Optimal Control of the Laplace-Beltrami Operator on Compact Surfaces-Concept and Numerical Treatment

Abstract

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, which is discretized by finite elements on a polyhedral approximation of $Γ$. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.