A robust family of exponential attractors for a linear time discretization of the Cahn-Hilliard equation with a source term

TL;DR

This paper constructs exponential attractors for a discretized Cahn-Hilliard equation with a source term, showing their convergence to the continuous system's attractor as the time step decreases, with bounds independent of the discretization.

Contribution

It introduces a robust family of exponential attractors for a linear IMEX discretization of the Cahn-Hilliard equation, demonstrating their convergence and bounded fractal dimension.

Findings

Exponential attractors exist for small enough time steps.

Attractors converge to the continuous system's attractor as time step approaches zero.

Fractal dimension of attractors is uniformly bounded independently of the time step.

Abstract

We consider a linear implicit-explicit (IMEX) time discretization of the Cahn-Hilliard equation with a source term, endowed with Dirichlet boundary conditions. For every time step small enough, we build an exponential attractor of the discrete-in-time dynamical system associated to the discretization. We prove that, as the time step tends to 0, this attractor converges for the symmmetric Hausdorff distance to an exponential attractor of the continuous-in-time dynamical system associated with the PDE. We also prove that the fractal dimension of the exponential attractor (and consequently, of the global attractor) is bounded by a constant independent of the time step. The results also apply to the classical Cahn-Hilliard equation with Neumann boundary conditions.