Global dynamics of a discontinuous Galerkin approximation to a class of reaction-diffusion equation
type: Item
instance of: scholarly article
title: Global dynamics of a discontinuous Galerkin approximation to a class of reaction-diffusion equation
zbMATH Open document ID: 0839.65102
DOI: 10.1016/0168-9274(95)00093-A
author: Yanyan Li
published in: Applied Numerical Mathematics
publication date: 1996-06-16T00:00:00Z
review text: The subject of this paper is the nonlinear equation (1) \(\dot U- \gamma\Delta U= - f(U)\) (called the Allen-Cahn equation), \(\gamma\) being a positive constant. The unknown function \(U(x, t)\) is considered for \(t> 0\) and \(x\) belonging to a bounded convex domain \(\Omega\subset \mathbb{R}^d\) \((d= 1,2,3)\) with smooth boundary \(\Gamma\). Assuming the initial condition (2) \(U(.,0)= U_0\) on \(\Omega\) and the boundary condition (3) \(U= 0\) on \(\Gamma\), the authors construct a fully discretized (with respect to both \(x\) and \(t\)) Galerkin scheme to solve approximately problem (1), (2), (3). This scheme is proved to be uniquely solvable. Further investigations are concerning the global dynamic structure of this scheme, namely its absorbing sets, global attractors and the related discrete semigroup operator.
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 826019
zbMATH Keywords: global dynamics
zbMATH Keywords: discontinuous Galerkin approximation
zbMATH Keywords: reaction-diffusion equation
zbMATH Keywords: Allen-Cahn equation
zbMATH Keywords: absorbing sets
zbMATH Keywords: global attractors
zbMATH Keywords: discrete semigroup operator
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
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full work available at URL: 0168-9274(95)00093-a
OpenAlex ID: W2069706222
description: scientific article; zbMATH DE number 826019