A Coarse Grid Space Construction Based on Local DtN Maps

Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. In this work we construct the coarse grid space using the low frequency modes of the subdomain DtN maps, and apply the obtained two-level preconditioner (the additive Schwarz method together with the new coarse grid) to the extended or the original linear system arising from an overlapping domain decomposition. The patch method is also added to further improve the convergence. Our method is suitable for the parallel implementation and its efficiency is demonstrated by numerical examples on problems with high heterogeneities for both manual and automatic partitionings. Some notations and definitions A coefficient matrix of the linear system Ax = b Z, Y full rank matrices which span the coarse grid subspaces. E E = Y AZ, Galerkin matrix or coarse-grid matrix Ξ Ξ = ZEY T , coarse-grid correction matrix in MG and DDM PD PD = I −AΞ = I −AZ(Y AZ)Y T QD QD = I − ΞA = I − Z(Y AZ)Y A PBNN PBNN = QDM PD + ZE Y T PADEF2 PADEF2 = QDM −1 + ZEZ