Multiscale analysis of heterogeneous domain decomposition methods for time-dependent advection–reaction–diffusion problems

Abstract

Domain decomposition methods which use different models in different subdomains are called heterogeneous domain decomposition methods. We are interested here in the case where there is an accurate but expensive model one should use in the entire domain, but for computational savings we want to use a cheaper model in parts of the domain where expensive features of the accurate model can be neglected. For the model problem of a time dependent advection–reaction–diffusion equation in one spatial dimension, we study approximate solutions of three different heterogeneous domain decomposition methods with pure advection reaction approximation in parts of the domain. Using for the first time a multiscale analysis to compare the approximate solutions to the solution of the accurate expensive model in the entire domain, we show that a recent heterogeneous domain decomposition method based on factorization of the underlying differential operator has better approximation properties than more classical variational or non-variational heterogeneous domain decomposition methods. We show with numerical experiments in two spatial dimensions that the performance of the algorithms we study is well predicted by our one dimensional multiscale analysis, and that our theoretical results can serve as a guideline to compare the expected accuracy of heterogeneous domain decomposition methods already for moderate values of the viscosity.