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.

  • Keywords

Sine-Gordon equations Quasi-Wilson element Semi-discrete and fully-discrete schemes Error estimate and superclose result

  • AMS Subject Headings

65N15 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-31-271, author = {}, 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 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 KW - Quasi-Wilson element KW - Semi-discrete and fully-discrete schemes KW - 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 & Ding Zhang. (2019). 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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