带变系数的多项时间分数阶扩散方程各向异性三角形元的超收敛分析  

Superconvergence Analysis of Anisotropic Linear Triangular Finite Element for Multi-term Time Fractional Diffusion Equations with Variable Coefficient

作　　者：王芬玲[1] 史艳华[1] 史争光 赵艳敏 WANG Fenling;SHI Yanhua;SHI Zhengguang;ZHAO Yanmin(School of Science,Xuchang University,Xuchang 461000,China;School of Economic Mathematics,Southwestern University of Finance and Economics,Chengdu 611130,China)

机构地区：[1]许昌学院数理学院,河南许昌461000 [2]西南财经大学经济数学学院,四川成都611130

出　　处：《应用数学》2022年第4期793-806,共14页Mathematica Applicata

基　　金：Supported by the National Natural Science Foundation of China(11971416);the Key Scientific Research Projects in Universities of Henan Province(21B110007,22A110022);the Foundation for University Key Young Teacher of Henan Province(2019GGJS214);the Scientific Research Innovation Team of Xuchang University(2022CXTD002)。

摘　　要：本文在空间方向上利用有限元方法,时间方向上利用经典的L1逼近格式,对一类带变系数的二维多项时间分数阶扩散方程建立了各向异性网格下的全离散格式.给出了全离散格式在L^(2)和H^(1)范数下稳定性的严格证明.利用线性三角形元的投影算子和插值算子之间的高精度分析结果,得到了在H1范数下具有O(h^(2)+τ^(2−α))的超逼近结果,这里h和τ分别是空间和时间步长.然后借助插值后处理技巧导出了超收敛分析,而该结果如果单独使用插值算子或者投影算子是无法得到的.最后,给出了一些数值结果证明了理论方法的有效性.By using finite element method in spatial direction and classical L1 approximation in temporal direction,a fully-discrete scheme is established for a class of two-dimensional multi-term time fractional diffusion equations with variable coefficient under anisotropic meshes.The stability properties of the approximate scheme are rigorously proved in L^(2)-norm and H^(1)-norm.With the help of high accuracy result between the projection operator and the interpolation operator of the linear triangular finite element,the superclose result with order O(h^(2)+τ^(2−α))in H1-norm is deduced,where h andτare the step sizes in space and time,respectively.Then the global superconvergence is presented by means of interpolation postprocessing technique,which is not deduced by the interpolation and projection alone.Finally,several numerical results are implemented to evaluate the efficiency of the theoretical results.

关 键 词：多项时间分数阶扩散方程 线性三角形元 各向异性网格 稳定性 超收敛 

分 类 号：O242.21[理学—计算数学]