Mathematical analysis of robustness of two-level domain decomposition methods with respect to inexact coarse solves

Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. The GenEO coarse space has been shown to lead to a robust two-level Schwarz preconditioner which scales well over multiple cores (Spillane et al. in Numer Math 126(4):741-770, 2014. 10.1007/s00211-013-0576-y; Dolean et al. in An introduction to domain decomposition methods: algorithms, theory and parallel implementation, SIAM, Philadelphia, 2015). The robustness is due to its good approximation properties for problems with highly heterogeneous material parameters. It is available in the finite element packages FreeFem++ (Hecht in J Numer Math 20(3-4):251-265, 2012), Feel++ (Prud'homme in Sci Program 14(2):81-110, 2006), Dune (Blatt et al. in Arch Numer Softw 4(100):13-29, 2016) and is implemented as a standalone library in HPDDM (Jolivet and Nataf in HPDDM: high-Performance Unified framework for Domain Decomposition methods, MPI-C++ library, 2014. ) and as such is available as well as a PETSc (Balay et al. in: Arge, Bruaset, Langtangen, (eds) Modern software tools in scientific computing, Birkhauser Press, Basel, 1997) preconditioner. But the coarse component of the preconditioner can ultimately become a bottleneck if the number of subdomains is very large and exact solves are used. It is therefore interesting to consider the effect of inexact coarse solves. In this paper, robustness of GenEO methods is analyzed with respect to inexact coarse solves. Interestingly, the GenEO-2 method introduced in Haferssas et al. (SIAM J Sci Comput 39(4):A1345-A1365, 2017. 10.1137/16M1060066) has to be modified in order to be able to prove its robustness in this context.