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：瞬态偏微分方程的Galerkin和Petrov-Galerkin投影降阶模型通常通过在空间连续或空间离散层面定义的降维与投影过程构建。两种情况下，常对所得降阶模型添加稳定化处理以提升方法的稳定性和精度——当降阶模型分辨率不足时，这种稳定化操作在对流主导系统中尤为常见。尽管这两种方法在某些设置下等价，但各自领域已发展出不同技术体系。本文以对流扩散反应方程的有限元法离散化（此时连续层面与离散层面存在对偶关系）为框架，系统阐释这两种方法路径，并比较两种背景下发展的基于残差的稳定化技术。在空间连续情形下，我们考察了Galerkin方法、流线迎风Petrov-Galerkin方法、Galerkin/最小二乘法以及对偶稳定化方法。对于最小二乘和对偶方法，分别分析了基于"先离散后稳定"和时空技术构建的公式体系。在空间离散情形下，我们考察了Galerkin方法、最小二乘Petrov-Galerkin方法和伴随Petrov-Galerkin方法。本文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，通过连续与离散过程发展的基于残差的稳定化方法相比标准Galerkin方法具有显著改进。