# Error estimates for Dirichlet control problems in polygonal domains

## TL;DR

This paper provides improved error estimates for finite element approximations of Dirichlet boundary control problems on polygonal domains, including non-convex cases, with detailed analysis of convergence rates.

## Contribution

It introduces new error estimates that improve upon existing results and extends analysis to non-convex polygonal domains for Dirichlet control problems.

## Key findings

- Enhanced convergence rates for control variables
- Error estimates applicable to non-convex domains
- Analysis of unconstrained and constrained control problems

## Abstract

The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasi-uniform meshes and superconvergence meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in non-convex domains are provided.