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HiMod Reduction of Advection–Diffusion–Reaction Problems with General Boundary Conditions

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Abstract

We extend the hierarchical model reduction procedure previously introduced in Ern et al. (in: Kunisch, Of, Steinbach (eds) Numerical mathematics and advanced applications, Springer, Berlin, pp 703–710, 2008) and Perotto et al. (Multiscale Model Simul 8(4):1102–1127, 2010) to deal with general boundary conditions, enforcing their prescription in the basis function set. This is achieved by solving a Sturm–Liouville Eigenvalue problem. We analyze this approach and provide a convergence analysis for the associated error in the case of a linear advection–diffusion–reaction problem in rectangles (2D) and slabs (3D). Numerical results confirm the theoretical investigation and the reliability of the proposed approach.

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Notes

  1. LifeV is an open source finite element library developed by MOX at Politecnico di Milano, Italy, the Department of Mathematics at EPFL, Switzerland and the Department of Mathematics and Computer Science at Emory University, USA (https://cmcsforge.epfl.ch/doxygen/lifev/).

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Acknowledgements

The second and the third authors acknowledge the support of NSF Grant DMS-1419060 “Hierarchical Model Reduction Techniques for Incompressible Fluid-Dynamics and Fluid-Structure Interaction Problems” for this research. The authors wish to thank also Prof. Sandro Salsa for fruitful discussions concerning the analysis of the method, and Pablo J. Blanco for hosting the first author for one month at Laboratório Nacional de Computacao Científica in Petrópolis, Brazil. This article has been partially supported by Istituto Nazionale di Alta Matematica “Francesco Severi” (IT) Project (GNCS2017 Metodi numerici avanzati combinati con tecniche di riduzione computazionale per PDEs parametrizzate ed applicazioni).

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Inria Paris, Sorbonne Universités UPMC Univ Paris 6, Paris, France

    Matteo C. Aletti

  2. MOX, Dipartimento di Matematica, Politecnico di Milano, P.zza L. da Vinci 32, 20133, Milan, Italy

    Simona Perotto

  3. Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr NE, S.te N416, Atlanta, GA, 30322, USA

    Alessandro Veneziani

  4. School of Advanced Studies IUSS, P.zza della Vittoria 15, 27100, Pavia, Italy

    Alessandro Veneziani

Authors
  1. Matteo C. Aletti
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  2. Simona Perotto
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  3. Alessandro Veneziani
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Correspondence to Simona Perotto.

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Aletti, M.C., Perotto, S. & Veneziani, A. HiMod Reduction of Advection–Diffusion–Reaction Problems with General Boundary Conditions. J Sci Comput 76, 89–119 (2018). https://doi.org/10.1007/s10915-017-0614-5

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  • DOI: https://doi.org/10.1007/s10915-017-0614-5

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