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时空分数阶扩散方程的扩展混合有限元方法

袁琼, 杨志伟, 付芳芳   

  1. 陆军炮兵防空兵学院, 合肥 230031
  • 收稿日期:2020-03-09 出版日期:2021-09-15 发布日期:2021-09-17

袁琼, 杨志伟, 付芳芳. 时空分数阶扩散方程的扩展混合有限元方法[J]. 数值计算与计算机应用, 2021, 42(3): 276-288.

Yuan Qiong, Yang Zhiwei, Fu FangFang. EXPANDED MIXED FINITE ELEMENT FORMULATION FOR TIME-SPACE FRACTIONAL DIFFUSION EQUATIONS[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(3): 276-288.

EXPANDED MIXED FINITE ELEMENT FORMULATION FOR TIME-SPACE FRACTIONAL DIFFUSION EQUATIONS

Yuan Qiong, Yang Zhiwei, Fu FangFang   

  1. Army Artillery Air Defense Academy, Hefei 230031, China
  • Received:2020-03-09 Online:2021-09-15 Published:2021-09-17
文章主要讨论了带有双边Riemann-Liouville分数阶导数的分数阶扩散方程.通过引入未知函数的通量$p=-K ({\theta_0}I_x^\beta+{(1-\theta)_x}I_1^\beta) Du$和导数$q=Du$作为中间变量,建立了相应的鞍点变分格式.基于鞍点格式构造了可同时高精度逼近未知函数,未知函数导数和分数阶通量的$L^1$全离散扩展混合有限元格式.在数值分析中,借助混合元投影算子的投影误差估计得到关于未知函数$u$的收敛阶为$O (\tau^{2-\alpha}+h^{\min\{k+1,s-1+{\beta\over2}\}}),$关于函数导数与分数阶数值通量$p$的收敛阶为$O (\tau^{2-\frac{3\alpha}{2}}+h^{\min\{k+1,s-1+{\beta\over2}\}}).$文中数值实验表明,所提出的$L^1$全离散扩展混合有限元格式具有理想的数值逼近效果.
In this thesis, we consider the following fractional-order conservative diffusion equation of order 2 -β with 0 < β < 1. By introducing the flux of the unknown function $p=-K({\theta _0}I_x^\beta + {(1-\theta)_x}I_1^\beta)Du$ and the derivative q=Du as intermediate variables, we establish the corresponding saddle point variation formulation and the expanded mixed finite element scheme. On the basis of the formulation we constructed the L1 fully discrete expanded mixed finite element scheme. In the numerical analysis, we adopt the idea of mathematical induction to replace the discrete Gronwall inequality. Through the complex analysis and argumentation, we establish the convergence theory of the fully discrete expanded mixed element method and obtain the convergence order of u is $O(\tau^{2-\alpha} +h^{\min\{k+1, s-1+{\beta\over2}\}}),$, the convergence order of the function derivative and the fractional numerical flux p is $O(\tau^{2-\frac{3\alpha}{2}} +h^{\min\{k+1, s-1+{\beta\over2}\}}).$ Numerical experiments in the thesis show that the proposed L1 fully discrete expanded mixed finite element scheme has an ideal numerical approximation effect.

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