分布阶扩散—波动方程的有限元解的误差估计  

ERROR ESTIMATION OF FINITE ELEMENT SOLUTION FOR A DISTRIBUTED-ORDER DIFFUSION-WAVE EQUATION

作　　者：高兴华 李宏 刘洋[2] Gao Xinghua;Li Hong;Liu Yang(School of Mathematical Sciences,Inner Mongolia Normal University,Hohhot 010022,China;School of Mathematical Sciences,Inner Mongolia University,Hohhot 010021,China)

机构地区：[1]内蒙古师范大学数学科学学院,呼和浩特010021 [2]内蒙古大学数学科学学院,呼和浩特010021

出　　处：《计算数学》2021年第4期493-505,共13页Mathematica Numerica Sinica

基　　金：国家自然科学基金(11761053);内蒙古自然科学基金(2021MS01018,2020MS01003);内蒙古草原英才和内蒙古自治区高等学校青年科技英才支持计划(NJYT-17-A07)资助。

摘　　要：本文考虑了分布阶时间分数阶扩散波动方程,其中时间分数阶导数是在Caputo意义上定义的,其阶次α,β分别属于(0,1)和(1,2).文中提出了在计算上行之有效的数值方法来模拟分布阶时间分数阶扩散波动方程.在时间上,通过中点求积公式把分布阶项转换为多项的时间分数阶导数项,并且利用L1和L2公式来近似Caputo分数阶导数;空间上使用Galerkin有限元方法进行离散.给出了基于H^(1)范数的有限元解的稳定性和误差估计的详细证明,最后的数值算例结果说明了理论分析的正确性以及有效性.In this paper,the distributed-order diffusion-wave equation is considered.The time fractional derivatives are defined in the Caputo sense,and their orders α,β belong to the intervals(0,1) and(1,2),respectively.Some computationally effective numerical methods are proposed for simulating the distributed-order time-fractional diffusion-wave equation.In the time direction,the mid-point quadrature rule is used to transform the distributed-order term into the multi-term time fractional terms,then L_(1) formula and L_(2) formula are chosen to approximate the Caputo fractional derivatives.Further,the spatial direction is discretized by the Galerkin finite element method.The stability and error estimation of the fully discrete scheme based on the H^(1) norm are proved.Finally,a numerical example is given to illustrate the correctness and effectiveness of the theoretical analysis.

关 键 词：分布阶时间分数阶扩散波动方程 中点求积公式 GALERKIN有限元方法 稳定性和误差估计 

分 类 号：O241.82[理学—计算数学]