# A finite element method for Dirichlet boundary control of elliptic partial differential equations

## TL;DR

This paper develops a new variational finite element approach for Dirichlet boundary control of elliptic PDEs, proving well-posedness and convergence, and validating with numerical examples.

## Contribution

It introduces a novel variational formulation that relates the state and adjoint state through boundary control, with proven well-posedness and convergence analysis for finite element approximations.

## Key findings

- Finite element approximations converge at order k-1/2.
- The variational formulation is well-posed in H^1 spaces.
- Numerical results confirm theoretical convergence rates.

## Abstract

This paper introduces a new variational formulation for Dirichlet boundary control problem of elliptic partial differential equations, based on observations that the state and adjoint state are related through the control on the boundary of the domain, and that such a relation may be imposed in the variational formulation of the adjoint state. Well-posedness (unique solvability and stability) of the variational problem is established in the $H^{1}(\Omega)\times H_{0}^{1}(\Omega)$ space for the respective state and adjoint state. A finite element method based on this formulation is analyzed. It is shown that the conforming $k-$th order finite element approximations to the state and the adjoint state, in the respective $L^{2}$ and $H^{1}$ norms converge at the rate of order $k-1/2$ on quasi-uniform mesh for conforming element of order $k$. Numerical examples are presented to validate the theory.