A new error analysis for parabolic Dirichlet boundary control problems

¹ The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen 518057, P.R. China
² School of Mathematics, Sichuan University, Chengdu 610064, P.R. China
³ NCMIS & LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China

^* Corresponding author: wgong@lsec.cc.ac.cn

Received: 2 January 2024
Accepted: 20 January 2025

Abstract

This paper investigates the finite element approximation of a parabolic Dirichlet boundary control problem, presenting a new a priori error estimate. We establish two main convergence results for both semi-discrete and fully discrete optimal control problems, under suitable assumptions. Specifically, we demonstrate convergence orders of O(k^¼) and O(k^{¾ − ɛ}) (∀ɛ > 0) for the temporal semi-discretization of control problems on polytopes and smooth domains, respectively. For control problems defined on polyhedra, we achieve a convergence rate of O(k^¼ + h^½) in the fully discrete setting. The contributions of this work are twofold. First, we provide an improved temporal convergence rate for parabolic Dirichlet boundary control problems on smooth domains, setting a foundation for further fully discrete error analysis. Second, we refine the existing fully discrete error estimate for boundary control problems on polyhedra by removing the artificial mesh size restriction k = O(h²). As an intermediate but essential result, we establish both the convergence order and stability of the finite element approximation for parabolic inhomogeneous boundary value problems. Importantly, these results hold under low regularity boundary conditions without imposing mesh size constraints.