A Novel Discretization Method for Semilinear Reaction-Diffusion Equation

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 (Omega) (partition of domain W with mesh size H) with a large time step size Theta and a linearized equations on a fine mesh T-h (Omega) (partition of domain Omega with mesh size h) using smaller time step size theta. Both theoretical and numerical results show that when h = H-2, theta = Theta(2), the novel two-grid numerical solution achieves the same approximate accuracy as that for the original seminlinear problem directly by finite element method with T-h (Omega) and theta.