2019/04/22 by Shaohong Du, Zhiqiang Cai, Du, Shaohong +1
Computer Science · Engineering · #49K20 #49M25 #65K10 #65N21 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1904.09783
openalex publication_date 2019/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 H1(Ω)× H01(Ω) 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 L2 and H1 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.