时间分数阶扩散方程的一类三次有限体积元方法
A Cubic Finite Volume Element Method for the Time Fractional Diffusion Equation
DOI: 10.12677/AAM.2022.118582, PDF,    科研立项经费支持
作者: 何斯日古楞, 高晶英：呼和浩特民族学院，数学与大数据学院，内蒙古 呼和浩特；澈力木格：呼和浩特民族学院，计算机科学与信息工程学院，内蒙古 呼和浩特
关键词: 时间分数阶扩散方程；三次有限体积元法；L1-公式；Time Fractional Diffusion Equation； Cubic Finite Volume Element Method； L1-Formula
摘要: 本文基于应力佳点对偶网格剖分以及分片三次Lagrange插值试探函数空间和分片常数检验函数空间的三次有限体积元法和Caputo导数的L1-逼近公式构造数值格式求解一维时间分数阶扩散方程，并证明了格式的L2范数在时间和空间方向分别 阶和四阶收敛误差估计。通过数值实验验证了理论分析结果以及所提格式的有效性。
Abstract: In this article, the one-dimensional time fractional diffusion equation is solved by a numerical scheme which is constructed by the L1-formula of approximating the Caputo fractional derivative and a cubic finite volume element method. The cubic finite volume element method is based on the optimal stress points dual partition, and the trial function space of piecewise cubic Lagrange inter-polation and the test function space of piecewise constant. The L2-norm error estimate of fourth or-der convergence in space and order convergence in time is proved. Numerical experiments are given to verify the effectiveness of the theoretical analysis results and the proposed scheme.
文章引用：何斯日古楞, 澈力木格, 高晶英. 时间分数阶扩散方程的一类三次有限体积元方法[J]. 应用数学进展, 2022, 11(8): 5529-5535. https://doi.org/10.12677/AAM.2022.118582

参考文献

[1] 李荣华, 陈仲英. 微分方程的广义差分法[M]. 长春: 吉林大学出版社, 1994.
[2] 李荣华, 祝丕奇. 二阶椭圆偏微分方程的广义差分方法(I)-三角网情形[J]. 高等学校计算数学学报, 1982, 4(2): 140-152.
[3] Tian, M. and Chen, Z. (1991) Quadratical Element Generalized Differential Methods for Elliptic Equations. Numerical Mathematics: A Journal of Chinese Universities, 13, 99-113.
[4] Liebau, F. (1996) The Finite Volume Element Method with Quadratic Basis Functions. Computing, 57, 281-299. [Google Scholar] [CrossRef] 
[5] Chen, Z., Wu, J. and Xu, Y. (2012) Higher-Order Finite Volume Meth-ods for Elliptic Boundary Value Problems. Advances in Computational Mathematics, 37, 191-253. [Google Scholar] [CrossRef] 
[6] Wang X. and Li, Y.H. (2016) L2 Error Estimates for High Order Finite Volume Methods on Triangular Meshes. SIAM Journal on Numerical Analysis, 54, 2729-2749. [Google Scholar] [CrossRef] 
[7] Zou, Q. (2017) An Unconditionally Stable Quadritic Finite Volume Scheme Over Triangular Meshes for Elliptic Equations. Journal of Scientific Computing, 70, 112-124. [Google Scholar] [CrossRef] 
[8] Wang, T.K. and Gu, Y. (2010) Superconvergent Biquadratic Fi-nite Volume Element Method for Two-Dimensional Poisson's Equations. Journal of Computational and Applied Math-ematics, 234, 447-460. [Google Scholar] [CrossRef] 
[9] Gao, G.H. and Wang, T.K. (2010) Cubic Superconvergent Finite Volume Element Method for One-dimensional Elliptic and Parabolic Equations. Journal of Computational and Applied Mathematics, 233, 2285-2301. [Google Scholar] [CrossRef] 
[10] 王星, 高广花, 王同科. 半线性抛物问题的一类三次有限体积元方法[J]. 辽宁工程技术大学学报(自然科学版), 2015, 34(2): 281-284.
[11] 于长华, 王晓玲, 李永海. 解两点边值问题的一类修改的三次有限体积元方法[J]. 计算数学, 2010, 32(4): 385-398.
[12] Yu, C.H. and Li, Y.H. (2011) Biquadratic Finite Volume Element Methods Based on Optimal Stress Points for Parabolic Problem. Journal of Compu-tational and Applied Mathematics, 236, 1055-1068. [Google Scholar] [CrossRef] 
[13] 杨凯丽, 何斯日古楞, 肖宇宇. 对流扩散方程的三次有限体积元法[J]. 内蒙古大学学报(自然科学版), 2021, 52(3): 250-256.
[14] 肖宇宇, 何斯日古楞, 杨凯丽. 对流扩散方程的时间间断时空有限体积元法[J]. 高校应用数学学报, 2021, 36(2): 179-192.
[15] Jiang, Y. J. and Ma, J.T. (2011) High-Order Finite Element Methods for Time-Fractional Partial Differential Equations. Journal of Computational Applied Mathematics, 235, 3285-3290. [Google Scholar] [CrossRef] 

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