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Journal of Scientific Computing

, Volume 75, Issue 1, pp 560–595 | Cite as

An Equilibrated Fluxes Approach to the Certified Descent Algorithm for Shape Optimization Using Conforming Finite Element and Discontinuous Galerkin Discretizations

  • Matteo GiacominiEmail author
Article

Abstract

The certified descent algorithm (CDA) is a gradient-based method for shape optimization which certifies that the direction computed using the shape gradient is a genuine descent direction for the objective functional under analysis. It relies on the computation of an upper bound of the error introduced by the finite element approximation of the shape gradient. In this paper, we present a goal-oriented error estimator which depends solely on local quantities and is fully-computable. By means of the equilibrated fluxes approach, we construct a unified strategy valid for both conforming finite element approximations and discontinuous Galerkin discretizations. The new variant of the CDA is tested on the inverse identification problem of electrical impedance tomography: both its ability to identify a genuine descent direction at each iteration and its reliable stopping criterion are confirmed.

Keywords

Shape optimization Certified descent algorithm A posteriori error estimator Equilibrated fluxes Conforming finite element Discontinuous Galerkin Electrical impedance tomography 

Mathematics Subject Classification

49Q10 65M60 65N15 65N21 

Notes

Acknowledgements

The author expresses his sincere gratitude to Alexandre Ern for the useful advices and to Olivier Pantz for many fruitful discussions and for carefully reading the manuscript. The author wishes to thank the anonymous reviewers for their comments that helped to greatly improve the manuscript. Part of this work has been developed during a stay of the author at the Laboratoire J. A. Dieudonné at Université de Nice-Sophia Antipolis whose support is kindly acknowledged.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.CMAP, Inria, Ecole polytechnique, CNRSUniversité Paris-SaclayPalaiseauFrance
  2. 2.DRIInstitut Polytechnique des Sciences AvancéesIvry-sur-SeineFrance
  3. 3.Laboratori de Càlcul Numèric, E.T.S. de Ingenieros de CaminosUniversitat Politècnica de Catalunya – BarcelonaTechBarcelonaSpain

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