Assess graduate-level mastery in evaluating convergence, stability, and optimality of multigrid and domain-decomposition preconditioners for large-scale indefinite and saddle-point linear systems from mixed finite element discretizations. Items should test theoretical analysis (spectral and field-of-values bounds, inf-sup stability, Schur-complement estimates, norm equivalence), algorithmic scalability and parameter-robustness, design and analysis of block/approximate-Schur preconditioners, and interpretation of numerical diagnostics (GMRES behavior, eigenvalue distributions, condition numbers). Tasks may require proving optimality or failure modes, proposing robustness fixes, and outlining numerical experiments to validate claims.