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计算数学  2018, Vol. 40 Issue (3): 299-312    DOI:
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多项时间分数阶扩散方程各向异性线性三角元的高精度分析
王芬玲1, 樊明智1, 赵艳敏1, 史争光2, 石东洋3
1. 许昌学院数学与统计学院, 许昌 461000;
2. 西南财经大学 经济数学学院, 成都 611130;
3. 郑州大学数学与统计学院, 郑州 450001
HIGH ACCURACY ANALYSIS OF ANISOTROPIC LINEAR TRIANGULAR ELEMENT FOR MULTI-TERM TIME FRACTIONAL DIFFUSION EQUATIONS
Wang Fenling1, Fan Mingzhi1, Zhan Yanmin1, Shi Zhengguang2, Shi Dongyang3
1. School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China;
2. School of Economic Matnematics, Southwestern University of Finance and Economic, Chengdu 611130, China;
3. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 475001, China
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摘要 在各向异性网格下,针对具有Caputo导数的二维多项时间分数阶扩散方程,给出了线性三角形元的高精度分析.首先,基于线性三角形元和改进的L1格式,建立了一个全离散逼近格式,并证明了其无条件稳定性;其次,利用有限元插值算子与Riesz投影算子之间的关系及相关的高精度结果,导出了超逼近性质.进而,借助于插值后处理技术得到了超收敛估计.值得指出的是,单独利用插值算子或Riesz投影都无法得到上述超逼近和超收敛结果.最后,利用数值算例验证了理论分析的正确性.此外,对一些常见的有限单元在该方程的数值逼近方面,作了进一步探讨.
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关键词多项时间分数阶扩散方程   各向异性三角形元   全离散格式   无条件稳定   超逼近和超收敛     
Abstract: High accuracy analysis of linear triangular element is proposed for two-dimensional multi-term time fractional diffusion equations with Caputo fractional derivative on anisotropic meshes. Firstly, based on linear triangular element and modified L1 scheme, a fully-discrete approximate scheme is established and the unconditional stability analysis is investigated. Secondly, by use of the relationship between the interpolation operator and Riesz projection operator, superclose property is derived by related known high accuracy results. Moreover, the superconvergence estimate is obtained through the interpolation postprocessing technique. It is worth mentioning that the above superclose and superconvergence results will not be derived by the interpolation operator and Riesz projection operator alone. Finally, numerical results are provided to confirm the validity of our theoretical analysis. Furthermore, some popular finite elements of numerical approximation for the focused equation are investigated.
Key wordsmulti-term time fractional diffusion equations   anisotropic triangular element   fully-discrete scheme   unconditional stability   superclose and superconvergence   
收稿日期: 2017-06-11;
基金资助:

国家自然科学基金(11101381;11471296);河南省教育厅项目(16A110022;17A110011).

通讯作者: 樊明智,Email:mathfanmz@163.com.     E-mail: mathfanmz@163.com
引用本文:   
. 多项时间分数阶扩散方程各向异性线性三角元的高精度分析[J]. 计算数学, 2018, 40(3): 299-312.
. HIGH ACCURACY ANALYSIS OF ANISOTROPIC LINEAR TRIANGULAR ELEMENT FOR MULTI-TERM TIME FRACTIONAL DIFFUSION EQUATIONS[J]. Mathematica Numerica Sinica, 2018, 40(3): 299-312.
 
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