Communications in Mathematical Sciences
Volume 20 (2022)
A finite element method for Dirichlet boundary control of elliptic partial differential equations
Pages: 1081 – 1102
This paper introduces a new variational formulation for Dirichlet boundary control problem of elliptic partial differential equations, based on an observation 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 new variational problem is established in the $H^1 (\Omega) \times H^1_0 (\Omega)$ spaces 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 meshes. Numerical examples are presented to validate the theory.
Dirichlet boundary control problem, new variational formulation, finite element method, a priori error estimates
2010 Mathematics Subject Classification
49K20, 49M25, 65K10, 65N21, 65N30
The first-named author’s work is supported in part by the Natural Science Foundation of Chongqing under Grant cstc2018jcyjAX490, the Education Science Foundation of Chongqing under Grant KJZD-K201900701, the Group Building Scientific Innovation Project for universities in Chongqing (CXQT21021), and the Joint Training Base Construction Projection for Graduate Students in Chongqing (JDLHPYJD202101 6).
Received 8 February 2021
Received revised 18 October 2021
Accepted 29 October 2021
Published 11 April 2022