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Galerkin and Petrov-Galerkin projection-based reduced-order models (ROMs) oftransient partial differential equations are typically obtained by performing adimension reduction and projection process that is defined at either thespatially continuous or spatially discrete level. In both cases, it is commonto add stabilization to the resulting ROM to increase the stability andaccuracy of the method; the addition of stabilization is particularly commonfor advection-dominated systems when the ROM is under-resolved. While these twoapproaches can be equivalent in certain settings, differing techniques haveemerged in both contexts. This work outlines these two approaches within thesetting of finite element method (FEM) discretizations (in which case a dualityexists between the continuous and discrete levels) of theconvection-diffusion-reaction equation, and compares residual-basedstabilization techniques that have been developed in both contexts. In thespatially continuous case, we examine the Galerkin, streamline upwindPetrov-Galerkin (SUPG), Galerkin/least-squares (GLS), and adjoint (ADJ)stabilization methods. For the GLS and ADJ methods, we examine formulationsconstructed from both the "discretize-then-stabilize" technique and thespace-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 providenumerical experiments, which demonstrate that residual-based stabilized methodsdeveloped via continuous and discrete processes yield substantial improvementsover standard Galerkin methods when the underlying FEM model is under-resolved.

Residual-based stabilized reduced-order models of the transient  convection-diffusion-reaction equation obtained through discrete and  continuous projection