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

TL;DR

This paper compares residual-based stabilization techniques for reduced-order models of the convection-diffusion-reaction equation, highlighting their effectiveness in improving stability and accuracy in under-resolved finite element discretizations.

Contribution

It systematically analyzes and compares continuous and discrete residual-based stabilization methods for reduced-order models of convection-diffusion-reaction equations.

Findings

Residual-based stabilized ROMs outperform standard Galerkin methods in under-resolved settings.

Continuous and discrete stabilization techniques yield comparable improvements.

Numerical experiments confirm the effectiveness of the proposed stabilization methods.

Abstract

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.