Convergence Analysis of Schwarz Waveform Relaxation for Nonlocal Diffusion Problems

Ke Li,Dali Guo,Yunxiang Zhao,Qiuye Sun

doi:10.1155/2021/5574403

Abstract

Diffusion equations with Riemann–Liouville fractional derivatives are Volterra integro-partial differential equations with weakly singular kernels and present fundamental challenges for numerical computation. In this paper, we make a convergence analysis of the Schwarz waveform relaxation (SWR) algorithms with Robin transmission conditions (TCs) for these problems. We focus on deriving good choice of the parameter involved in the Robin TCs, at the continuous and fully discretized level. Particularly, at the space-time continuous level, we show that the derived Robin parameter is much better than the one predicted by the well-understood equioscillation principle. At the fully discretized level, the problem of determining a good Robin parameter is studied in the convolution quadrature framework, which permits us to precisely capture the effects of different temporal discretization methods on the convergence rate of the SWR algorithms. The results obtained in this paper will be preliminary preparations for our further study of the SWR algorithms for integro-partial differential equations.

Highlights

Fractional kinetic equations have been proved useful to model the anomalous slow diffusion [1]
(2) Compute and compare the values of R􏽥 max(q) ≔ maxω∈[(π/T),(π/Δt)]R􏽥 (w, q), i.e., the convergence factor of the Schwarz waveform relaxation (SWR) algorithm with q qopt and q qbest obtained through numerical optimizations
For a specified choice of η, the convergence factors associated with the two time-integrators are almost equal. ese can be regarded as theoretical predictions, and we provide numerical results for validation

Summary

Introduction

Fractional kinetic equations have been proved useful to model the anomalous slow diffusion (subdiffusion) [1]. The so-called fractional cable equation was introduced for modeling the anomalous diffusion in spiny neuronal dendrites [18,19,20]: Cmztu(x, t) Dβt 􏼢4RdAz2xu(x, t)􏼣 −. Our interest is to apply the Schwarz waveform relaxation (SWR) algorithms to solve the fractional diffusion equations. Much less results are known about the algorithms applied to fractional order PDEs; according to our best knowledge, this is the first time that the SWR algorithms are analyzed for fractional problems. For the integer order PDEs, optimizing the Robin parameter is deeply studied in [30,35,36,37] and the main finding is that a good parameter can be computed by solving one or two simple nonlinear equations.

The Model Problem and the SWR Algorithms

Convergence Analysis at the Continuous Level

Convergence Analysis at the Discrete Level

Numerical Results

Conclusions

Conflicts of Interest