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

TL;DR

This paper analyzes the convergence behavior of waveform relaxation algorithms for time-fractional equations, revealing how fractional order influences convergence rates and providing optimal parameters for different scenarios.

Contribution

It introduces a detailed convergence analysis for Dirichlet-Neumann and Neumann-Neumann algorithms applied to fractional equations, highlighting the impact of fractional order and diffusion coefficients.

Findings

Convergence rate varies with fractional order, from superlinear near zero to almost finite step near two.

Optimal relaxation parameters depend on diffusion coefficients in subdomains.

Numerical experiments confirm theoretical convergence estimates.

Abstract

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.