Abdelkrim Chakib

On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W 1, p (Ω) with a constant that is independent of Ω ∈ 𝒰, where 𝒰 is a set of uniformly bounded and uniformly Lipschitz domains in ℝ n . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ2(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.

A shape optimization formulation of weld pool determination

In this paper, we propose a shape optimization formulation for a problem modeling a process of welding. We show the existence of an optimal solution. The finite element method is used for the discretization of the problem. The

Study of a Numerical Approach for a Transient Flow Problem in Porous Media

In this work, we deal with the numerical study of the new approximation method proposed in [7] for a transient flow problem in porous media. The stationary problem, obtained from a time discretization of this transient problem, is considered as an optimal shape design formulation. We prove the existence of the solution of the discrete optimal shape problem obtained from finite element discretization. We study the convergence and give numerical results showing the efficiency of the proposed approach.