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

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 are related through domain, such relation may be imposed in state. Well-posedness (unique solvability stability) is established $H^{1}(\Omega)\times H_{0}^{1}(\Omega)$ space respective A finite element method this analyzed. It shown conforming $k-$th order approximations to state, $L^{2}$ $H^{1}$ norms converge at rate $k-1/2$ quasi-uniform mesh $k$. Numerical examples presented validate theory.