Nonlinear Galerkin Methods

Marion, Martine; Témam, Roger

doi:10.1137/0726063

Cited by 338 publications

(195 citation statements)

References 9 publications

Supporting

Mentioning

192

Contrasting

Unclassified

Order By: Relevance

“…By now, it is a well established fact that, for the dissipative parabolic evolution equations, the nonlinear Galerkin method, which is based on the theory of Inertial Manifolds [32] and Approximate Inertial Manifolds (AIM) (see, e.g., [27], [33], [53], [63], [76] and references therein), is a more accurate numerical method than the standard Galerkin method (cf. [23], [39], [47], [48]).…”

Section: Approximationmentioning

confidence: 99%

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

Cao¹,

Kevrekidis²,

Titi³

2001

Indiana Univ. Math. J.

View full text Add to dashboard Cite

ABSTRACT. This paper introduces an explicit numerical criterion for the stabilization of steady state solutions of the NavierStokes equations (NSE) with linear feedback control. Given a linear feedback controller that stabilizes a steady state solution to the closed-loop standard Galerkin (or nonlinear Galerkin) NSE discretization, it is shown that, if the number of modes involved in the computation is large enough, this controller stabilizes a nearby steady state of the closed-loop NSE. We provide an explicit estimate, in terms of the physical parameters, for the number of modes required in order for this criterion to hold. Moreover, we provide an estimate for the distance between the stabilized numerical steady state and the actually stabilized steady state of the closed-loop Navier-Stokes equations. More accurate approximation procedures, based on the concept of postprocessing the Galerkin results, are also presented. All the criterion conditions are imposed on the computed numerical solution, and no a priori knowledge is required about the steady state solution of the full PDE. This criterion holds for a large class of unbounded linear feedback operators and can be slightly modified to include certain nonlinear parabolic systems such as reaction-diffusion systems.

show abstract

Section: Approximationmentioning

confidence: 99%

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

Cao¹,

Kevrekidis²,

Titi³

2001

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…It is classical that (2.5)-(2.6) posses a unique solution (u,p) (see [17,18,21]). We conclude this section by recalling some regularity results ofthe solution of (2.5)-(2.6).…”

Section: The Equation Of N Avier-stokes Typementioning

confidence: 99%

“…In particular, the small scale component is often obtained as a nonlinear functional of the large scale component. These questions have mainly been addressed in the case of spectral Fourier discretizations( see [4][5][6]15,17,20] and the references therein) . The case of finite element approximations for general nonlinear evolution equations are considered in [18][19].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a modified finite element nonlinear Galerkin method

Miao

Mattheij

et al. 2004

Numerische Mathematik

View full text Add to dashboard Cite

“…The existence of approximate inertial manifolds has been proved for the two-dimensional Navier-Stokes équations [4,13] and also for reaction-diffusion équations in high space dimension [8] (for the latest équations non existence results of inertial manifolds are known when n = 4 [7]). Let us also mention that the concept of approximate inertial manifolds leads to new numerical schemes, well adapted to the long term intégration of évolution équations [9].…”

Section: Introductionmentioning

confidence: 99%