时间分数阶扩散方程分裂正定混合元有限元方法  

A SPLITTING POSITIVE DEFINITE MIXED FINITE ELEMENT METHOD FOR TIME FRACTIONAL DIFFUSION EQUATIONS

作　　者：唐跃龙[1] 华玉春[1] Tang Yuelong;Hua Yuchun(Hunan University of Science and Engineering,Yongzhou 425199)

机构地区：[1]湖南科技学院,永州425199

出　　处：《高等学校计算数学学报》2022年第4期299-310,共12页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金项目(11401201);湖南省自然科学基金项目(2020JJ4323);湖南省教育厅科学研究项目(20A211,20C0854);湖南科技学院科学研究项目(18XKY063,20XKY059)。

摘　　要：1引言众所周知,由于经典混合有限元方法所用到的两个混合有限元离散空间通常需要满足LBB条件[1],使得可选择的逼近空间非常有限,同时计算成本很高.因此许多研究者考虑对其改进,希望避免LBB条件的限制,比较成功的改进思路主要有两个,一是1998年Pani在文[2]中提出的H1混合有限元方法,该方法所用的两个混合有限元逼近空间不要求满足LBB条件,且能同时得到未知函数及其通量的HH1模误差估计,但对原问题解的正则性有了更高的要求.Based on the splitting positive definite mixed variational form,we construct a fully discrete splitting positive definite mixed finite element approximation scheme for time fractional diffusion equations,where the space derivative are approximated by splitting positive mixed finite element and the time fractional derivative is evaluated by the L1 approximation on graded mesh.Due to the introduction of splitting technique,the LBB condition between the two mixed finite approximation spaces is not necessary.Therefore,the selection of mixed element approximation space is more abundant.By utilizing the regularity of the exact solution and the approximation properties of mixed finite element spaces and some projection operators,we obtain the stability and convergence results of the splitting positive definite mixed element method for the time fractional diffusion equation.Two numerical examples are provided to validate our theoretical results.

关 键 词：混合有限元方法 未知函数 解的正则性 离散空间 逼近空间 可选择的 计算成本 改进思路 

分 类 号：O241.1[理学—计算数学]