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.

翻译：在本篇文章中,我们研究了Drichlet-Neumann和Neumann-Neumann的波形放松算法在1D和2D中用于常规域的时间折射子扩散和扩散波方程式的趋同行为,在1D和2D中,无维扩散系数在不同子域中采用不同的恒定值。我们首先观察到,不同的扩散系数导致不同的放松参数,以便实现最佳趋同。我们的分析用这些最佳放松参数估计,当时间衍生物的分序接近于零时,这些算法的慢超线趋同速度,当时间衍生物的分序接近于2时,而超线性趋同速度随着分序的增加而加快。因此,我们成功地赶上了时间衍生物分序变化的趋同速度的转变,并用数字实验加以验证。