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In this article, we have studied the convergence behavior of theDirichlet-Neumann and Neumann- Neumann waveform relaxation algorithms fortime-fractional sub-diffusion and diffusion-wave equations in 1D & 2D forregular domains, where the dimensionless diffusion coefficient takes differentconstant values in different subdomains. We first observe that differentdiffusion coefficients lead to different relaxation parameters for optimalconvergence. Using these optimal relaxation parameters, our analysis estimatesthe slow superlinear convergence of the algorithms when the fractional order ofthe time derivative is close to zero, almost finite step convergence when theorder is close to two, and in between, the superlinear convergence becomesfaster as fractional order increases. So, we have successfully caught thetransition of convergence rate with the change of fractional order of the timederivative in estimates and verified them with numerical experiments.

Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for time fractional sub-diffusion and diffusion-wave equations