Generalized Norm-Resolvent and Pseudospectral Convergence of Galerkin Projections for Non-Self-Adjoint Operators, with a Reproducible Non-Normal Benchmark

We establish a Galerkin framework for approximating spectra and ε-pseudospectra of closed, densely defined non-self-adjoint operators with compact resolvent. Discrete stability together with projected graph consistency yields generalized norm-resolvent convergence, uniform on compact subsets of the resolvent set, and hence local pseudospectral convergence. Under a non-degeneracy condition on the resolvent-norm level set, we obtain the quantitative estimate d_H≤δ_n/γ. For the Dirichlet advection-diffusion operator Tu=-u^''+bu^', we prove projected consistency for the sine-Galerkin scheme with rate O(n^(-1) ) and derive an explicit stability bound outside the numerical range from W(T_n )⊂W(T). Numerical experiments quantify Hausdorff convergence, resolvent-norm error, eigenvalue convergence, boundary-gradient behaviour, and Péclet-number dependence, with observed log-log slopes near -4. A closed-form 2 × 2 Jordan-block benchmark is reproduced to 1.5 × 〖.10〗^(-9), independently validating the numerical implementation.