Optimal control of elliptic PDEs on surfaces of codimension 1

2014/11/18 by Charles Brett, Andreas Dedner, Brett, C. +3
Engineering · Mathematics · #65N15 #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1411.4974

openalex publication_date 2014/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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 L² error estimates for the control and present numerical results that agree with these. A comparison is also made to point control problems.

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