TY - JOUR
T1 - A Novel Discretization Method for Semilinear Reaction-Diffusion Equation
AU - Chen , Luoping
AU - Chen , Yanping
JO - Advances in Applied Mathematics and Mechanics
VL - 2
SP - 409
EP - 423
PY - 2018
DA - 2018/10
SN - 10
DO - http://doi.org/10.4208/aamm.OA-2017-0011
UR - https://global-sci.org/intro/article_detail/aamm/12218.html
KW - Two-level discretization method, semilinear reaction-diffusion equation, convergence analysis.
AB - <p style="text-align: justify;">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 $\mathcal{T}_H(\Omega)$ (partition of domain $\Omega$ with mesh size $H$) with a large time step size $\Theta$ and a linearized equations on a fine mesh $\mathcal{T}_h(\Omega)$ (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 semilinear problem directly by finite element method with $\mathcal{T}_h(\Omega)$ and $\theta$.</p>