非线性Schrdinger方程新混合元方法的高精度分析  

HIGH ACCURACY ANALYSIS OF A NEW MIXED FINITE ELEMENT METHOD FOR NONLINEAR SCHRDINGER EQUATION

作　　者：赵艳敏[1,2] 石东洋[2] 王芬玲[1] 

机构地区：[1]许昌学院数学与统计学院,河南许昌461000 [2]郑州大学数学与统计学院,郑州450001

出　　处：《计算数学》2015年第2期162-178,共17页Mathematica Numerica Sinica

基　　金：国家自然科学基金(11101381;11101384;11271430);许昌学院杰出青年骨干人才培养计划资助项目

摘　　要：基于双线性元及其梯度所属空间,建立了非线性Schrdinger方程的自由度少且易满足B-B条件的新混合元格式.首先,利用双线性元的高精度分析和导数转移技巧,在半离散格式下,导出了原始变量在H^1模及流量在L^2模意义下的超逼近性质,进而,借助于插值后处理算子,得到了整体超收敛结果.最后,对向后:Euler和Crank-Nicolson-Galerkin全离散格式分别给出了原始变量的H^1模及L^2模和流量的L^2模误差分析,并通过数值算例,表明逼近格式是高效的.Based on spaces of bilinear finite element and its gradient, a new mixed finite element approximate formulation with less degree of freedoms is established for nonlinear SchrSdinger equation, which can satisfy B-B condition easily. Firstly, under semi-discrete scheme, superclose properties of original variable in H^1-norm and flux in L^2-norm are derived by use of high accuracy analysis of bilinear finite element and derivative transferring technique. Moreover, the global superconvergence result is obtained by interpolation postprocessing operators. Finally, the error analysis of original variable in H^1-norm and L^2-norm and flux in L^2-norm for backward Euler and Crank-Nicolson-Galerkin fully-discrete schemes are presented, respectively. And it is shown that the proposed approximate schemes are effective by numerical examples.

关 键 词：非线性SCHRODINGER方程 双线性元 新混合元方法 超逼近和超收敛 半离散和全离散格式 

分 类 号：O241.82[理学—计算数学]