OPTIMAL CONTROL OF THE LAPLACE-BELTRAMI OPERATOR ON COMPACT SURFACES： CONCEPT AND NUMERICAL TREATMENT

We consider optimal control problems of elliptic PDEs on hypersurfaces F in 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 F. 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.