Sharp Ascent--Descent Spectral Stability under Strong Resolvent Convergence

We establish sharp stability results for of non--selfadjoint the ascent and descent spectra under strong resolvent convergence (SRS), a natural framework for finite element approximations of non-selfadjoint and singularly perturbed operators. The key quantitative hypothesis is the reduced minimum modulus $\gamma(T-\lambda)>0$, which guarantees closed range and enables the transfer of the Kaashoek -- Taylor criteria via gap convergence of operator graphs. At the essential level, B--Fredholm theory extends stability to powers $(T-\lambda)^m$ provided $\gamma((T-\lambda)^j)>0$ for all $1\le j\le m$. We introduce a computable finite-element diagnostic $\gamma_h = \sigma_{\min}(M^{-1/2}(A_h-\lambda M)M^{-1/2})$, which serves as a practical surrogate for $\gamma(T-\lambda)$ and remains uniformly positive even in convection-dominated regimes when stabilized schemes (e.g., SUPG) are employed. Numerical experiments confirm that $\liminf_{h\to0}\gamma_h>0$ is both necessary and sufficient for spectral stability, while a Volterra-type counterexample demonstrates the indispensability of the closed-range condition for powers. The analysis clarifies why norm resolvent convergence fails for rough or singular limits, and how SRS-combined with quantitative control of $\gamma_h$--rescues ascent--descent stability in realistic computational settings.