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.

翻译：加辽金和彼得罗夫-加辽金投影降阶模型（ROMs）通常通过在空间连续或空间离散层面进行降维与投影过程而获得。在这两种情形下，常需对所得ROM添加稳定化处理以提升方法的稳定性和精度；对于对流主导系统中分辨率不足的ROM，添加稳定化尤为常见。尽管这两种方法在某些情况下等价，但二者已发展出不同的技术体系。本文在有限元离散框架下（此时连续与离散层面存在对偶性）阐述了针对对流-扩散-反应方程的这两种方法，并比较了两种框架下发展的基于残差的稳定化技术。在空间连续情形中，我们考察了加辽金、流线迎风彼得罗夫-加辽金（SUPG）、加辽金/最小二乘（GLS）和伴随（ADJ）稳定化方法。对于GLS和ADJ方法，我们分别分析了基于"先离散再稳定化"和时空技术的构造形式。在空间离散情形中，我们考察了加辽金、最小二乘彼得罗夫-加辽金（LSPG）和伴随彼得罗夫-加辽金（APG）方法。我们总结了现有分析方法，并通过数值实验证明：当底层有限元模型分辨率不足时，基于残差的连续与离散稳定化方法相比标准加辽金方法均能实现显著改进。