A Local Discontinuous Galerkin Method for Dirichlet Boundary Control Problems

TL;DR

This paper introduces a local discontinuous Galerkin method for solving Dirichlet boundary control problems in convection-diffusion equations, providing error estimates and numerical validation.

Contribution

It develops a novel discretization approach using local discontinuous Galerkin methods for boundary control problems with theoretical error analysis.

Findings

Error estimates for the control and state variables

Numerical results confirming theoretical convergence

Effective handling of control constraints

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.