Discretisation of global attractors for lattice dynamical systems
The authors investigate the existence of numerical attractors for lattice dynamical system (LDS) \[\frac{du_i}{dt}=\nu(u_{i-1}-2u_i+u_{i+1})+f(u_i)+g_i,\ \ \ i\in\mathbb{Z},\ \ \nu>0,\tag{1}\] where \((g_i)_{i\in\mathbb{Z}}\in{\ell}^2\), \({\ell}^2=\big\{u=(u_i)_{i\in\mathbb{Z}}:\sum_{i\in\mathbb{Z}}u_i^2<\infty\big\}\) is a Hilbert space, the nonlinearity \(f\colon\mathbb{R}\to\mathbb{R}\) is a continuously differentiable function satisfying inequality \(f(s)s\le-\alpha s^2\) for some \(\alpha>0\). The LDS (1) can be written as the infinite dimensional ordinary differential equation in \({\ell}^2\) \[\frac{du}{dt}=Au+F(u)+g,\tag{2}\] where \(A\colon{\ell}^2\to{\ell}^2\), \((Au)_i=\nu(u_{i-1}-2u_i+u_{i+1})\), \(F\colon{\ell}^2\to{\ell}^2\) is the Nemytskii operator, \(F(u)=(f(u_i))_{i\in\mathbb{Z}}\), \(g=(g_i)_{i\in\mathbb{Z}}\). For given initial data \(u(0) = u_0\in{\ell}^2\) the existence and uniqueness of a global solution of the ODE system (2) follows by standard argument (see [\textit{P. W. Bates} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, No. 1, 143--153 (2001; Zbl 1091.37515)]). The semi-dynamical system generated by (2) possesses a global attractor. By the implicit Euler scheme with constant time step \(h>0\), the ODE (2) is reduced to \[u^{(h)}_{n+1}=u^{(h)}_{n}+hAu^{(h)}_{n+1} +hF(u^{(h)}_{n+1})+hg, \ \ n\in\mathbb{N}_0, \tag{3}\] where \(u^{(h)}_{n}=\big(u^{(h)}_{n,i}\big)_{i\in\mathbb{Z}}\). The finite dimensional truncations of (3) are considered. It is proved that the finite dimensional numerical attractors converge upper semicontinuously to the global attractor of the original lattice model (1) as the discretisation step size tends to zero.
scientific article; zbMATH DE number 7229187