Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet--to--Neumann maps

Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. For smooth problems, the theory and practice of such two-level methods is well established, but this is not the case for problems with complicated variation and high contrasts in the coefficients. Stable coarse spaces for high contrast problems are also important purely for approximation purposes, when it is not desirable to resolve all the fine scale variations in the problem. In a previous study, two of the authors introduced a coarse space adapted to highly heterogeneous coefficients using the low frequency modes of the subdomain DtN maps. In this work, we present a rigorous analysis of a two-level overlapping additive Schwarz method (ASM) with this coarse space, which provides an automatic criterion for the number of modes that need to be added per subdomain to obtain a convergence rate of the order of the constant coefficient case. Our method is suitable for parallel implementation and its efficiency is demonstrated by numerical examples on some challenging problems with high heterogeneities for automatic partitionings.