Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction-Subdiffusion Equations
Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction-Subdiffusion Equations
复制标题
非线性时间分数阶反应-细分扩散方程的线性伽辽金方法的无条件最优误差估计
DOI：
10.1007/s10915-018-0642-9
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发表时间：
2018
影响因子：
2.5
通讯作者：
Zhang Zhimin
中科院分区：
文献类型：
--
作者：
Li Dongfang;Zhang Jiwei;Zhang Zhimin
本文讨论了线性化Galerkin有限元方法数值求解某些多维分数阶反应次扩散方程的无条件最优误差估计，而对于多维非线性抛物型问题的数值逼近的经典分析通常要求对时间步长的限制，这依赖于空间网格的大小。要得到无条件最优的误差估计，关键是要得到数值解在范数意义下的有界性。为此，我们引入了时间离散的椭圆型方程，构造了非局部问题的能量函数，并适当地处理了误差求和问题。与整数阶非线性问题相比，时间分数阶导数中的非局部卷积给数值格式的开发和分析带来了很大的困难。数值算例验证了我们的理论结果。
This paper is concerned with unconditionally optimal error estimates of linearized Galerkin finite element methods to numerically solve some multi-dimensional fractional reaction–subdiffusion equations, while the classical analysis for numerical approximation of multi-dimensional nonlinear parabolic problems usually require a restriction on the time-step, which is dependent on the spatial grid size. To obtain the unconditionally optimal error estimates, the key point is to obtain the boundedness of numerical solutions in the-norm. For this, we introduce a time-discrete elliptic equation, construct an energy function for the nonlocal problem, and handle the error summation properly. Compared with integer-order nonlinear problems, the nonlocal convolution in the time fractional derivative causes much difficulties in developing and analyzing numerical schemes. Numerical examples are given to validate our theoretical results.