Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system

Abstract

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

On the modified of the one-dimensional Cahn-Hilliard equation with a source term

“…An abstract result was first derived, based on the construction in [12], and it was then applied to the backward Euler scheme. The same approach was successfully applied for a time splitting scheme in [4], for a discretized Ginzburg Landau equation in [3] and for a space semidiscretization of the Allen-Cahn equation in [28]. In these papers, the nonlinearity was treated implicitly.…”

On the Caginalp for a Conserve Phase-Field with a Polynomial Potentiel of Order 2<i>p</i> - 1

“…These equations are known as the conserved phase-field model (see [21]- [30]) based on type II heat conduction and with two temperatures (see [3] and [4]), conservative in the sense that, when endowed with Neumann boundary conditions, the spatial average of the order parameter is a conserved quantity. Indeed, in that case, integrating (18) over the spatial domain Ω , we have the conservation of mass,…”

Convergence of Exponential Attractors for a Finite Element Approximation of the Allen–Cahn Equation

“…An abstract construction of a robust family of exponential attractors was first proposed, and then applied in every space dimension to the backward Euler time semidiscretization of the Allen-Cahn equation with a polynomial nonlinearity. It was also applied in [3] to the case of a time-splitting discretization of the Caginalp phase-field system.…”