Optimal Control of the Laplace-Beltrami Operator on Compact Surfaces-Concept and Numerical Treatment (Journal of Computational Mathematics, 2018)
We consider optimal control problems of elliptic PDEs on hypersurfaces $Γ$ in $\mathbb{R}^n$ for $n$=2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, whichis discretized by finite elements on a polyhedral approximation of $Γ$. The discrete optimalcontrol problem is formulated on the approximating surface and is solved numerically witha semi-smooth Newton algorithm. We derive optimal a priori error estimates for problemsincluding control constraints and provide numerical examples confirming our analyticalfindings.
Keywords: Elliptic optimal control problem; Laplace-Beltrami operator; Surfaces; Control constraints; Error estimates; Semi-smooth Newton method.