时间分数阶子扩散和扩散- wave equations的Dirichlet-Neumann和Neumann-Neumann波形松弛算法
Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for time fractional sub-diffusion and diffusion-wave equations
摘要：
在本文中，我们研究了时间分数阶子扩散方程和扩散波方程的Dirichlet-Neumann和Neumann-Neumann波形松弛算法在规则区域上的收敛性，其中无量纲扩散系数在不同的子区域上取不同的常数值.我们首先观察到，不同的扩散系数导致不同的松弛因子的最佳收敛。使用这些最佳松弛因子，我们的分析估计缓慢的超线性收敛的时间导数的分数阶接近零时，几乎有限步收敛的阶数接近2时，和之间，超线性收敛变得更快的分数阶的增加。因此，我们成功地捕捉到了估计的收敛速度随时间导数分数阶数变化的过渡，并通过数值实验进行了验证。
英文摘要：
In this article, we have studied the convergence behavior of the Dirichlet-Neumann and Neumann- Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion-wave equations in 1D & 2D for regular domains, where the dimensionless diffusion coefficient takes different constant values in different subdomains. We first observe that different diffusion coefficients lead to different relaxation parameters for optimal convergence. Using these optimal relaxation parameters, our analysis estimates the slow superlinear convergence of the algorithms when the fractional order of the time derivative is close to zero, almost finite step convergence when the order is close to two, and in between, the superlinear convergence becomes faster as fractional order increases. So, we have successfully caught the transition of convergence rate with the change of fractional order of the time derivative in estimates and verified them with numerical experiments.
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