# Boundary concentrated finite elements for optimal control problems with distributed observation

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## Abstract

We consider the discretization of an optimal boundary control problem with distributed observation by the boundary concentrated finite element method. If the constraint is a \(H^{1+\delta }(\Omega )\) regular elliptic PDE with smooth differential operator and source term, we prove for the two dimensional case that the discretization error in the \(L_2\) norm decreases like \(N^{-\delta }\), where \(N\) is the number of unknowns. Our approach is suitable for solving a wide class of problems, among them piecewise defined data and tracking functionals acting only on a subdomain of \(\Omega \). We present several numerical results.

## Keywords

Optimal control Elliptic partial differential equation Higher-order discretization Boundary-concentrated finite elements A priori error estimates## Notes

### Acknowledgments

This work was funded by the Austrian Science Fund (FWF) Grant P23484-N18.

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