D. Graham and Ioannis G. Kevrekidis
Computers & Chemical Engineering 20 (5) 495 (1996)
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LOCALIZATION AND APPROXIMATION OF ATTRACTORS FOR THE KURAMOTO-SIVASHINSKY EQUATIONS

Yujiang Wu and Benyu Guo
Acta Mathematica Scientia 20 (2) 145 (2000)
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Low dimensional approximate inertial manifolds for the kuramoto-sivashinsky equation

Wenhan Chen
Numerical Functional Analysis and Optimization 14 (3-4) 265 (1993)
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Stability study, error estimation, and condition number for semi-implicit schemes using incremental unknowns

Francois Pouit
Numerical Methods for Partial Differential Equations 12 (6) 743 (1996)
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Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem

Yinnian He and Kaitai Li
Numerische Mathematik 98 (4) 647 (2004)
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On the regularity for the Boussinesq equations in a bounded domain

Weiwei Hu, Igor Kukavica and Mohammed Ziane
Journal of Mathematical Physics 54 (8) 081507 (2013)
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Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates

Hussain A Ibdah, Cecilia F Mondaini and Edriss S Titi
IMA Journal of Numerical Analysis 40 (4) 2584 (2020)
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Long Time Dynamics of Forced Critical SQG

Peter Constantin, Andrei Tarfulea and Vlad Vicol
Communications in Mathematical Physics 335 (1) 93 (2015)
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Fourier nonlinear Galerkin approximation for the two dimensional Navier-Stokes equations

Hou Yanren
Applied Mathematics and Mechanics 17 (9) 879 (1996)
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Thierry Dubois, François Jauberteau and Roger Temam
371 116 (1990)
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A. Armaou and P.D. Christofides
2 2310 (2002)
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Homogenization Based LES for Turbulent Combustion

Christer Fureby
Flow, Turbulence and Combustion 84 (3) 459 (2010)
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Approximate inertial manifolds and effective viscosity in turbulent flows

Ciprian Foias, Oscar P. Manley and Roger Temam
Physics of Fluids A: Fluid Dynamics 3 (5) 898 (1991)
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The Generalized Nonlinear Galerkin Method Applied to Hydrodynamics of the Open Sea

B. DI MARTINO and P. ORENGA
International Journal of Computational Fluid Dynamics 14 (4) 305 (2001)
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Multi-level spectral galerkin method for the navier-stokes problem I : spatial discretization

Yinnian He, Kam-Moon Liu and Weiwei Sun
Numerische Mathematik 101 (3) 501 (2005)
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Thierry Dubois, Francois Jauberteau and Roger M. Temam
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Using approximate inertial manifold approach to model turbulent non-premixed combustion

Maryam Akram and Venkat Raman
Physics of Fluids 33 (3) 035125 (2021)
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A parallel spectral Fourier-Nonlinear Galerkin algorithm for simulation of turbulence

A. Averbuch, L. Ioffe, M. Israeli and L. Vozovoi
Numerical Methods for Partial Differential Equations 13 (6) 699 (1997)
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Postprocessing the Galerkin Method: The Finite-Element Case

Bosco GarcÍa-Archilla and Edriss S. Titi
SIAM Journal on Numerical Analysis 37 (2) 470 (1999)
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Convergence of a chaotic attractor with increased spatial resolution of the Ginzburg-Landau equation

M.S. Jolly, R. Témam and C. Xiong
Chaos, Solitons & Fractals 5 (10) 1833 (1995)
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F. Bouchon and T. Dubois
7 275 (1999)
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Postprocessing for Stochastic Parabolic Partial Differential Equations

Gabriel J. Lord and Tony Shardlow
SIAM Journal on Numerical Analysis 45 (2) 870 (2007)
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A simple postrocess procedure for Galerkin method

Hou Yanren and Li Kaitai
Applied Mathematics and Mechanics 21 (1) 79 (2000)
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DPIV/T-driven convective heat transfer simulation

X. Ma, G.E. Karniadakis, H. Park and M. Gharib
International Journal of Heat and Mass Transfer 45 (17) 3517 (2002)
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Invariant helical subspaces for the Navier-Stokes equations

A. Mahalov, E. S. Titi and S. Leibovich
Archive for Rational Mechanics and Analysis 112 (3) 193 (1990)
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On the Question of Turbulence Modeling by Approximate Inertial Manifolds and the Nonlinear Galerkin Method

John G. Heywood and Rolf Rannacher
SIAM Journal on Numerical Analysis 30 (6) 1603 (1993)
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Konstantin Kemenov and Suresh Menon
9 49 (2004)
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Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise

I. D. Chueshov
Journal of Dynamics and Differential Equations 7 (4) 549 (1995)
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Slow Manifolds and Invariant Sets of the Primitive Equations

R. Temam and D. Wirosoetisno
Journal of the Atmospheric Sciences 68 (3) 675 (2011)
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Subgrid modelling and the interaction of small and large wavelengths in turbulent flows

T. Dubois, F. Jauberteau, M. Marion and R. Temam
Computer Physics Communications 65 (1-3) 100 (1991)
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Separation of variables in the Stokes problem application to its finite element multiscale approximation

O. Goubet
ESAIM: Mathematical Modelling and Numerical Analysis 28 (3) 243 (1994)
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Induced trajectories and approximate inertial manifolds

Roger Temam
ESAIM: Mathematical Modelling and Numerical Analysis 23 (3) 541 (1989)
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Limiting Behavior for an Iterated Viscosity

Ciprian Foias, Michael S. Jolly and Oscar P. Manley
ESAIM: Mathematical Modelling and Numerical Analysis 34 (2) 353 (2000)
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Two-grid finite-element schemes for the transient Navier-Stokes problem

Vivette Girault and Jacques-Louis Lions
ESAIM: Mathematical Modelling and Numerical Analysis 35 (5) 945 (2001)
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An Adaptive Multi-level method for Convection Diffusion Problems

Martine Marion and Adeline Mollard
ESAIM: Mathematical Modelling and Numerical Analysis 34 (2) 439 (2000)
DOI: 10.1051/m2an:2000150
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Approximate inertial manifolds for the pattern formation Cahn-Hilliard equation

Martine Marion
ESAIM: Mathematical Modelling and Numerical Analysis 23 (3) 463 (1989)
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Bifurcations of finite difference schemes and their approximate inertial forms

Rolf Bronstering and Min Chen
ESAIM: Mathematical Modelling and Numerical Analysis 32 (6) 715 (1998)
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