  Advanced Krylov subspace methods are investigated for the solution of large
sparse linear systems arising from stiff adjoint-based aerodynamic shape
optimization problems. A special attention is paid to the flexible inner-outer
GMRES strategy combined with most relevant preconditioning and deflation
techniques. The choice of this specific class of Krylov solvers for challenging
problems is based on its outstanding convergence properties. Typically in our
implementation the efficiency of the preconditioner is enhanced with a domain
decomposition method with overlapping. However, maintaining the performance of
the preconditioner may be challenging since scalability and efficiency of a
preconditioning technique are properties often antagonistic to each other. In
this paper we demonstrate how flexible inner-outer Krylov methods are able to
overcome this critical issue. A numerical study is performed considering either
a Finite Volume (FV), or a high-order Discontinuous Galerkin (DG)
discretization which affect the arithmetic intensity and memory-bandwith of the
algebraic operations. We consider test cases of transonic turbulent flows with
RANS modelling over the two-dimensional supercritical OAT15A airfoil and the
three-dimensional ONERA M6 wing. Benefits in terms of robustness and
convergence compared to standard GMRES solvers are obtained. Strong scalability
analysis shows satisfactory results. Based on these representative problems a
discussion of the recommended numerical practices is proposed.
