Optimal control of elliptic PDEs on surfaces of codimension 1
It provides theoretical guarantees for a numerically simpler discretization approach to optimal control problems with surface integrals, benefiting researchers in PDE-constrained optimization.
The paper develops and analyzes an elliptic optimal control problem with a fidelity term on a hypersurface, proving a priori L^2 error estimates for the control and validating them with numerical results.
We consider an elliptic optimal control problem where the objective functional contains an integral along a surface of codimension 1, also known as a hypersurface. In particular, we use a fidelity term that encourages the state to take certain values along a curve in 2D or a surface in 3D. In the discretisation of this problem, which uses piecewise linear finite elements, we allow the hypersurface to be approximated e.g. by a polyhedral hypersurface. This can lead to simpler numerical methods, however it complicates the numerical analysis. We prove a priori $L^2$ error estimates for the control and present numerical results that agree with these. A comparison is also made to point control problems.