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Volume 26, Issue 3
Optimal Error Estimates of a Linearized Crank-Nicolson Galerkin FEM for the Kuramoto-Tsuzuki Equations

Dongfang Li, Waixiang Cao, Chengjian Zhang & Zhimin Zhang

DOI: 10.4208/cicp.OA-2018-0208

Commun. Comput. Phys., 26 (2019), pp. 838-854.

Published online: 2019-04

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  • Abstract

This paper is concerned with unconditionally optimal error estimate of the linearized Galerkin finite element method for solving the two-dimensional and three-dimensional Kuramoto-Tsuzuki equations, while the classical analysis for these nonlinear problems always requires certain time-step restrictions dependent on the spatial mesh size. The key to our analysis is to obtain the boundedness of the numerical approximation in the maximum norm, by using error estimates in certain norms in the different time level, the corresponding Sobolev embedding theorem, and the inverse inequality. Numerical examples in both 2D and 3D nonlinear problems are given to confirm our theoretical results.

  • Keywords

Unconditionally optimal error estimates, linearized Galerkin finite element method, Kuramoto-Tsuzuki equation, high-dimensional nonlinear problems.

  • AMS Subject Headings

65N30, 65N12, 65N15, 35B45

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-26-838, author = {Dongfang Li, Waixiang Cao, Chengjian Zhang and Zhimin Zhang}, title = {Optimal Error Estimates of a Linearized Crank-Nicolson Galerkin FEM for the Kuramoto-Tsuzuki Equations}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {3}, pages = {838--854}, abstract = {

This paper is concerned with unconditionally optimal error estimate of the linearized Galerkin finite element method for solving the two-dimensional and three-dimensional Kuramoto-Tsuzuki equations, while the classical analysis for these nonlinear problems always requires certain time-step restrictions dependent on the spatial mesh size. The key to our analysis is to obtain the boundedness of the numerical approximation in the maximum norm, by using error estimates in certain norms in the different time level, the corresponding Sobolev embedding theorem, and the inverse inequality. Numerical examples in both 2D and 3D nonlinear problems are given to confirm our theoretical results.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0208}, url = {http://global-sci.org/intro/article_detail/cicp/13149.html} }
TY - JOUR T1 - Optimal Error Estimates of a Linearized Crank-Nicolson Galerkin FEM for the Kuramoto-Tsuzuki Equations AU - Dongfang Li, Waixiang Cao, Chengjian Zhang & Zhimin Zhang JO - Communications in Computational Physics VL - 3 SP - 838 EP - 854 PY - 2019 DA - 2019/04 SN - 26 DO - http://doi.org/10.4208/cicp.OA-2018-0208 UR - https://global-sci.org/intro/article_detail/cicp/13149.html KW - Unconditionally optimal error estimates, linearized Galerkin finite element method, Kuramoto-Tsuzuki equation, high-dimensional nonlinear problems. AB -

This paper is concerned with unconditionally optimal error estimate of the linearized Galerkin finite element method for solving the two-dimensional and three-dimensional Kuramoto-Tsuzuki equations, while the classical analysis for these nonlinear problems always requires certain time-step restrictions dependent on the spatial mesh size. The key to our analysis is to obtain the boundedness of the numerical approximation in the maximum norm, by using error estimates in certain norms in the different time level, the corresponding Sobolev embedding theorem, and the inverse inequality. Numerical examples in both 2D and 3D nonlinear problems are given to confirm our theoretical results.

Dongfang Li, Waixiang Cao, Chengjian Zhang and Zhimin Zhang. (2019). Optimal Error Estimates of a Linearized Crank-Nicolson Galerkin FEM for the Kuramoto-Tsuzuki Equations. Communications in Computational Physics. 26 (3). 838-854. doi:10.4208/cicp.OA-2018-0208
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