时空分数阶扩散方程的扩展混合有限元方法  

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

作　　者：袁琼 杨志伟 付芳芳 Yuan Qiong;Yang Zhiwei;Fu FangFang(Army Artillery Air Defense Academy,Hefei 230031,China)

机构地区：[1]陆军炮兵防空兵学院,合肥230031

出　　处：《数值计算与计算机应用》2021年第3期276-288,共13页Journal on Numerical Methods and Computer Applications

摘　　要：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(θ_(0)I_(x)^(β)+(1-θ)_(x)I_(1)^(β))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(τ^(2-α)+hmin^({k+1,s-1+β/2})),关于函数导数与分数阶数值通量p的收敛阶为O(τ^(2-3α/2)+hmin^({k+1,s-1+β/2}))文中数值实验表明,所提出的L1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-βwith 0<β<1.By introducing the flux of the unknown function p=-K(θ_(0)I_(x)^(β)+(1-θ)_(x)I_(1)^(β))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(τ^(2-α)+hmin^({k+1,s-1+β/2})),the convergence order of the function derivative and the fractional numerical flux p is O(τ^(2-3α/2)+hmin^({k+1,s-1+β/2})).Numerical experiments in the thesis show that the proposed L^(1) fully discrete expanded mixed finite element scheme has an ideal numerical approximation effect.

关 键 词：时空分数阶扩散方程 L^(1)全离散扩展混合有限元格式 数值分析 数值实验 

分 类 号：O241.82[理学—计算数学]