(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
In this work, we investigate a novel two-level discretization method for semilinear reaction-diffusion equations. Motivated by the two-grid method for nonlinear partial differential equations (PDEs) introduced by Xu [18] on physical space, our discretization method uses a two-grid finite element discretization method for semilinear partial differential equations on physical space and a two-level finite difference method for the corresponding time space. Specifically, we solve a semilinear equations on a coarse mesh T_H(Ω) (partition of domain Ω with mesh size H) with a large time step size Θ and a linearized equations on a fine mesh T_h(Ω) (partition of domain Ω with mesh size h) using smaller time step size θ. Both theoretical and numerical results show that when h=H^2, θ=Θ^2, the novel two-grid numerical solution achieves the same approximate accuracy as that for the original semilinear problem directly by finite element method with T_h(Ω) and θ.