Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems
Maurizio Grasselli Morgan Pierre
Following a result of Chill and Jendoubi in the continuous case, we study the asymptotic behavior of sequences $(U^n)_n$ in $R^d$ which satisfy the following backward Euler scheme:

$\varepsilon\frac{(U^{n+1}-2U^n+U^{n-1}}{\Delta t^2} +\frac{U^{n+1}-U^n}{\Delta t}+\nabla F(U^{n+1})=G^{n+1}, n\ge 0, $

where $\Delta t>0$ is the time step, $\varepsilon\ge 0$, $(G^{n+1})_n$ is a sequence in $ R^d$ which converges to $0$ in a suitable way, and $F\in C^{1,1}_{l o c}(R^d, R)$ is a function which satisfies a Łojasiewicz inequality. We prove that the above scheme is Lyapunov stable and that any bounded sequence $(U^n)_n$ which complies with it converges to a critical point of $F$ as $n$ tends to $\infty$. We also obtain convergence rates. We assume that $F$ is semiconvex for some constant $c_F\ge 0$ and that $1/\Delta t

keywords: proximal method backward Euler scheme gradient-like systems Łojasiewicz inequality
Convergence to equilibrium for the backward Euler scheme and applications
Benoît Merlet Morgan Pierre
We prove that, under natural assumptions, the solution of the backward Euler scheme applied to a gradient flow converges to an equilibrium, as time goes to infinity. Optimal convergence rates are also obtained. As in the continuous case, the proof relies on the well known Lojasiewicz inequality. We extend these results to the $\theta$-scheme with $\theta\in [1/2, 1]$, and to the semilinear heat equation. Applications to semilinear parabolic equations, such as the Allen-Cahn or Cahn-Hilliard equation, are given
keywords: stability Lojasiewicz inequality backward Euler scheme convergence rates. $\theta$-scheme
Stable discretizations of the Cahn-Hilliard-Gurtin equations
Sami Injrou Morgan Pierre
We study space and time discretizations of the Cahn-Hilliard-Gurtin equations with a polynomial nonlinearity. We first consider a space semi-discrete version of the equations, and we prove in particular that any solution converges to a steady state (as in the continuous case). Then, we study the numerical stability of the fully discrete scheme obtained by applying the Euler backward scheme to the space semi-discrete problem. In particular, we show that this fully discrete problem is unconditionally stable. Numerical simulations in one space dimension conclude the paper.
keywords: Numerical stability Euler backward scheme {\L}ojasiewicz inequality
Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation
Michel Pierre Morgan Pierre
The main goal of this paper is to prove existence of global solutions in time for an Allen-Cahn-Gurtin model of pseudo-parabolic type. Local solutions were known to ``blow up" in some sense in finite time. It is proved that the equation is actually governed by a monotone-like operator. It turns out to be multivalued and measure-valued. The measures are singular with respect to the Lebesgue measure. This operator allows to extend the local solutions globally in time and to fully solve the evolution problem. The asymptotic behavior is also analyzed.
keywords: singular measures. Pseudo-parabolic capacity global existence monotone operator Allen-Cahn blow-up
Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system
Narcisse Batangouna Morgan Pierre

We consider a time semi-discretization of the Caginalp phase-field model based on an operator splitting method. For every time-step parameter $ \tau $, we build an exponential attractor $ \mathcal{M}_\tau $ of the discrete-in-time dynamical system. We prove that $ \mathcal{M}_\tau $ converges to an exponential attractor $\mathcal{M}_0$ of the continuous-in-time dynamical system for the symmetric Hausdorff distance as $ \tau $ tends to $0$. We also provide an explicit estimate of this distance and we prove that the fractal dimension of $ \mathcal{M}_\tau $ is bounded by a constant independent of $ \tau $.

keywords: Operator splitting fractional step method global attractor exponential attractor
A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions
Laurence Cherfils Madalina Petcu Morgan Pierre
We consider a finite element space semi-discretization of the Cahn-Hilliard equation with dynamic boundary conditions. We prove optimal error estimates in energy norms and weaker norms, assuming enough regularity on the solution. When the solution is less regular, we prove a convergence result in some weak topology. We also prove the stability of a fully discrete problem based on the backward Euler scheme for the time discretization. Some numerical results show the applicability of the method.
keywords: backward Euler scheme semilinear parabolic problem Finite element method error estimates Łojasiewicz inequality. linearized Crank-Nicolson scheme
A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term
Maurizio Grasselli Nicolas Lecoq Morgan Pierre
We prove the Lyapunov stability of a time and space discretization of the Cahn-Hilliard equation with inertial term. The space discretization is a mixed (or "splitting") nite element method with numerical integration which includes a standard nite di erence approximation. The time discretization is the backward Euler scheme. The smallness assumption on the time step does not depend on the mesh step.
keywords: discrete negative norms numerical integration Cahn-Hilliard equation mixed nite elements Second-order gradient-like ow Lojasiewicz inequality

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