Two fast finite difference schemes for elliptic Dirichlet boundary control problems

2019 ◽  
Vol 61 (1-2) ◽  
pp. 481-503
Author(s):  
Jun Liu
Keyword(s):  
Finite Difference ◽  
Boundary Control ◽  
Dirichlet Boundary ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Control Problems ◽  
Dirichlet Boundary Control ◽  
Boundary Control Problems

scholarly journals Crank-Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problem

2021 ◽  
Author(s):  
Caijie Yang ◽  
Tongjun Sun
Keyword(s):  
Finite Difference ◽  
Control Problem ◽  
Boundary Control ◽  
Dirichlet Boundary ◽  
Second Order ◽  
Difference Schemes ◽  
Optimality System ◽  
Finite Difference Schemes ◽  
Dirichlet Boundary Control ◽  
Boundary Control Problem

In this paper, we adopt the optimize-then-discretize approach to solve parabolic optimal Dirichlet boundary control problem. First, we derive the first-order necessary optimality system, which includes the state, co-state equations and the optimality condition. Then, we propose Crank-Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible and second-order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present a numerical example to validate our theoretical analysis.

Energy corrected FEM for optimal Dirichlet boundary control problems

2018 ◽  
Vol 139 (4) ◽  
pp. 913-938 ◽  
Author(s):  
Lorenz John ◽  
Piotr Swierczynski ◽  
Barbara Wohlmuth
Keyword(s):  
Boundary Control ◽  
Dirichlet Boundary ◽  
Control Problems ◽  
Dirichlet Boundary Control ◽  
Boundary 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.

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