Direct discontinuous Galerkin method for solving nonlinear time fractional diffusion equation with weak singularity solution
Direct discontinuous Galerkin method for solving nonlinear time fractional diffusion equation with weak singularity solution
复制标题
DOI：
10.1016/j.aml.2019.106111
复制
发表时间：
2020-04
期刊：
影响因子：
--
通讯作者：
Jincheng Ren;Chaobao Huang;Na An
中科院分区：
文献类型：
--
作者：
Jincheng Ren;Chaobao Huang;Na An
本文考虑了阶为α∈(0,1)的Caputo分数阶导数的非线性时间分数扩散方程。利用时间上的梯度网格上著名的卡普托导数的1型公式，空间上均匀网格上的直接不连续伽辽金（DDG）方法，以及非线性项的牛顿线性化逼近方法，构造了一个完全离散的DDG格式。它的误差在每一个时间水平t n是有界的l2 （Ω）范数通过一个非平凡的投影到一个未知的解到有限元空间。然后，通过选择合适的分级网格，证明了最优误差估计。数值实验验证了我们的分析是准确的。
In this work, the nonlinear time fractional diffusion equation with Caputo fractional derivative of order α∈(0, 1) is considered. By the well-known L1-type formula of Caputo derivative on a graded mesh in time, a direct discontinuous Galerkin (DDG) method on a uniform mesh in space, and the Newton linearization method approximation of the nonlinear term, a fully discrete DDG scheme is constructed. Its error at each time level t n is bounded in the L 2 (Ω) norm by means of a non-trivial projection of an unknown solution into the finite element space. Then, the optimal error estimate is proved by choosing a suitable graded mesh. Numerical experiments are presented to verify that our analysis is sharp.