Optimization and Convergence of Numerical Attractors for Discrete-Time Quasi-Linear Lattice System

Abstract

Existence and connection of numerical attractors for discrete-time p -Laplace lattice systems via the implicit Euler scheme are proved. The numerical attractors are shown to have an optimized bound, which leads to the continuous convergence of the numerical attractors when the graph of the nonlinearity closes to the vertical axis or when the external force vanishes. A new type of Taylor expansion without Fréchet derivatives is established and applied to show the discretization error of order two, which is crucial to prove that the numerical attractors converge upper semicontinuously to the global attractor of the original continuous-time system as the step size of the time goes to zero. It is also proved that the truncated numerical attractors for finitely dimensional systems converge upper semicontinuously to the numerical attractor and the lower semicontinuity holds in special cases.