分布阶扩散-波动方程的有限元解的误差估计

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$alpha，beta$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.

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

In this paper, the distributed-order diffusion-wave equation is considered. The time fractional derivatives are defined in the Caputo sense, and their orders $alpha$, $beta$ 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 $L1$ formula and $L2$ 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.