Residual based a posteriori error estimation for Dirichlet boundary control problems

Hamdullah Yücel

doi:10.1051/proc/202171185

Abstract

We study a residual–based a posteriori error estimate for the solution of Dirichlet boundary control problem governed by a convection diffusion equation on a two dimensional convex polygonal domain, using the local discontinuous Galerkin (LDG) method with upwinding for the convection term. With the usage of LDG method, the control variable naturally exists in the variational form due to its mixed finite element structure. We also demonstrate the application of our a posteriori error estimator for the adaptive solution of these optimal control problems.

Highlights

We investigate a numerical approximation of Dirichlet boundary control problems governed by a convection diffusion equation: minimize
Primal–dual weighted error estimates were derived in [28] for Dirichlet boundary control problem governed by a convection diffusion equation with control constraints
We intent to contribute a residual–based a-posteriori error estimates for the solution of Dirichlet boundary control problem governed by a convection diffusion equation, using the local discontinuous Galerkin method with upwinding for the convection term

Summary

Introduction

We investigate a numerical approximation of Dirichlet boundary control problems governed by a convection diffusion equation: minimize. The convergence properties of discretization methods applied to the optimal control problems can be substantially different from the convergence properties of discretization methods applied to a single convection dominated PDEs due to the transport of the information in the optimality system with opposite directions In such kind of problems (1)-(2), the specific difficulty is that the control variable is not involved in the variational form of the standard finite element setting. Primal–dual weighted error estimates were derived in [28] for Dirichlet boundary control problem governed by a convection diffusion equation with control constraints. We intent to contribute a residual–based a-posteriori error estimates for the solution of Dirichlet boundary control problem governed by a convection diffusion equation, using the local discontinuous Galerkin method with upwinding for the convection term.

Discretization of Model Problem

A Posteriori Error Estimates

Numerical Experiments