Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of classC0 in Hilbert spaces

Guermond, Jean‐Luc

doi:10.1002/1098-2426(200101)17:1<1::aid-num1>3.0.co;2-1

Cited by 22 publications

(17 citation statements)

References 22 publications

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“…We review this point of view in the present section, which is organized as follows: first, we describe popular two-scale subgrid viscosity models; then, we review scale similarity models; finally, we show that numerical implementations of some two-scale models are very similar to a subgrid stabilization technique that has been introduced in the literature to solve non-coercive PDE's, [25,27,26]. This investigation led us to draw the following conclusions:…”

Section: Two-scale Methodsmentioning

confidence: 88%

“…If ε is O(1), the Galerkin technique is optimal, but if ε is small, the coercivity is not strong enough to guarantee that the Galerkin approximation is satisfactory. It is shown in [26] that by using the same two-level framework as above and by perturbing the Galerkin technique with the same bilinear form b h as above, Theorem 6.1 still holds. The remarkable result here is that the estimates are uniform with respect to ε, and optimal convergence is obtained in the graph norm of A.…”

Section: The Discrete Settingmentioning

confidence: 99%

“…As a consequence, when approximating this type of equation supplemented with non-smooth data, the approximate solution exhibits spurious node-to-node oscillations. The two-level subgrid viscosity technique developed in Guermond [25,27,26] is one possible cure to this problem.…”

Section: Subgrid Viscosity Stabilizationmentioning

confidence: 99%

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Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows

Guermond

Oden

Prudhomme

2004

Journal of Mathematical Fluid Mechanics

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The main objective of this paper is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We review several LES models for which we attempt to provide mathematical justifications. For instance, some filtering techniques and nonlinear viscosity models are found to be regularization techniques that transform the possibly ill-posed Navier-Stokes equation into a well-posed set of PDE's. Spectral eddyviscosity methods are also considered. We show that these methods are not spectrally accurate, and, being quasi-linear, that they fail to be regularizations of the Navier-Stokes equations. We then propose a new spectral hyper-viscosity model that regularizes the Navier-Stokes equations while being spectrally accurate. We finally review scale-similarity models and two-scale subgrid viscosity models. A new energetically coherent scale-similarity model is proposed for which the filter does not require any commutation property nor solenoidality of the advection field. We also show that two-scale methods are mathematically justified in the sense that, when applied to linear non-coercive PDE's, they actually yield convergence in the graph norm. Mathematics Subject Classification (2000). 65N30, 65M, 76D05, 76F.

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Section: Two-scale Methodsmentioning

confidence: 88%

Section: The Discrete Settingmentioning

confidence: 99%

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Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows

Guermond

Oden

Prudhomme

2004

Journal of Mathematical Fluid Mechanics

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“…In the following we refer to numerical results obtained by schemes belonging to the considered class of stabilisation methods. The application of subgrid modelling (gradients of fluctuations) to scalar transport equations of convection-diffusion type has been numerically studied in [15,[19][20][21][22][23] in the two-level context for continous piecewise linear and quadratic elements.…”

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confidence: 99%

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Matthies¹,

Skrzypacz²,

Tobiska³

2007

ESAIM: M2AN

170

183

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Abstract. The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory, we extend the results of Braack and Burman for the standard two-level version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. Moreover, our general theory allows to derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads to much more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscale approach.Mathematics Subject Classification. 65N12, 65N30, 76D05.

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“…Sharper results can be derived by using the advective derivative in the weighting function. Following [17], the proof of the error estimate then hinges on discrete inf-sup stability, assuming additional regularity in time of the exact solution. As a result, the desired h k+ 1 2 error estimate is achieved for any polynomial degree, any value for λ, and shape-regular meshes.…”

Section: Definition Of F K (Thick Lines)mentioning

confidence: 99%

Weighting the Edge Stabilization

Ern¹,

Guermond²

2013

SIAM J. Numer. Anal.

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Abstract. A modification of the edge stabilization technique is proposed to improve the behavior of the method when solving conservation equations with non-smooth data and/or non-smooth solutions. The key ingredient is tempering the edge stabilization in regions of large gradients through appropriate weights. The new method is shown to preserve the convergence properties of the original method on smooth solutions and numerical tests indicate that it performs better on non-smooth solutions.

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Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of classC0 in Hilbert spaces

Cited by 22 publications

References 22 publications

Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows

Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Weighting the Edge Stabilization

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