一类非线性广义Sine-Gordon方程的有限元超收敛分析  

Superconvergence Analysis of a Finite Element to a Class of Generalized Nonlinear Sine-Gordon Equations

作　　者：梁洪亮[1] 赵树理[1] 

机构地区：[1]商丘师范学院数学系,河南商丘476000

出　　处：《四川师范大学学报（自然科学版）》2012年第5期610-614,共5页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(10971203);河南省自然科学基金(092300410148);河南省教育厅自然科学基金(2011A110013)资助项目

摘　　要：Sine-Gordon方程在许多重要的数学物理问题上都有着重要的应用,其数值解的研究已有许多结果,但都是在正则网格下的.在各向异性网格下,利用双线性有限元方法研究了一类更广泛的二维非线性广义Sine-Gordon方程.首先,讨论其在半离散格式下解的收敛性,得到了和在传统的正则网格下相一致的收敛性结果;其次,在不借助Ritz投影的情况下,利用插值算子的特殊性质得到了解u的超逼近性质;最后,通过构造一个具有各向异性特征的插值后处理算子导出了关于u的整体超收敛结果.Sine-Gordon equations play an important role in many important mathematical physics problems. There are many results about their numerical solutions which are obtained under traditional regular meshes. In this paper, the generalized nonlinear Sine-Gor- don equations are discussed by using bilinear finite element under anisotropic meshes. Firstly, the same convergence result as tradition- al one under regular meshes about these equations solution is obtained. Secondly, the superapproximatiou of the solution u is obtained by virtue of the interpolation operator but Ritz projection. Finally, based on the interpolated postprocessing technique, the global super- convergence is derived.

关 键 词：非线性Sine-Gordon方程 双线性有限元 各向异性 半离散 超收敛 

分 类 号：O242.21[理学—计算数学]