删除或更新信息,请邮件至freekaoyan#163.com(#换成@)

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

本站小编 Free考研考试/2021-12-27

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
收稿日期:2017-06-11出版日期:2018-09-15发布日期:2018-08-08
通讯作者:樊明智,Email:mathfanmz@163.com.

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


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
Received:2017-06-11Online:2018-09-15Published:2018-08-08







摘要



编辑推荐
-->


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.
MR(2010)主题分类:
65N30
65Z05

分享此文:


()

[1] Samko S G,Kilbas A A,Marichev O I.Fractional Integrals and Derivatives:Theory and Applications[M].Gordon and Breach Science Publishers:Amsterdam,1993.

[2] Miller K S,Ross B.An introdution to fractional calculus and fractional differential equations[M].John Wiley:New York,1993.

[3] Scalas E,Gorenflo R,Mainardi F.Fractional calculus and continuous-time finance[J].Physica A,2000,284(1-4):376-384.

[4] Benson D A,Wheatcraft S W,Meerchaert M M.Application of a fractional advection-dispersion equation[J].Water Resources Research, 2000,36(2):1403-1412.

[5] Chechkin A V,Gorenflo R,Sokolov I M.Fractional diffusion in inhomogeneous media[J].Physica A,2005,38(42):679-684.

[6] Kilbas A A,Srivastava H M,Trujillo J J.Theory and Applications of Fractional Differential Equations[M].Elsevier:Amsterdam,2006.

[7] Bueno-Orovio A,Kay D,Grav V,et al.Fractional diffusion models of cardiac electrical propagation:role of structural heterogeneity in dispersion of repolarization[J].Journal of the Royal Society, Interface,2014,11(97):20140352.

[8] Podlubny I.Fractional Differential Equations[M].Academic Press:San Diego,1999.

[9] Li Z Y,Liu Y K,Yamamoto M.Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients[J].Applied Mathematics and Computation,2015,257(15):381-397.

[10] Luchko Y.Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation[J].Journal of Mathematical Analysis and Applications,2011,374(2):538-548.

[11] Mohammed A R,Luchko Y.Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives[J]. Applied Mathematics and Computation,2015,257(15):40-51.

[12] Ming C Y,Liu F W,Zheng L C,Ian T,Vo A.Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelasticfluid[J].Computersand & Mathematics with Applications,2016,72(9):2084-2097.

[13] Li G S,Chun L S,Jia X Z,Du D H.Numerical solution to the multi-term time fractional diffusion equation in a finite domain[J].Numerical Mathematics Theory Methods & Applications,2016,9(3):337-357.

[14] Jin B T,Raytcho L,Liu Y K,Zhou Z.The Galerkin finite element method for a multi-term time-fractional diffusion equation[J].Journal of Computational Physics,2015,281:825-843.

[15] Liu Y,Du Y W,Li H.Finite Difference/Finite Element Method for a Nonlinear Time-Fractional Fourth-Order Reaction Diffusion Problem[J]. Computers & Mathematics with Applications.2015,70(4):573-591.

[16] Liu Y,Fang Z C,Li H.A mixed finite element method for a time-fractional fourth-order partial differential equation[J]. Applied Mathematics and Computation,2014,243(15):703-717.

[17] 张铁.抛物型积分-微分方程有限元近似的超收敛性质[J].高等学校计算数学学报.2001,23(3):193-201.

[18] Chen H T,Lin Q,Zhou J M,Wang H.Uniform error estimates for triangle finite element solutions of advection diffusion equations[J].Advance Computational Mathematics.2013,38(1):83-100.

[19] Lin Q,Wang H,Zhang S H.Uniform optimal-order estimates for finite element methods foradvection-diffusion equations[J].Journal of Systems Science and Complexity,2009,22(4):555-559.

[20] 林群,王宏,周俊明,张书华,陈宏焘.对流扩散方程三角形有限元解的一致估计[J].数学的实践与认识,2011,41(19):173-184.

[21] 石东洋,梁慧.各向异性网格下线性三角形的超收敛分析[J].工程数学学报,2007,24(3):487-493.

[22] 石东洋,王芬玲,赵艳敏.非线性sine-Gordon方程的各向异性线性元高精度分析新模式[J].计算数学,2014,36(3):245-256.

[23] Shi D Y,Wang P L,Zhao Y M.Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation[J].Applied Mathematics Letters,2014,38(38):129-134.

[24] Zhao Y M,Bu W P,Huang J F.Finite element method for two-dimensional space-fractional advection-dispersion equations[J]. Applied Mathematics and Computation,2015,257(15):553-565.

[25] .Jiang Y J,Ma J T.High-order finite element methods for time-fractional partial differential equations[J]. Joural of Computation and Applied Mathematics,2011,235(11):3285-3290.

[26] Bu W P,Liu X T,Tang Y F,Yang J Y.Finite element multigrid method for multi-term time fractional advection diffusion equations[J].Intemational Journal of Modeling Simulation, & Scientific Computing,2015, 6(1):1540001.

[27] Bu W P,Xiao A G,Zeng W.Finite difference/finite element methods for distributed-order time fractional diffusion equations[J].Journal of Scientific Computing,2017,72(1):422-441.

[28] 林群,严宁宁.高效有限元构造与分析[M].河北大学出版社:保定,1996.

[29] 石东洋,梁慧.一个新的非常规Hermite型各向异性矩形元的超收敛分析及外推[J].计算数学,2005,27(4):369-382.

[30] Chen S C,Shi D Y.Accuracy analysis for quasi-Wison element[J].Acta Mathematica Scientia,2000,20(1):44-48.

[31] Chen S C,Shi D Y,Zhao Y C.Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes[J].IMA Journal of Numerical Analysis,2004,24(1):77-95.

[32] Shi D Y,Wang F L,Zhao Y M.Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations[J].Acta Mathematicae Applicatae Sinica,2013,29(2):403-414.

[33] Shi D Y,Pei L F.Nonconforming quadrilateral finite element method for a class of nonlinear sine-Gordon equations[J]. Applied Mathematics and Computation,2013,219(17):9447-9460.

[34] Knobloch P,Tobiska L.The P1rmmod element:a new nonconforming finite element for convection-diffusion problems[J]. SIAM Journal on Numerical Analysis,2003,41(2):436-456.

[35] Shi D Y,Liang H.Superconvergence analysis Wilson element on anisotropic meshes[J].Applied Mathematics and Mechanics,2007, 28(1):119-125.

[36] 石东洋,郝晓斌.Sobolev型方程各向异性Carey元的高精度分析[J]. 工程数学学报,2009,26(6):1021-1026.

[1]贾东旭, 盛志强, 袁光伟. 扩散方程一种无条件稳定的保正并行有限差分方法[J]. 计算数学, 2019, 41(3): 242-258.
[2]王俊俊, 李庆富, 石东洋. 非线性抛物方程混合有限元方法的高精度分析[J]. 计算数学, 2019, 41(2): 191-211.
[3]刘金存, 李宏, 刘洋, 何斯日古楞. 非线性分数阶反应扩散方程组的间断时空有限元方法[J]. 计算数学, 2016, 38(2): 143-160.
[4]石东洋, 张厚超, 王瑜. 一类非线性四阶双曲方程扩展的混合元方法的超收敛分析[J]. 计算数学, 2016, 38(1): 65-82.
[5]杭旭登. Du Fort-Frankel格式及DFF-I并行格式的稳定性[J]. 计算数学, 2015, 37(3): 273-285.
[6]赵艳敏, 石东洋, 王芬玲. 非线性Schrödinger方程新混合元方法的高精度分析[J]. 计算数学, 2015, 37(2): 162-178.
[7]石东洋, 王芬玲, 樊明智, 赵艳敏. sine-Gordon方程的最低阶各向异性混合元高精度分析新途径[J]. 计算数学, 2015, 37(2): 148-161.
[8]石东洋, 王芬玲, 赵艳敏. 非线性sine-Gordon方程的各向异性线性元高精度分析新模式[J]. 计算数学, 2014, 36(3): 245-256.
[9]李宏, 罗振东, 安静, 孙萍. Sobolev方程的全离散有限体积元格式及数值模拟[J]. 计算数学, 2012, 34(2): 163-172.
[10]方志朝, 李宏, 刘洋. 四阶强阻尼波动方程的混合控制体积法[J]. 计算数学, 2011, 33(4): 409-422.
[11]安静, 孙萍, 罗振东, 黄晓鸣. 非定常Stokes方程的稳定化全离散有限体积元格式[J]. 计算数学, 2011, 33(2): 213-224.
[12]田向军,谢正辉,罗振东,朱江. 非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅲ):时间二阶精度的全离散格式[J]. 计算数学, 2004, 26(3): 257-276.
[13]程晓良,叶兴德. 形状记忆合金问题的有限元逼近[J]. 计算数学, 2000, 22(1): 41-48.

--> -->
阅读次数
全文







摘要





Cited

Shared






PDF全文下载地址:

http://www.computmath.com/jssx/CN/article/downloadArticleFile.do?attachType=PDF&id=234
相关话题/数学 计算 分数 工程 统计学院