Dynamics of parabolic equations via the finite element method. I: Continuity of the set of equilibria.
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)].
scientific article; zbMATH DE number 6623114