一类四阶抛物方程一个低阶非协调混合元方法的超收敛分析  

Superconvergence Analysis of a Low Order Nonconforming Mixed Finite Element Method for a Class of Fourth-order Parabolic Equations

作　　者：杨晓侠[1] 石东洋[2] 张芳[2] 

机构地区：[1]平顶山学院数学与信息科学学院,河南平顶山467000 [2]郑州大学数学与统计学院,河南郑州450001

出　　处：《应用数学》2016年第2期370-380,共11页Mathematica Applicata

基　　金：国家自然科学基金(11271340)

摘　　要：对一类四阶抛物方程利用EQ_1^(rot)元和零阶Raviart-Thomas元提出一个低阶非协调混合元逼近格式.首先证明半离散格式逼近解的存在唯一性.其次,基于上述两个单元的高精度分析,利用对时间变量的导数转移技巧并借助插值后处理技术,在半离散格式下得到了原始变量u,中间变量v=—△u的H^1-模意义下以及流量=—▽u的L^2-模意义下O(h^2)阶的超逼近性质和超收敛结果.最后,证明向后Euler全离散格式逼近解的存在唯一性,并通过采用一个新的分裂技巧,导出u和v在H^1-模意义下以及在L^2-模意义下关于h的无条件的O(h^2+τ)阶的超逼近性质和超收敛结果.这里,h及τ分别表示空间剖分参数和时间步长.In this paper,with the help of EQ_1^（rot） and zero order Raviart-Thomas elements,a low order nonconforming mixed finite element approximation scheme is proposed for a class of fourth-order parabolic equations.Firstly,the existence and uniqueness of approximation solution are proved for semidiscrete scheme.Secondly,based on the high accuracy analysis of the about two elements,using derivative delivery technique with respect to the time variable,through interpolated postprocessing approach,the superclose properties and superconvergence results with order O（h^2） for both the primitive solution u and the intermediate variable v = —△u in H^1-norm,flux  = —▽u in L^2-norm are obtained,respectively.Finally,for backward Euler full-discrete scheme,the existence and uniqueness of approximation solution are shown.At the same time,by use of a new splitting technique,the superclose properties and superconvergence results with order O（h^2 ＋τ） for both u and v in H^1-norm, in L^2-norm are derived unconditionally with respect to h.Here,h and τ are parameters of the subdivision in space and time step,respectively.

关 键 词：四阶抛物方程 非协调混合元方法 半离散和全离散格式 超收敛 

分 类 号：O242.21[理学—计算数学]