"Provably" correct spectral calculations in hydrodynamic stability

1. Linear stability theory is a cornerstone of hydrodynamic stability. Solving eigenvalue problems numerically requires discretizing the underlying partial differential equations, and many numerical methods (finite differences, finite elements, collocation, Galerkin, etc...) have been developed to do this. In essence, these reduce a continuous problem to a linear (generalized) eigenvalue problem. The resulting eigenvalues and eigenvectors must then be investigated to distinguish the "real" from the "spurious" eigenvalues. This is known to be probelmatic for systems with a spectrum that is not purely discrete, but in fact the problem is in general difficult. Recent results (e.g. Colbrook & Hansen 2022) relate spectral computations to the Solvability Complexity Index. Using such ideas one can develop methods that provably converge. Here we examine how these methods can be applied to hydrodynamic stability problems. We consider two approaches: one based on the injection moduus and one based on Beyn's method for infinite-dimenisional problems. With the help of some examples, we examine how such approaches need to be formulated, in particular the question of appropriate norms, and the methods' advantages and disadvantages.