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 1u),∇vh) and the consistency error can be estimated as order O(h 2 ) in broken H 1 − norm/L 2 − norm when u ∈ H 3 ()/H 4 (), where I 1u is the bilinear interpolation of u,vh 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 + � 2 ) is obtained for the rectangular partition when u ∈ H 4 (), 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.