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The topological asymptotic for the Navier-Stokes equations

Samuel Amstutz (2005)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional...

The topological asymptotic for the Navier-Stokes equations

Samuel Amstutz (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional...

Two-grid finite-element schemes for the transient Navier-Stokes problem

Vivette Girault, Jacques-Louis Lions (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size H . In the second step, the problem is linearized by substituting into the non-linear term, the velocity 𝐮 H computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size h . This approach is motivated by the fact that,...

Two-grid finite-element schemes for the transient Navier-Stokes problem

Vivette Girault, Jacques-Louis Lions (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size H. In the second step, the problem is linearized by substituting into the non-linear term, the velocity uH computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size h. This approach is motivated by the fact that,...

Two-Layer Flow with One Viscous Layer in Inclined Channels

O. K. Matar, G. M. Sisoev, C. J. Lawrence (2008)

Mathematical Modelling of Natural Phenomena

We study pressure-driven, two-layer flow in inclined channels with high density and viscosity contrasts. We use a combination of asymptotic reduction, boundary-layer theory and the Karman-Polhausen approximation to derive evolution equations that describe the interfacial dynamics. Two distinguished limits are considered: where the viscosity ratio is small with density ratios of order unity, and where both density and viscosity ratios are small. The evolution equations account for the presence of...

Two-level stabilized nonconforming finite element method for the Stokes equations

Haiyan Su, Pengzhan Huang, Xinlong Feng (2013)

Applications of Mathematics

In this article, we present a new two-level stabilized nonconforming finite elements method for the two dimensional Stokes problem. This method is based on a local Gauss integration technique and the mixed nonconforming finite element of the N C P 1 - P 1 pair (nonconforming linear element for the velocity, conforming linear element for the pressure). The two-level stabilized finite element method involves solving a small stabilized Stokes problem on a coarse mesh with mesh size H and a large stabilized Stokes...

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