Patrick Sodré | Searcharxiv

arXiv2015

Optimal Control of a Free Boundary Problem with Surface Tension Effects: A Priori Error Analysis

We present a finite element method along with its analysis for the optimal control of a model free boundary problem with surface tension effects, formulated and studied in \cite{HAntil_RHNochetto_PSodre_2014a}. The state system couples the Laplace equation in the bulk with the Young-Laplace equation on the free boundary to account for surface tension. We first prove that the state and adjoint system have the requisite regularity for the error analysis (strong solutions). We discretize the state, adjoint and control variables via piecewise linear finite elements and show optimal $O(h)$ error estimates for all variables, including the control. This entails using the second order sufficient optimality conditions of \cite{HAntil_RHNochetto_PSodre_2014a}, and the first order necessary optimality conditions for both the continuous and discrete systems. We conclude with two numerical examples which examine the various error estimates.

math.OC↗

arXiv2014

Optimal Control of a Free Boundary Problem: Analysis with Second Order Sufficient Conditions

We consider a PDE-constrained optimization problem governed by a free boundary problem. The state system is based on coupling the Laplace equation in the bulk with a Young-Laplace equation on the free boundary to account for surface tension, as proposed by P.\ Saavedra and L.\ R.\ Scott \cite{PSaavedra_RScott_1991}. This amounts to solving a second order system both in the bulk and on the interface. Our analysis hinges on a convex control constraint such that the state constraints are always satisfied. Using only first order regularity we show that the control to state operator is twice continuously Fréchet differentiable. We improve slightly the regularity of the state variables and exploit it to show existence of a control together with second order sufficient optimality conditions.

math.OC↗