Long-time behavior of the two-grid finite element method for fully discrete semilinear evolution equations with positive memory

Based on two-grid discretizations, two fully discrete finite element algorithms for semilinear parabolic integro-differential equations with positive memory are proposed. With the backward Euler scheme for the temporal discretization, the basic idea of the space two-grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step. To further decreases the amount of computational work, a space–time two-grid algorithm based on a coarse space grid with large time stepsize ΔTΔT and a finer space grid with small time stepsize ΔtΔt for the evolutional equations is proposed in this paper. The sharp long-time stability and error estimates for the standard finite element method, the space two-grid finite element method, and the space–time two-grid finite element method are derived. It is showed that the two-grid algorithms’ long-time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid.