scholarly journals Residual based a posteriori error estimation for Dirichlet boundary control problems

2021 ◽  
Vol 71 ◽  
pp. 185-195
Author(s):  
Hamdullah Yücel
Keyword(s):  
Boundary Control ◽  
Dirichlet Boundary ◽  
Posteriori Error ◽  
Control Problems ◽  
Error Estimator ◽  
Polygonal Domain ◽  
A Posteriori ◽  
Dirichlet Boundary Control ◽  
A Posteriori Error ◽  
Ldg Method

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.

scholarly journals A convergent adaptive finite element method for elliptic Dirichlet boundary control problems

2018 ◽  
Vol 39 (4) ◽  
pp. 1985-2015 ◽  
Author(s):  
Wei Gong ◽  
Wenbin Liu ◽  
Zhiyu Tan ◽  
Ningning Yan
Keyword(s):  
Finite Element ◽  
Boundary Control ◽  
Dirichlet Boundary ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Control Problems ◽  
Adaptive Finite Element ◽  
Dirichlet Boundary Control ◽  
A Posteriori Error ◽  
Boundary Control Problems

Abstract This paper concerns the adaptive finite element method for elliptic Dirichlet boundary control problems in the energy space. The contribution of this paper is twofold. First, we rigorously derive efficient and reliable a posteriori error estimates for finite element approximations of Dirichlet boundary control problems. As a by-product, a priori error estimates are derived in a simple way by introducing appropriate auxiliary problems and establishing certain norm equivalence. Secondly, for the coupled elliptic partial differential system that resulted from the first-order optimality system, we prove that the sequence of adaptively generated discrete solutions including the control, the state and the adjoint state, guided by our newly derived a posteriori error indicators, converges to the true solution along with the convergence of the error estimators. We give some numerical results to confirm our theoretical findings.

Equivalent a Posteriori Error Estimator of Spectral Approximation for Control Problems with Integral Control-State Constraints in One Dimension

2016 ◽  
Vol 8 (3) ◽  
pp. 464-484
Author(s):  
Fenglin Huang ◽  
Yanping Chen ◽  
Xiulian Shi
Keyword(s):  
State Constraints ◽  
Posteriori Error ◽  
Control Problems ◽  
Error Estimator ◽  
Spectral Approximation ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Integral Control ◽  
Posteriori Error Estimator ◽  
A Posteriori Error

Abstract.In this paper, we investigate the Galerkin spectral approximation for elliptic control problems with integral control and state constraints. Firstly, an a posteriori error estimator is established,which can be acted as the equivalent indicatorwith explicit expression. Secondly, appropriate base functions of the discrete spacesmake it is probable to solve the discrete system. Numerical test indicates the reliability and efficiency of the estimator, and shows the proposed method is competitive for this class of control problems. These discussions can certainly be extended to two- and three-dimensional cases.

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