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Volume 10, Issue 5
A New Error Analysis of Nonconforming $EQ^{rot}_1$ FEM for Nonlinear BBM Equation

Yanhua Shi, Yanmin Zhao, Fenling Wang & Dongyang Shi

Adv. Appl. Math. Mech., 10 (2018), pp. 1227-1246.

Published online: 2018-07

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  • Abstract

Nonconforming $EQ^{rot}_1$ element is applied to solving a kind of nonlinear Benjamin-Bona-Mahony (BBM for short) equation both for space-discrete and fully discrete schemes. A new important estimate is proved, which improves the result of previous works with the exact solution $u$ belonging to $H^2(Ω)$ instead of $H^3(Ω)$. And then, the supercloseness and global superconvergence estimates in broken $H^1$ norm are obtained for space-discrete scheme. Further, the superclose estimates are deduced for backward Euler and Crank-Nicolson schemes. To confirm our theoretical analysis, numerical experiments for backward Euler scheme are executed. It seems that the results presented herein have never been seen for nonconforming FEMs in the existing literature.

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@Article{AAMM-10-1227, author = {Yanhua and Shi and and 9655 and and Yanhua Shi and Yanmin and Zhao and and 9653 and and Yanmin Zhao and Fenling and Wang and and 9654 and and Fenling Wang and Dongyang and Shi and and 9918 and and Dongyang Shi}, title = {A New Error Analysis of Nonconforming $EQ^{rot}_1$ FEM for Nonlinear BBM Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {5}, pages = {1227--1246}, abstract = {

Nonconforming $EQ^{rot}_1$ element is applied to solving a kind of nonlinear Benjamin-Bona-Mahony (BBM for short) equation both for space-discrete and fully discrete schemes. A new important estimate is proved, which improves the result of previous works with the exact solution $u$ belonging to $H^2(Ω)$ instead of $H^3(Ω)$. And then, the supercloseness and global superconvergence estimates in broken $H^1$ norm are obtained for space-discrete scheme. Further, the superclose estimates are deduced for backward Euler and Crank-Nicolson schemes. To confirm our theoretical analysis, numerical experiments for backward Euler scheme are executed. It seems that the results presented herein have never been seen for nonconforming FEMs in the existing literature.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0264}, url = {http://global-sci.org/intro/article_detail/aamm/12596.html} }
TY - JOUR T1 - A New Error Analysis of Nonconforming $EQ^{rot}_1$ FEM for Nonlinear BBM Equation AU - Shi , Yanhua AU - Zhao , Yanmin AU - Wang , Fenling AU - Shi , Dongyang JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1227 EP - 1246 PY - 2018 DA - 2018/07 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2017-0264 UR - https://global-sci.org/intro/article_detail/aamm/12596.html KW - AB -

Nonconforming $EQ^{rot}_1$ element is applied to solving a kind of nonlinear Benjamin-Bona-Mahony (BBM for short) equation both for space-discrete and fully discrete schemes. A new important estimate is proved, which improves the result of previous works with the exact solution $u$ belonging to $H^2(Ω)$ instead of $H^3(Ω)$. And then, the supercloseness and global superconvergence estimates in broken $H^1$ norm are obtained for space-discrete scheme. Further, the superclose estimates are deduced for backward Euler and Crank-Nicolson schemes. To confirm our theoretical analysis, numerical experiments for backward Euler scheme are executed. It seems that the results presented herein have never been seen for nonconforming FEMs in the existing literature.

Yanhua Shi, Yanmin Zhao, Fenling Wang & Dongyang Shi. (1970). A New Error Analysis of Nonconforming $EQ^{rot}_1$ FEM for Nonlinear BBM Equation. Advances in Applied Mathematics and Mechanics. 10 (5). 1227-1246. doi:10.4208/aamm.OA-2017-0264
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