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

Dongyang Shi & Ding Zhang

10.4208/jcm.1212-m3897

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

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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.

  • History

Published online: 2013-06

  • AMS Subject Headings

65N15, 65N30.

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