Communications in Mathematical Sciences

Volume 20 (2022)

Number 4

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

Pages: 1081 – 1102

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n4.a6

Authors

Shaohong Du (School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, China)

Zhiqiang Cai (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Abstract

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.

Keywords

Dirichlet boundary control problem, new variational formulation, finite element method, a priori error estimates

2010 Mathematics Subject Classification

49K20, 49M25, 65K10, 65N21, 65N30

Received 8 February 2021

Received revised 18 October 2021

Accepted 29 October 2021

Published 11 April 2022