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Un résultat de convergence d'ordre deux en temps pour l'approximation des équations de Navier–Stokes par une technique de projection incrémentale

Jean-Luc Guermond (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The Navier–Stokes equations are approximated by means of a fractional step, Chorin–Temam projection method; the time derivative is approximated by a three-level backward finite difference, whereas the approximation in space is performed by a Galerkin technique. It is shown that the proposed scheme yields an error of 𝒪 ( δ t 2 + h l + 1 ) for the velocity in the norm of l2(L2(Ω)d), where l ≥ 1 is the polynomial degree of the velocity approximation. It is also shown that the splitting error of projection schemes based...

Uniform stabilization of a viscous numerical approximation for a locally damped wave equation

Arnaud Münch, Ademir Fernando Pazoto (2007)

ESAIM: Control, Optimisation and Calculus of Variations

This work is devoted to the analysis of a viscous finite-difference space semi-discretization of a locally damped wave equation in a regular 2-D domain. The damping term is supported in a suitable subset of the domain, so that the energy of solutions of the damped continuous wave equation decays exponentially to zero as time goes to infinity. Using discrete multiplier techniques, we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size)...

Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications

Farah Abdallah, Serge Nicaise, Julie Valein, Ali Wehbe (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we consider the approximation of second order evolution equations. It is well known that the approximated system by finite element or finite difference is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial...

Uniformly exponentially stable approximations for a class of second order evolution equations

Karim Ramdani, Takéo Takahashi, Marius Tucsnak (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are...

Unique solvability and stability analysis of a generalized particle method for a Poisson equation in discrete Sobolev norms

Yusuke Imoto (2019)

Applications of Mathematics

Unique solvability and stability analysis is conducted for a generalized particle method for a Poisson equation with a source term given in divergence form. The generalized particle method is a numerical method for partial differential equations categorized into meshfree particle methods and generally indicates conventional particle methods such as smoothed particle hydrodynamics and moving particle semi-implicit methods. Unique solvability is derived for the generalized particle method for the...

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