Approximation of Nonconforming Quasi-Wilson Element for Sine-Gordon Equations

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.