Optimal Control of the Laplace-Beltrami operator on compact surfaces - concept and numerical treatment

TL;DR

This paper develops a numerical framework for optimal control problems involving the Laplace-Beltrami operator on surfaces, providing error estimates and confirming results through numerical experiments.

Contribution

It introduces a finite element discretization and a semismooth Newton method for solving surface PDE control problems with error analysis.

Findings

Optimal error estimates for control problems on surfaces

Effective numerical solution via semismooth Newton algorithm

Validation through numerical experiments

Abstract

We consider optimal control problems of elliptic PDEs on hypersurfaces in 2- or 3-dimensional Euclidean space. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of the surface. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semismooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.