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Isogeometric analysis and proper orthogonal decomposition for parabolic problems

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Abstract

We investigate the combination of Isogeometric Analysis (IGA) and proper orthogonal decomposition (POD) based on the Galerkin method for model order reduction of linear parabolic partial differential equations. For the proposed fully discrete scheme, the associated numerical error features three components due to spatial discretization by IGA, time discretization with the \(\theta \)-scheme, and eigenvalue truncation by POD. First, we prove a priori error estimates of the spatial IGA semi-discrete scheme. Then, we show stability and prove a priori error estimates of the space-time discrete scheme and the fully discrete IGA-\(\theta \)-POD Galerkin scheme. Numerical tests are provided to show efficiency and accuracy of NURBS-based IGA for model order reduction in comparison with standard finite element-based POD techniques.

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Acknowledgments

The authors acknowledge Dr. Peng Chen and Dr. Toni Lassila for helpful discussions on model order reduction and Prof. Fabio Nobile for insights on error estimates. The help of Federico Negri with the library MLife developed by Prof. Fausto Saleri is acknowledged. The use of the IGA library GeoPDEs [18] is also acknowledged. S. Zhu greatly appreciates the warm hospitality of Prof. Quarteroni and other group members throughout his visit as a postdoctoral researcher at CMCS-MATHICSE-EPFL.

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Correspondence to Shengfeng Zhu.

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S. Zhu was financially supported in part by the National Natural Science Foundation of China under Grant 11201153 and a postdoctoral scholarship from China Scholarship Council.

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Zhu, S., Dedè, L. & Quarteroni, A. Isogeometric analysis and proper orthogonal decomposition for parabolic problems. Numer. Math. 135, 333–370 (2017). https://doi.org/10.1007/s00211-016-0802-5

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