A Novel Discretization Method for Semilinear Reaction-Diffusion Equation (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 $\mathcal{T}_H(Ω)$ (partition of domain $Ω$ with mesh size $H$) with a large time step size $Θ$ and a linearized equations on a fine mesh $\mathcal{T}_h(Ω)$ (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 $\mathcal{T}_h(Ω)$ and $θ$.