An equilibrated fluxes approach to the certified descent algorithm for shape optimization using conforming finite element and discontinuous Galerkin discretizations

TL;DR

This paper introduces a goal-oriented error estimator based on equilibrated fluxes for the certified descent algorithm in shape optimization, applicable to finite element and discontinuous Galerkin methods, improving reliability and efficiency.

Contribution

It develops a unified equilibrated fluxes approach for error estimation in the certified descent algorithm, applicable to multiple discretization methods, enhancing shape optimization accuracy.

Findings

Effective identification of genuine descent directions.

Reliable stopping criterion demonstrated.

Applicable to electrical impedance tomography.

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.