Ahmad Makki

Lebanese American University | LAU · Department of Computer Science and Mathematical

We analyze here a finite element space semidiscretization of the Allen-Cahn/Cahn- Hilliard system coupled with heat equation and based on the Maxwell-Cattaneo law. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates in energy norm...

The numerical analysis of the coupled Cahn-Hilliard/Allen-Cahn system endowed with dynamic boundary conditions is studied in this article. We consider a semi-discretisation in space using a finite element method and we derive error estimates between the exact and the approximate solution. Then, using the backward Euler scheme for the time variable,...

Our aim in this article is to study generalizations of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the associated system. In particular, we prove the existence of the g...

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen-Cahn/Cahn-Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

We propose a finite element discretization of a singular elliptic problem with discontinuous coefficients. We use a "regularize then discretize" approach. We show that our method converges in 1, 2 and 3 space dimensions. We also perform numerical simulations in two space dimensions with FreeFem++, using an adaptive mesh strategy to deal with the si...

Our aim in this article is to study the asymptotic behavior of a Cahn-Hilliard/ Allen-Cahn system coupled with a heat equation based on the type III heat conduction law with singular potentials. We also show further regularity results and we prove a strict separation property (from the pure states) in one space dimension.

Our aim in this article is to study generalizations of the noncon-served Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global a...

Our aim in this article is to study the long time behavior, in terms of finite dimensional attractors, of a class of doubly nonlinear Allen–Cahn equations with singular potentials. In particular, we prove the existence of the global attractor which has finite fractal dimension.

We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a r...

This thesis is situated in the context of the theoretical and numerical analysis of models in phase separation which take into account the anisotropic effects. This is relevant, for example, for the development of crystals in their liquid matrix for which the effects of anisotropy are very strong. We study the existence, uniqueness and the regulari...

Our aim in this paper is to prove the existence and uniqueness of solutions to Cahn-Hilliard and Allen-Cahn type equations based on a modification of the Ginzburg-Landau free energy proposed in [12] (see also [16]) which takes into account strong anisotropy effects. In particular, the free energy contains a regularization term, called Willmore regu...

Our aim is to prove the existence and uniqueness of solutions for one-dimensional Cahn-Hilliard and Allen-Cahn type equations based on a modification of the Ginzburg-Landau free energy proposed in [8]. In particular, the free energy contains an additional term called Willmore regularization and takes into account strong anisotropy effects.