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Stability of Constant Solutions to the Navier-Stokes System in ℝ³

Piotr Bogusław Mucha (2001)

Applicationes Mathematicae

The paper examines the initial value problem for the Navier-Stokes system of viscous incompressible fluids in the three-dimensional space. We prove stability of regular solutions which tend to constant flows sufficiently fast. We show that a perturbation of a regular solution is bounded in W r 2 , 1 ( ℝ ³ × [ k , k + 1 ] ) for k ∈ ℕ. The result is obtained under the assumption of smallness of the L₂-norm of the perturbing initial data. We do not assume smallness of the W r 2 - 2 / r ( ℝ ³ ) -norm of the perturbing initial data or smallness of the...

Stability of oscillating boundary layers in rotating fluids

Nader Masmoudi, Frédéric Rousset (2008)

Annales scientifiques de l'École Normale Supérieure

We prove the linear and non-linear stability of oscillating Ekman boundary layers for rotating fluids in the so-called ill-prepared case under a spectral hypothesis. Here, we deal with the case where the viscosity and the Rossby number are both equal to ε . This study generalizes the study of [23] where a smallness condition was imposed and the study of [26] where the well-prepared case was treated.

Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations

Petr Knobloch, Lutz Tobiska (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, a general technique is developed to enlarge the velocity space V h 1 of the unstable -element by adding spaces V h 2 such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q2/Q1-element and the 4Q1/Q1-element. However, our new elements are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Q1...

Stabilization of a 1-D tank modeled by the shallow water equations

Christophe Prieur, Jonathan de Halleux (2002)

Journées équations aux dérivées partielles

We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by the shallow water equations. By means of a Lyapunov approach, we deduce control laws to stabilize the fluid's state and the tank's position. Although global asymptotic stability is yet to be proved, we numerically simulate the system and observe the stabilization for different control situations.

Stabilization of a non standard FETI-DP mortar method for the Stokes problem

E. Chacón Vera, T. Chacón Rebollo (2014)

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

In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove...

Stabilized Galerkin finite element methods for convection dominated and incompressible flow problems

Gert Lube (1994)

Banach Center Publications

In this paper, we analyze a class of stabilized finite element formulations used in computation of (i) second order elliptic boundary value problems (diffusion-convection-reaction model) and (ii) the Navier-Stokes problem (incompressible flow model). These stabilization techniques prevent numerical instabilities that might be generated by dominant convection/reaction terms in (i), (ii) or by inappropriate combinations of velocity/pressure interpolation functions in (ii). Stability and convergence...

Stabilized Galerkin methods for magnetic advection

Holger Heumann, Ralf Hiptmair (2013)

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

Taking the cue from stabilized Galerkin methods for scalar advection problems, we adapt the technique to boundary value problems modeling the advection of magnetic fields. We provide rigorous a priori error estimates for both fully discontinuous piecewise polynomial trial functions and -conforming finite elements.

Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension

Malte Braack, Andreas Prohl (2013)

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

The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapor phase in the presence of capillarity effects close to phase boundaries. Standard numerical discretizations are known to violate discrete versions of inherent energy inequalities, thus leading to spurious dynamics of computed solutions close to static equilibria (e.g., parasitic currents). In this work, we propose a time-implicit discretization of...

Stable upwind schemes for the magnetic induction equation

Franz G. Fuchs, Kenneth H. Karlsen, Siddharta Mishra, Nils H. Risebro (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be...

Staggered schemes for all speed flows

Raphaèle Herbin, Walid Kheriji, Jean-Claude Latche (2012)

ESAIM: Proceedings

We review in this paper a class of schemes for the numerical simulation of compressible flows. In order to ensure the stability of the discretizations in a wide range of Mach numbers and introduce sufficient decoupling for the numerical resolution, we choose to implement and study pressure correction schemes on staggered meshes. The implicit version of the schemes is also considered for the theoretical study. We give both algorithms for the barotropic Navier-Stokes equations, for the full Navier-Stokes...

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