A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control

Bibliographic Details
Title:	A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control
Authors:	Zoccolan, Fabio, Strazzullo, Maria, Rozza, Gianluigi
Publication Year:	2023
Collection:	Computer Science Mathematics
Subject Terms:	Mathematics - Numerical Analysis, 49J20, 49M41, 65M60
More Details:	In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the P\'eclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times. Comment: 27 pages, 36 figures, 4 tables
Document Type:	Working Paper
DOI:	10.1515/cmam-2023-0171
Access URL:	http://arxiv.org/abs/2301.01973
Accession Number:	edsarx.2301.01973
Database:	arXiv

More Details
DOI:	10.1515/cmam-2023-0171