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A Local Discontinuous Galerkin Method for Dirichlet Boundary Control Problems

Peter Benner, Michael Hinze, Hamdullah Yücel

TL;DR

This work analyzes Dirichlet boundary control for a 2D convection–diffusion equation on convex polygonal domains using a local discontinuous Galerkin discretization with $P_1$ flux and potential approximations. It develops and compares two control discretization strategies—variational discretization and full discretization—and proves a priori error estimates for the state, adjoint, and control, showing a worst-case rate of $O(h^{1/2})$ under modest regularity. The analysis relies on LDG projection tools and a Carstensen-type quasi-interpolation, without requiring stability of the discrete solution operator, and confirms the theory with extensive numerical experiments across analytic, singular, and polygonal scenarios, including convection-dominated regimes. The results support the method’s robustness and suggest future work on adaptive refinement via a posteriori error estimators.

Abstract

In this paper, we consider control constrained $L^2-$Dirichlet boundary control of a convection-diffusion equation on a two dimensional convex polygonal domain. We discretize the control problem based on the local discontinuous Galerkin method with piecewise linear ansatz functions for the flux and potential. We derive a priori error estimates for the full as well as for the variational discrete control approximation. We present a selection of numerical results to demonstrate the performance of our approach and to underpin the theoretical findings.

A Local Discontinuous Galerkin Method for Dirichlet Boundary Control Problems

TL;DR

This work analyzes Dirichlet boundary control for a 2D convection–diffusion equation on convex polygonal domains using a local discontinuous Galerkin discretization with flux and potential approximations. It develops and compares two control discretization strategies—variational discretization and full discretization—and proves a priori error estimates for the state, adjoint, and control, showing a worst-case rate of under modest regularity. The analysis relies on LDG projection tools and a Carstensen-type quasi-interpolation, without requiring stability of the discrete solution operator, and confirms the theory with extensive numerical experiments across analytic, singular, and polygonal scenarios, including convection-dominated regimes. The results support the method’s robustness and suggest future work on adaptive refinement via a posteriori error estimators.

Abstract

In this paper, we consider control constrained Dirichlet boundary control of a convection-diffusion equation on a two dimensional convex polygonal domain. We discretize the control problem based on the local discontinuous Galerkin method with piecewise linear ansatz functions for the flux and potential. We derive a priori error estimates for the full as well as for the variational discrete control approximation. We present a selection of numerical results to demonstrate the performance of our approach and to underpin the theoretical findings.
Paper Structure (10 sections, 4 theorems, 87 equations, 4 figures, 6 tables)

This paper contains 10 sections, 4 theorems, 87 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Regularity and optimality system
  3. Local discontinuous Galerkin formulation
  4. Error estimates
  5. Discussion of Theorem \ref{['thm:nodisc_adjoint_representation']} and Theorem \ref{['thm:fully-discrete']}:
  6. Numerical Experiments
  7. Example 1
  8. Example 2
  9. Example 3
  10. Conclusions

Key Result

Theorem 2.2

Suppose that Assumption assumption_data is satisfied and $y^d \in H^{t}(\Omega)$ with $0 \leq t < 1$. Then, for $s\in [\frac{1}{2}, \min\{\frac{1}{2}+t,\pi/\theta -1/2\})$ we for the solution of the optimality system p13a-p13c have as well as

Figures (4)

  • Figure 1: Example \ref{['Ex1']}: Computed solutions of state $y$, adjoint $z$, and control $u$ (from left to right) with $\epsilon=1$.
  • Figure 2: Example \ref{['Ex_singular']}: Computed solutions of state $y$ and adjoint $z$ with $\epsilon=1$.
  • Figure 3: Example \ref{['Ex_polygonal']}: Domain with $\theta=\frac{5}{6} \pi$.
  • Figure 4: Example \ref{['Ex_polygonal']}: Computed solutions of state $y$ and adjoint $z$ with $\epsilon=1$.

Theorems & Definitions (7)

  • Theorem 2.2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 4.3
  • Remark 4.4