In this paper we analyze a two-dimensional shape optimization problem, governed by Stokes equations that are defined on a domain with a part of the boundary that is described as the graph of the control function. The state problem formulation is mapped onto a reference domain, which is independent of the control function, and the analysis is mainly led on such domain. The existence of an optimal control function is proved, and optimality conditions are derived. After the analytical inspection of the problem, finite element discretization is considered for both the control function and the state variables, and a priori convergence error estimates are derived. Numerical experiments assess the validity of the theoretical results.
DOI : 10.1051/m2an/2014060
Mots clés : Shape optimization, Stokes problem, reference domain, convergence rates, finite elements
@article{M2AN_2015__49_4_921_0,
author = {Fumagalli, Ivan and Parolini, Nicola and Verani, Marco},
title = {Shape {Optimization} for {Stokes} flows: a finite element convergence analysis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {921--951},
publisher = {EDP-Sciences},
volume = {49},
number = {4},
year = {2015},
doi = {10.1051/m2an/2014060},
mrnumber = {3371898},
zbl = {1320.76068},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2014060/}
}
TY - JOUR AU - Fumagalli, Ivan AU - Parolini, Nicola AU - Verani, Marco TI - Shape Optimization for Stokes flows: a finite element convergence analysis JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 921 EP - 951 VL - 49 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2014060/ DO - 10.1051/m2an/2014060 LA - en ID - M2AN_2015__49_4_921_0 ER -
%0 Journal Article %A Fumagalli, Ivan %A Parolini, Nicola %A Verani, Marco %T Shape Optimization for Stokes flows: a finite element convergence analysis %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 921-951 %V 49 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2014060/ %R 10.1051/m2an/2014060 %G en %F M2AN_2015__49_4_921_0
Fumagalli, Ivan; Parolini, Nicola; Verani, Marco. Shape Optimization for Stokes flows: a finite element convergence analysis. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 921-951. doi : 10.1051/m2an/2014060. http://archive.numdam.org/articles/10.1051/m2an/2014060/
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