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Advances in Numerical Analysis
Volume 2013 (2013), Article ID 974284, 8 pages
http://dx.doi.org/10.1155/2013/974284
Research Article

A Proper-Orthogonal Decomposition Variational Multiscale Approximation Method for a Generalized Oseen Problem

Department of Mathematics, North Carolina A & T State University, Greensboro, NC 27411, USA

Received 6 June 2013; Accepted 25 October 2013

Academic Editor: Yinnian He

Copyright © 2013 John Paul Roop. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the variational multiscale (VMS) stabilization for the reduced-order modeling of incompressible flows. It is well known that the proper orthogonal decomposition (POD) technique in reduced-order modeling experiences numerical instability when applied to complex flow problems. In this case a POD discretization naturally separates out structures which corresponding to the energy cascade on large and small scales, in order, a VMS approach is natural. In this paper, we provide the mathematical background necessary for implementing VMS to a POD-Galerkin model of a generalized Oseen problem. We provide theoretical evidence which indicates the consistency of utilizing a VMS approach in the stabilization of reduced order flows. In addition we provide numerical experiments indicating that VMS improves fidelity in reproducing the qualitative properties of the flow.