Numerical Analysis for Dirichlet Optimal Control Problems on Convex Polyhedral Domains

TL;DR

This work extends a priori error analysis for Dirichlet boundary control of the Laplace equation to three-dimensional convex polyhedral domains, showing that FE discretization errors depend on the domain's largest interior edge angle via $ω_Ω$. The authors develop a weighted-regularity framework, including a normal trace theorem for convex polyhedra, and establish discrete weighted $H^1$ regularity for the state, enabling two discretization schemes—variational discretization and cellwise linear discretization—to yield convergent error estimates for the boundary control. They prove that, under mild regularity of the desired state $u_d$, the control error satisfies $\|\bar q-\bar q_h\|_{L^2(∂Ω)} ≤ c h^{1/2+s}$ (for $0<s<\min(π/ω_Ω-1,1/2)$), with first-order convergence up to a $|\ln h|$ factor when $ω_Ω<2π/3$, matching the 2D theory in spirit but with novel 3D techniques. Numerical experiments corroborate the theoretical rates and illustrate the practicality of the approach for convex polyhedral domains.

Abstract

In this paper error analysis for finite element discretizations of Dirichlet boundary control problems is developed. For the first time, optimal discretization error estimates are established in the case of three dimensional polyhedral and convex domains. The convergence rates solely depend on the size of largest interior edge angle. These results are comparable to those for the two dimensional case. However, the approaches from the two dimensional setting are not directly extendable such that new techniques have to be used. The theoretical results are confirmed by numerical experiments.

Figures (2)

• Figure 1: Domain $\Omega_\omega$ for $\omega\in\set{\frac{\pi}{2},\frac{2\pi}{3},\frac{3\pi}{4}}$
• Figure 2: Error $\lVert\bar{q}-\bar{q}_h\rVert_{L^2(\partial \Omega)}$

Theorems & Definitions (51)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 1
• proof
• Theorem 4
• proof