Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators^∗

¹ CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France.
matteo.giacomini@polytechnique.edu
² DRI Institut Polytechnique des Sciences Avancées, 15-21 rue M. Grandcoing, 94200 Ivry-sur-Seine, France.
karim.trabelsi@ipsa.fr
³ Laboratoire J.A. Dieudonné UMR 7351 CNRS-Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France.
olivier.pantz@polytechnique.org

Received: 14 September 2015
Revised: 16 March 2016
Accepted: 13 April 2016

Abstract

In this paper we introduce a novel certified shape optimization strategy – named Certified Descent Algorithm (CDA) – to account for the numerical error introduced by the Finite Element approximation of the shape gradient. We present a goal-oriented procedure to derive a certified upper bound of the error in the shape gradient and we construct a fully-computable, constant-free a posteriori error estimator inspired by the complementary energy principle. The resulting CDA is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion. After validating the error estimator, some numerical simulations of the resulting certified shape optimization strategy are presented for the well-known inverse identification problem of Electrical Impedance Tomography.