The Crank-Nicolson-Galerkin FEM for a nonlocal parabolic system

Abstract

A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonlocal nonlinear coupled system of the reaction-diffusion problem is presented here.  For sufficiently smooth solutions, the maximal error in the L²-norm possesses the optimal rate of convergence O(δ² + h^r+1) (where h is the mesh size and δ is the time step size with r ≥ 1) without any time step restriction.  Some important results on the energy decay and vanishing of the solutions in finite time are also presented.  To confirm our theoretical analysis, some numerical experiments are performed using Matlab.