分布阶波方程全离散有限元方法的高精度分析新途径  

A New Approach of High Accuracy Analysis of Fully Discrete Finite Element Method for Distributed Order Fractional Wave Equations

作　　者：任金城 石东洋 REN Jincheng;SHI Dongyang(College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450045, China;College of Mathematics and statistics, Zhengzhou University, Zhengzhou 450001, China)

机构地区：[1]河南财经政法大学数学与信息科学学院,郑州450046 [2]郑州大学数学与统计学院,郑州450001

出　　处：《应用数学学报》2019年第3期410-424,共15页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(11601119,11671369);河南省高校创新人才支持计划(18HASTIT027);河南财经政法大学青年拔尖人才资助计划资助项目

摘　　要：本文研究了时间分布阶波方程的全离散有限元数值逼近及其高精度误差分析的新途径。首先,基于L1公式离散Caput。时间分数阶导数,构造了时间分布阶波方程的有限元全离散格式,证明了格式的无条件稳定性。然后,利用双线性元的Ritz投影算子Rh和插值算子Ih之间的高精度误差估计,再借助于插值后处理技术得到了在全离散格式下单独利用插值或投影所无法得到的超逼近和超收敛结果。进一步地,将该方法应用于变系数分布阶波方程,也证明了格式的无条件稳定性和超收敛性。最后,对一些常见的单元作了进一步探讨。In this paper, a new approach of numerical fully discrete scheme based on the finite element approximation for the distributed order time fractional diffusion equations is developed and high accuracy error analysis is provided. Firstly, based on the L1 for mula for the approximation of the time distributed order fractional derivative, the fully discrete finite element scheme is derived and the unconditional stability of the scheme is obtained. Secondly, by use of the supercolse estimate between the Ritz projection operator Rh and interpolation operator Ih of the bilinear element and the interpolated post-processing technique, the superclose and superconvergence results for the fully discrete scheme are obtained, which can't be deduced by the interpolation or Ritz projection alone. Furthermore, the proposed method is applied to the equations with variable cofficienet and the unconditio nal st ability and superconvergent eatimates are also proved. Finally, some popular finite elements are investigated.

关 键 词：分布阶 分数阶导数 有限元方法 全离散格式 超收敛 

分 类 号：O242.21[理学—计算数学]