A posteriori error estimates for adaptive finite element discretizations of boundary control problems

2006 ◽  
Vol 14 (1) ◽  
pp. 57-82 ◽  
Author(s):  
R. H. W. Hoppe ◽  
Y. Iliash ◽  
C. Iyyunni ◽  
N. H. Sweilam
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Boundary Control ◽  
Control Problems ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
Boundary Control Problems ◽  
Finite Element Discretizations ◽  
Element Discretizations

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.

A Posteriori Error Estimates of Lowest Order Raviart-Thomas Mixed Finite Element Methods for Bilinear Optimal Control Problems

2012 ◽  
Vol 2 (2) ◽  
pp. 108-125 ◽  
Author(s):  
Zuliang Lu ◽  
Yanping Chen ◽  
Weishan Zheng
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Error Estimates ◽  
Optimal Control Problems ◽  
Mixed Finite Element ◽  
A Posteriori Error Estimates ◽  
Control Problems ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
Element Discretization

AbstractA Raviart-Thomas mixed finite element discretization for general bilinear optimal control problems is discussed. The state and co-state are approximated by lowest order Raviart-Thomas mixed finite element spaces, and the control is discretized by piecewise constant functions. A posteriori error estimates are derived for both the coupled state and the control solutions, and the error estimators can be used to construct more efficient adaptive finite element approximations for bilinear optimal control problems. An adaptive algorithm to guide the mesh refinement is also provided. Finally, we present a numerical example to demonstrate our theoretical results.

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