Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Abstract

The solution of large sparse linear systems is one of the most time consuming kernels in many 4 numerical simulations. The domain decomposition community has developed many efficient and robust methods in 5 the last decades. While many of these solvers fall into the abstract Schwarz (aS) framework, their robustness has 6 originally been demonstrated on a case-by-case basis. In this paper, we propose a bound for the condition number 7 of all deflated aS methods provided that the coarse grid consists of the assembly of local components that contain 8 the kernel of some local operators. We show that classical results from the literature on particular instances of 9 aS methods can be retrieved from this bound. We then show that such a coarse grid correction can be explicitly 10 obtained algebraically via generalized eigenproblems, leading to a condition number independent of the number of 11 domains. This result can be readily applied to retrieve or improve the bounds previously obtained via generalized 12 eigenproblems in the particular cases of Neumann-Neumann (NN), Additive Schwarz (AS) and optimized Robin but 13 also generalizes them when applied with approximate local solvers. Interestingly, the proposed methodology turns 14 out to be a comparison of the considered particular aS method with generalized versions of both NN and AS for 15 tackling the lower and upper part of the spectrum, respectively. We furthermore show that the application of the 16 considered grid corrections in an additive fashion is robust in the AS case although it is not robust for aS methods in 17 general. In particular, the proposed framework allows for ensuring the robustness of the AS method applied on the 18 Schur complement (AS/S), either with deflation or additively, and with the freedom of relying on an approximate 19 local Schur complement. Numerical experiments illustrate these statements.