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

Abstract

We consider a space semidiscretization of the Allen-Cahn equation by conforming Lagrange finite elements. For every mesh parameter h, we build an exponential attractor M h of the dynamical system associated to the approximate equations. We prove that, as h tends to 0, M h converges for the symmetric Hausdorff distance to an exponential attractor M0 of the dynamical system associated to the Allen-Cahn equation. We also provide an explicit estimate of this distance and we prove that the fractal dimension of M h and of the global attractor is bounded by a constant independent of h. Our proof is adapted from the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semigroup. Here, for the first time, the perturbation is a space discretization. The case of a time semidiscretization has been analyzed in a previous paper.

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

“…The stability of the global attractor as the time step or the mesh step tends to 0 has been considered, e.g., in [12,49,50,124], for discretized Allen-Cahn or Cahn-Hilliard equations. Exponential attractors have also been explored in [44,95,116,117]. These aspects could be further investigated.…”

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.…”