  Galerkin and Petrov-Galerkin projection-based reduced-order models (ROMs) of
transient partial differential equations are typically obtained by performing a
dimension reduction and projection process that is defined at either the
spatially continuous or spatially discrete level. In both cases, it is common
to add stabilization to the resulting ROM to increase the stability and
accuracy of the method; the addition of stabilization is particularly common
for advection-dominated systems when the ROM is under-resolved. While these two
approaches can be equivalent in certain settings, differing techniques have
emerged in both contexts. This work outlines these two approaches within the
setting of finite element method (FEM) discretizations (in which case a duality
exists between the continuous and discrete levels) of the
convection-diffusion-reaction equation, and compares residual-based
stabilization techniques that have been developed in both contexts. In the
spatially continuous case, we examine the Galerkin, streamline upwind
Petrov-Galerkin (SUPG), Galerkin/least-squares (GLS), and adjoint (ADJ)
stabilization methods. For the GLS and ADJ methods, we examine formulations
constructed from both the "discretize-then-stabilize" technique and the
space-time technique. In the spatially discrete case, we examine the Galerkin,
least-squares Petrov-Galerkin (LSPG), and adjoint Petrov-Galerkin (APG)
methods. We summarize existing analyses for these methods, and provide
numerical experiments, which demonstrate that residual-based stabilized methods
developed via continuous and discrete processes yield substantial improvements
over standard Galerkin methods when the underlying FEM model is under-resolved.
