Numerical Analysis for Neumann Optimal Control Problems on Convex Polyhedral Domains

TL;DR

This paper establishes finite element error estimates for Neumann boundary control problems on convex polyhedral domains, showing convergence rates depend on domain geometry and introducing new techniques for boundary estimates.

Contribution

It provides new error estimates for discretized Neumann control problems on convex polyhedral domains, including boundary control convergence rates depending on interior edge angles.

Findings

Optimal convergence rates depend on the largest interior edge angle.

Boundary control converges at rate two in the L2-norm, independent of domain angles.

Numerical experiments confirm theoretical error estimates.

Abstract

This paper is concerned with finite element error estimates for Neumann boundary control problems posed on convex and polyhedral domains. Different discretization concepts are considered and for each optimal discretization error estimates are established. In particular, for a full discretization with piecewise linear and globally continuous functions for the control and standard linear finite elements for the state optimal convergence rates for the controls are proven which solely depend on the largest interior edge angle. To be more precise, below the critical edge angle of $2\pi/3$, a convergence rate of two (times a log-factor) can be achieved for the discrete controls in the $L^2$-norm on the boundary. For larger interior edge angles the convergence rates are reduced depending on their size, which is due the impact of singular (domain dependent) terms in the solution. The results are comparable to those for the two dimensional case. However, new techniques in this context are used to prove the estimates on the boundary which also extend to the two dimensional case. Moreover, it is shown that the discrete states converge with a rate of two in the $L^2$-norm in the domain independent of the interior edge angles, i.e. for any convex and polyhedral domain. It is remarkable that this not only holds for a full discretization using piecewise linear and globally continuous functions for the control, but also for a full discretization using piecewise constant functions for the control, where the discrete controls only converge with a rate of one in the $L^2$-norm on the boundary at best. At the end, the theoretical results are confirmed by several numerical experiments.