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Volume 11, Issue 3
Crank-Nicolson Method of a Two-Grid Finite Volume Element Algorithm for Nonlinear Parabolic Equations

Yunjie Gong, Chuanjun Chen, Yuzhi Lou & Guanyu Xue

East Asian J. Appl. Math., 11 (2021), pp. 540-559.

Published online: 2021-05

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  • Abstract

A two-grid finite volume element algorithm based on Crank-Nicolson scheme for nonlinear parabolic equations is proposed. In this method, the nonlinear problem is solved on a coarse grid of size $H$ and a linear problem is considered on a fine grid of size $h$ by using the coarse-grid solution and one Newton iteration. This helps to improve the computing efficiency while keeping the accuracy. It is proved that the two-grid method can achieve asymptotically optimal error estimates in spaces and second order accuracy in time. Numerical results are consistent with the theoretical findings.

  • AMS Subject Headings

65N12, 65M60

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-11-540, author = {Gong , YunjieChen , ChuanjunLou , Yuzhi and Xue , Guanyu}, title = {Crank-Nicolson Method of a Two-Grid Finite Volume Element Algorithm for Nonlinear Parabolic Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2021}, volume = {11}, number = {3}, pages = {540--559}, abstract = {

A two-grid finite volume element algorithm based on Crank-Nicolson scheme for nonlinear parabolic equations is proposed. In this method, the nonlinear problem is solved on a coarse grid of size $H$ and a linear problem is considered on a fine grid of size $h$ by using the coarse-grid solution and one Newton iteration. This helps to improve the computing efficiency while keeping the accuracy. It is proved that the two-grid method can achieve asymptotically optimal error estimates in spaces and second order accuracy in time. Numerical results are consistent with the theoretical findings.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.090820.311220}, url = {http://global-sci.org/intro/article_detail/eajam/19140.html} }
TY - JOUR T1 - Crank-Nicolson Method of a Two-Grid Finite Volume Element Algorithm for Nonlinear Parabolic Equations AU - Gong , Yunjie AU - Chen , Chuanjun AU - Lou , Yuzhi AU - Xue , Guanyu JO - East Asian Journal on Applied Mathematics VL - 3 SP - 540 EP - 559 PY - 2021 DA - 2021/05 SN - 11 DO - http://doi.org/10.4208/eajam.090820.311220 UR - https://global-sci.org/intro/article_detail/eajam/19140.html KW - Crank-Nicolson method, two-grid algorithm, finite volume element method, error estimates, nonlinear parabolic equations. AB -

A two-grid finite volume element algorithm based on Crank-Nicolson scheme for nonlinear parabolic equations is proposed. In this method, the nonlinear problem is solved on a coarse grid of size $H$ and a linear problem is considered on a fine grid of size $h$ by using the coarse-grid solution and one Newton iteration. This helps to improve the computing efficiency while keeping the accuracy. It is proved that the two-grid method can achieve asymptotically optimal error estimates in spaces and second order accuracy in time. Numerical results are consistent with the theoretical findings.

Yunjie Gong, Chuanjun Chen, Yuzhi Lou & Guanyu Xue. (2021). Crank-Nicolson Method of a Two-Grid Finite Volume Element Algorithm for Nonlinear Parabolic Equations. East Asian Journal on Applied Mathematics. 11 (3). 540-559. doi:10.4208/eajam.090820.311220
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