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Advances in Computational Mathematics

, Volume 44, Issue 6, pp 1941–1978 | Cite as

POD reduced-order modeling for evolution equations utilizing arbitrary finite element discretizations

  • Carmen Gräßle
  • Michael Hinze
Article
  • 70 Downloads

Abstract

The main focus of the present work is the inclusion of spatial adaptivity for the snapshot computation in the offline phase of model order reduction utilizing proper orthogonal decomposition (POD-MOR) for nonlinear parabolic evolution problems. We consider snapshots which live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. From a numerical point of view, this leads to the problem that the usual POD procedure which utilizes a singular value decomposition of the snapshot matrix, cannot be carried out. In order to overcome this problem, we here construct the POD model/basis using the eigensystem of the correlation matrix (snapshot Gramian), which is motivated from a continuous perspective and is set up explicitly, e.g., without the necessity of interpolating snapshots into a common finite element space. It is an advantage of this approach that the assembly of the matrix only requires the evaluation of inner products of snapshots in a common Hilbert space. This allows a great flexibility concerning the spatial discretization of the snapshots. The analysis for the error between the resulting POD solution and the true solution reveals that the accuracy of the reduced-order solution can be estimated by the spatial and temporal discretization error as well as the POD error. Finally, to illustrate the feasibility of our approach, we present a test case of the Cahn–Hilliard system utilizing h-adapted hierarchical meshes and two settings of a linear heat equation using nested and non-nested grids.

Keywords

Model order reduction Proper orthogonal decomposition Adaptive finite element discretization Partial differential equation Evolution equations 

Mathematics Subject Classification (2010)

35K90 65K05 35K05 35K55 65N30 

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Notes

Acknowledgements

We like to thank Christian Kahle for providing many C++ libraries which we could use for the coding. The authors gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft through the priority program SPP1962 entitled “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Universität HamburgHamburgGermany

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