Convergence of Exponential Attractors for a Finite Element Approximation of the Allen–Cahn Equation. (10th December 2018)

• Record Type: Journal Article
• Authors: Pierre, Morgan
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 16(2018)
• Page Start: 1755
• Page End: 1784
• Publication Date: 2018-12-10
• Subjects: Allen–Cahn equation -- exponential attractor -- finite element method -- global attractor -- nonsmooth data error estimate; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1497651
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: We consider a space semidiscretization of the Allen–Cahn equation by continuous piecewise linear finite elements. For every mesh parameter h, we build an exponential attractor of the dynamical system associated with the approximate equations. We prove that, as h tends to 0, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated with the Allen–Cahn equation. We also prove that the fractal dimension of the exponential attractor 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, the perturbation is a space discretization. The case of a time semidiscretization has been analyzed in a previous paper.