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Coupled Model and Grid Adaptivity in Hierarchical Reduction of Elliptic Problems

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Abstract

In this paper we propose a surrogate model for advection–diffusion–reaction problems characterized by a dominant direction in their dynamics. We resort to a hierarchical model reduction where we couple a modal representation of the transverse dynamics with a finite element approximation along the mainstream. This different treatment of the dynamics entails a surrogate model enhancing a purely 1D description related to the leading direction. The coefficients of the finite element expansion along this direction introduce a generally non-constant description of the transversal dynamics. Aim of this paper is to provide an automatic adaptive approach to locally select the dimension of the modal expansion as well as the finite element step in order to satisfy a prescribed tolerance on a goal functional of interest.

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Notes

  1. From (13)–(14), the further upper bound \(|J(e_{\mathbf{m}^+})| \le \beta _\mathbf{m}/(1 -\beta _\mathbf{m})\, |J(\delta u_{\mathbf{mm}^+})|\) trivially follows.

  2. We usually set \(\widetilde{m}^+=m+2\); due to the parity of sinusoidal functions, the simplest choice \(\widetilde{m}^+=m+1\) is completely useless when dealing with solutions symmetric with respect to fiber \(\Omega _{1D}\).

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Acknowledgments

The authors wish to thank warmly Alexandre Ern for many discussions and fundamental suggestions he gave along the entire preparation of the manuscript.

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Authors and Affiliations

  1. MOX, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 , Milan, Italy

    Simona Perotto

  2. Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr., Atlanta, GA, 30322, USA

    Alessandro Veneziani

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  1. Simona Perotto
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  2. Alessandro Veneziani
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Correspondence to Simona Perotto.

Additional information

This work has been partially supported by the PRIN 2010–2011 project “Innovative methods for water resources management under hydro-climatic uncertainty scenarios”.

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Perotto, S., Veneziani, A. Coupled Model and Grid Adaptivity in Hierarchical Reduction of Elliptic Problems. J Sci Comput 60, 505–536 (2014). https://doi.org/10.1007/s10915-013-9804-y

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