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Volume 31, Issue 3
Approximation of Nonconforming Quasi-Wilson Element for Sine-Gordon Equations

Dongyang Shi & Ding Zhang

J. Comp. Math., 31 (2013), pp. 271-282.

Published online: 2013-06

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  • Abstract

In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product $(∇(u-I_h^1u),∇ v_h)$ and the consistency error can be estimated as order $O(h^2)$ in broken $H^1$-norm/$L^2$-norm when  $u$ ∈ $H^3(Ω)/H^4(Ω)$, where $I_h^1u$ is the bilinear interpolation of $u, v_h$ belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order $O(h^2)$ for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order $O(h^2+τ^2)$ is obtained for the rectangular partition when $u$ ∈ $H^4(Ω)$, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.

  • AMS Subject Headings

65N15, 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-31-271, author = {Dongyang Shi and Ding Zhang}, title = {Approximation of Nonconforming Quasi-Wilson Element for Sine-Gordon Equations}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {3}, pages = {271--282}, abstract = {

In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product $(∇(u-I_h^1u),∇ v_h)$ and the consistency error can be estimated as order $O(h^2)$ in broken $H^1$-norm/$L^2$-norm when  $u$ ∈ $H^3(Ω)/H^4(Ω)$, where $I_h^1u$ is the bilinear interpolation of $u, v_h$ belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order $O(h^2)$ for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order $O(h^2+τ^2)$ is obtained for the rectangular partition when $u$ ∈ $H^4(Ω)$, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1212-m3897}, url = {http://global-sci.org/intro/article_detail/jcm/9734.html} }
TY - JOUR T1 - Approximation of Nonconforming Quasi-Wilson Element for Sine-Gordon Equations AU - Dongyang Shi & Ding Zhang JO - Journal of Computational Mathematics VL - 3 SP - 271 EP - 282 PY - 2013 DA - 2013/06 SN - 31 DO - http://doi.org/10.4208/jcm.1212-m3897 UR - https://global-sci.org/intro/article_detail/jcm/9734.html KW - Sine-Gordon equations, Quasi-Wilson element, Semi-discrete and fully-discrete schemes, Error estimate and superclose result. AB -

In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product $(∇(u-I_h^1u),∇ v_h)$ and the consistency error can be estimated as order $O(h^2)$ in broken $H^1$-norm/$L^2$-norm when  $u$ ∈ $H^3(Ω)/H^4(Ω)$, where $I_h^1u$ is the bilinear interpolation of $u, v_h$ belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order $O(h^2)$ for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order $O(h^2+τ^2)$ is obtained for the rectangular partition when $u$ ∈ $H^4(Ω)$, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.

Dongyang Shi and Ding Zhang. (2013). Approximation of Nonconforming Quasi-Wilson Element for Sine-Gordon Equations. Journal of Computational Mathematics. 31 (3). 271-282. doi:10.4208/jcm.1212-m3897
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