Applied Numerical Mathematics ( IF 2.8 ) Pub Date : 2021-01-11 , DOI: 10.1016/j.apnum.2021.01.001 Jinhong Jia , Hong Wang , Xiangcheng Zheng
We develop a preconditioned fast divided-and-conquer finite element approximation for the initial-boundary value problem of variable-order time-fractional diffusion equations. Due to the impact of the time-dependent variable order, the coefficient matrix of the resulting all-at-once system does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires computational complexity and memory with M and N being the numbers of degrees of freedom in space and time, respectively. Furthermore, a preconditioner is introduced to reduce the number of iterations caused by the bad condition number of the coefficient matrix. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.
中文翻译:
多维空间中变阶时间分数阶扩散方程的预处理快速有限元逼近
针对可变阶时间分数阶扩散方程的初边值问题,我们开发了一种预处理的快速除数有限元逼近。由于时间相关变量阶的影响,最终的一次性系统的系数矩阵不具有类似于Toeplitz的结构。在本文中,我们通过将Toeplitz矩阵之和与对角矩阵相乘,得出系数矩阵的快速近似。我们证明近似值与原始问题渐近一致,这需要 计算复杂度和 存储器与中号和Ñ分别为的自由度的数量在空间和时间。此外,引入了预处理器以减少由系数矩阵的不良条件数引起的迭代次数。数值实验表明了该方法的有效性和有效性。