Galerkin approximation with Proper Orthogonal Decomposition: new error estimates and illustrative examples - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles ESAIM: Mathematical Modelling and Numerical Analysis Year : 2012

Galerkin approximation with Proper Orthogonal Decomposition: new error estimates and illustrative examples

Abstract

We propose a numerical analysis of Proper Orthogonal Decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and -- whereas a general uniform continuity bound seems out of reach -- we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound -- and the effectiveness of the POD approach altogether -- for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.
Fichier principal
Vignette du fichier
podpaper.pdf (654.95 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-00654539 , version 1 (22-12-2011)

Identifiers

Cite

Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie. Galerkin approximation with Proper Orthogonal Decomposition: new error estimates and illustrative examples. ESAIM: Mathematical Modelling and Numerical Analysis, 2012, 46 (4), pp.731-757. ⟨10.1051/m2an/2011053⟩. ⟨hal-00654539⟩
220 View
815 Download

Altmetric

Share

Gmail Facebook Twitter LinkedIn More