The combination of POD model reduction with adaptive finite element methods in the context of phase field models

The combination of POD model reduction with adaptive finite element methods in the context of... In the present paper we derive a reduced order model utilizing proper orthogonal decomposition (POD‐ROM), for which we utilize adaptively obtained spatial snapshots. In a fully discrete setting this contains the challenge that the snapshots are vectors of different lengths. In order to handle this issue, we interprete the snapshots as elements of a common Hilbert space and consider the POD method from an infinite‐dimensional perspective. Thus, the inner product of pairs of snapshots can be computed explicitely, which enables us to build the reduced order model. This approach is applied to a phase field model which is described by a Cahn‐Hilliard equation. In the numerical examples we illustrate our appoach and compare a nonsmooth with a smooth free energy concerning the influence on the quality of the solution to the ROM. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Proceedings in Applied Mathematics & Mechanics Wiley

The combination of POD model reduction with adaptive finite element methods in the context of phase field models

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Publisher
Wiley
Copyright
Copyright © 2017 Wiley Subscription Services
ISSN
1617-7061
eISSN
1617-7061
D.O.I.
10.1002/pamm.201710014
Publisher site
See Article on Publisher Site

Abstract

In the present paper we derive a reduced order model utilizing proper orthogonal decomposition (POD‐ROM), for which we utilize adaptively obtained spatial snapshots. In a fully discrete setting this contains the challenge that the snapshots are vectors of different lengths. In order to handle this issue, we interprete the snapshots as elements of a common Hilbert space and consider the POD method from an infinite‐dimensional perspective. Thus, the inner product of pairs of snapshots can be computed explicitely, which enables us to build the reduced order model. This approach is applied to a phase field model which is described by a Cahn‐Hilliard equation. In the numerical examples we illustrate our appoach and compare a nonsmooth with a smooth free energy concerning the influence on the quality of the solution to the ROM. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Journal

Proceedings in Applied Mathematics & MechanicsWiley

Published: Jan 1, 2017

References

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