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Volume 10, Issue 2
A Novel Discretization Method for Semilinear Reaction-Diffusion Equation

Luoping Chen & Yanping Chen

Adv. Appl. Math. Mech., 10 (2018), pp. 409-423.

Published online: 2018-10

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  • Abstract

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 $\theta$.

  • AMS Subject Headings

65M10, 78A48

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COPYRIGHT: © Global Science Press

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@Article{AAMM-10-409, author = {Chen , Luoping and Chen , Yanping}, title = {A Novel Discretization Method for Semilinear Reaction-Diffusion Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {2}, pages = {409--423}, abstract = {

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 $\theta$.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0011}, url = {http://global-sci.org/intro/article_detail/aamm/12218.html} }
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 -

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 $\theta$.

Luoping Chen & Yanping Chen. (2020). A Novel Discretization Method for Semilinear Reaction-Diffusion Equation. Advances in Applied Mathematics and Mechanics. 10 (2). 409-423. doi:10.4208/aamm.OA-2017-0011
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