Dynamics of parabolic equations via the finite element method. I: Continuity of the set of equilibria.
type: Item
instance of: scholarly article
title: Dynamics of parabolic equations via the finite element method. I: Continuity of the set of equilibria.
zbMATH Open document ID: 1372.35154
published in: Journal of Differential Equations
publication date: 2016-09-05T00:00:00Z
review text: The authors prove the continuity of the set of equilibria and of their linearized unstable manifolds for finite element approximations of semilinear parabolic initial value problems of the form \[ \begin{aligned} \partial_t u&=L u+f(u)\;(0,\infty)\times \Omega,\\ u|_{\partial\Omega}&=0 \; (0,\infty)\times\partial\Omega, \\ u|_{t=0}&=u_0\in H_0^1(\Omega) \; \{0\}\times\Omega,\end{aligned}\eqno{(1)} \] in a bounded, smooth domain \(\Omega\subset\mathbb{R}^n\), \(n\geq 2\). Here, \(L\) is a second order uniformly strongly elliptic operator with smooth coefficients, and the nonlinearity \(f\) is assumed to be subcritical and dissipative. More precisely, problem \((1)\) can be written in the abstract form \[ \begin{aligned}\dot u+A u&=F(u), \quad t>0, \\ u_0&\in H_0^1=X^{\frac{1}{2}},\end{aligned}\eqno{(2)} \] where the operator \(A=-L\) with domain \(H_0^1(\Omega)\cap H^2(\Omega)\) in \(L^2(\Omega)\), \(X^\alpha, \alpha>0,\) denote the associated fractional power spaces and \(F(u(t))x=f(u(t,x))\) the Nemitskii operator. Under suitable assumptions on the domain \(\Omega\) a discretization of \((2)\) is given by \[ \begin{aligned}\dot u_h+A_h u_h&=F_h(u_h) \quad t>0, \\ u_h(0)&=u_h^0\in X_h^{\frac{1}{2}}, \end{aligned}\eqno{(3)} \] where \(A_h\) is the finite element approximation of \(A\) and \(F_h\) the projection of \(F\) onto a finite dimensional space \(X_h^{\frac{1}{2}},\) \(h\in(0,1].\) The main results are the convergence of the set of equilibria of the discretized problem \((3)\) to the set of equilibria of problem \((2)\) as \(h\) tends to zero, and the continuity of the linearized unstable manifolds of equilibrium points. The proof is based on abstract convergence results obtained in [\textit{J. M. Arrieta} et al., J. Differ. Equations 231, No. 2, 551--597 (2006; Zbl 1110.35028)].
reviewed by: Stefanie Sonner
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 6623114
zbMATH Keywords: semilinear parabolic equations
zbMATH Keywords: finite element approximation
zbMATH Keywords: continuity of equilibrium points
zbMATH Keywords: unstable manifolds
author: Rodiak N. Figueroa-López
author: G. Lozada-Cruz
MaRDI profile type: Publication
full work available at URL: j.jde.2016.07.023
OpenAlex ID: W2510284844
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DOI: 10.1016/J.JDE.2016.07.023
description: scientific article; zbMATH DE number 6623114