抛物型积分微分方程新混合元格式的超逼近分析  

SUPERCLOSE ANALYSIS OF A NEW MIXED FINITE ELEMENT SCHEME FOR PARABOLIC INTERGRO-DIFFERENTIAL EQUATIONS

作　　者：赵艳敏[1] 石东伟[2] 王芬玲[1] 

机构地区：[1]许昌学院数学与统计学院,许昌461000 [2]河南科技学院数学系,新乡453003

出　　处：《系统科学与数学》2016年第4期591-604,共14页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金(11101381;11271340);许昌学院杰出青年骨干人才培养计划;河南省教育厅自然学科基金(14A110009)资助课题

摘　　要：基于双二次元及其梯度空间,建立了抛物型积分微分方程的一种新混合有限元逼近格式.在不需要Ritz-Volterra投影的前提下,直接利用双二次元插值的高精度结果及关于时间变量的导数转移技巧,在半离散格式下,得到了原始变量u和中间变量p=▽u+integral from n=0 to t▽u(s)ds分别关于H^1模和L^2模的O(h^4)阶超逼近结果,相比插值误差估计,提高了二阶精度.与此同时,对向后Euler格式,导出了u和p分别在H^1模与L^2模意义下的O(h^4+τ)阶超逼近;对Crank-Nicolson-Galerkin格式,在L^2模意义下证明了u和p分别具有O(h^4+τ~2)和O(h^3+τ~2)阶的超逼近性质.其中,h,τ分别表示空间剖分参数和时间步长,t代表时间变量.Based on spaces of biquadratic finite element and its gradient, a new mixed finite element approximate formulation is established for parabolic integrodifferential equations. Directly, by use of high accuracy results for interpolation of biquadratic finite element and derivative transferring technique with respect to the time variable, the superclose results with O（h4） order of original variable u in Hi-norm and intermediate variable p = u ＋ ∫ t 0 u（s）ds in L2-norm are obtained under semidiscrete scheme without Ritz-Volterra projection, which are two orders higher than interpolation error estimates. At the same time, we arrive at the superclose properties with O（h4 ＋ T） order of u in Hi-norm and p in L2-norm for backward Euler scheme. And then, for Crank-Nicolson-Galerkin fully-discrete scheme, it is proved that u and have superclose properties with orders O（h4 ＋ r2） and O（h3 ＋ r2）, respectively, in L2-norm. In this paper, h and T are parameter of subdivisions in space and time step, respectively; and t denotes the time variable.

关 键 词：抛物型积分微分方程 新混合元 双二次元 超逼近 半离散及全离散格式. 

分 类 号：O241.82[理学—计算数学]