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Volume 40, Issue 3
An Improved Two-Grid Technique for the Nonlinear Time-Fractional Parabolic Equation Based on the Block-Centered Finite Difference Method

Xiaoli Li, Yanping Chen & Chuanjun Chen

DOI: 10.4208/jcm.2011-m2020-0124

J. Comp. Math., 40 (2022), pp. 453-471.

Published online: 2022-02

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  • Abstract

A combined scheme of the improved two-grid technique with the block-centered finite difference method is constructed and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size $H$ and two linear problems based on the coarse-grid solutions and one Newton iteration is considered on a fine grid of size $h$. We provide the rigorous error estimate, which demonstrates that our scheme converges with order $\mathcal{O}(\Delta t^{2-\alpha}+h^2+H^4)$ on non-uniform rectangular grid. This result indicates that the improved two-grid method can obtain asymptotically optimal approximation as long as the mesh sizes satisfy $h=\mathcal{O}(H^2).$ Finally, numerical tests confirm the theoretical results of the presented method.

  • Keywords

Improved two-grid, Time-fractional parabolic equation, Nonlinear, Error estimates, Numerical experiments.

  • AMS Subject Headings

26A33, 65M06, 65M12, 65M15, 65M55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xiaolisdu@163.com (Xiaoli Li)

yanpingchen@scnu.edu.cn (Yanping Chen)

cjchen@ytu.edu.cn (Chuanjun Chen)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-453, author = {Li , XiaoliChen , Yanping and Chen , Chuanjun}, title = {An Improved Two-Grid Technique for the Nonlinear Time-Fractional Parabolic Equation Based on the Block-Centered Finite Difference Method}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {3}, pages = {453--471}, abstract = {

A combined scheme of the improved two-grid technique with the block-centered finite difference method is constructed and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size $H$ and two linear problems based on the coarse-grid solutions and one Newton iteration is considered on a fine grid of size $h$. We provide the rigorous error estimate, which demonstrates that our scheme converges with order $\mathcal{O}(\Delta t^{2-\alpha}+h^2+H^4)$ on non-uniform rectangular grid. This result indicates that the improved two-grid method can obtain asymptotically optimal approximation as long as the mesh sizes satisfy $h=\mathcal{O}(H^2).$ Finally, numerical tests confirm the theoretical results of the presented method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2011-m2020-0124}, url = {http://global-sci.org/intro/article_detail/jcm/20246.html} }
TY - JOUR T1 - An Improved Two-Grid Technique for the Nonlinear Time-Fractional Parabolic Equation Based on the Block-Centered Finite Difference Method AU - Li , Xiaoli AU - Chen , Yanping AU - Chen , Chuanjun JO - Journal of Computational Mathematics VL - 3 SP - 453 EP - 471 PY - 2022 DA - 2022/02 SN - 40 DO - http://doi.org/10.4208/jcm.2011-m2020-0124 UR - https://global-sci.org/intro/article_detail/jcm/20246.html KW - Improved two-grid, Time-fractional parabolic equation, Nonlinear, Error estimates, Numerical experiments. AB -

A combined scheme of the improved two-grid technique with the block-centered finite difference method is constructed and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size $H$ and two linear problems based on the coarse-grid solutions and one Newton iteration is considered on a fine grid of size $h$. We provide the rigorous error estimate, which demonstrates that our scheme converges with order $\mathcal{O}(\Delta t^{2-\alpha}+h^2+H^4)$ on non-uniform rectangular grid. This result indicates that the improved two-grid method can obtain asymptotically optimal approximation as long as the mesh sizes satisfy $h=\mathcal{O}(H^2).$ Finally, numerical tests confirm the theoretical results of the presented method.

Li , XiaoliChen , Yanping and Chen , Chuanjun. (2022). An Improved Two-Grid Technique for the Nonlinear Time-Fractional Parabolic Equation Based on the Block-Centered Finite Difference Method. Journal of Computational Mathematics. 40 (3). 453-471. doi:10.4208/jcm.2011-m2020-0124
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