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:Galerkin 和 Petrov-Galerkin 基于静态部分差异方程式的降序预测模型(ROMs)通常通过在空间连续或空间离散水平上确定一个尺寸降序和投影过程(FEM)来获得。在这两种情况下,通常的做法是在由此产生的ROM上增加稳定性,以提高该方法的稳定性和准确性;在ROM解解度不足时,对以静态为主的系统增加稳定性是特别常见的。虽然这两种方法在某些环境下可以等同,但在两种情况下都出现了不同的技术。 这项工作概述了在设定固定元素方法(FEM)中采用的这两种方法(在空间连续连续连续或离散水平之间存在双重性)的分解和投影过程。 在空间连续的情况下,我们检查Galerkin, 精简上风Petrov-Galerkin (SUPG)、 Galerkin/East-quarres (GLS) (GLS), 以及用于我们不断开发的GLO-S-S-S-S-S-deal-deal-S-deal-de 和Outeral-de-deal-de-de-de-Seral-S-de-Servial-de-de 这两种方法。这些是这些我们持续地检验和Silal-s-de-de-de-de-de-de-de-de-de-de-de-de-de-S-de-de-de-de-de-de-de-de-de-deal-deal-deal-de-de-de-de,这些方法。