scholarly journals A Finite Element Method for Elliptic Dirichlet Boundary Control Problems

2020 ◽  
Vol 20 (4) ◽  
pp. 827-843
Author(s):  
Michael Karkulik
Keyword(s):  
Finite Element ◽  
Boundary Control ◽  
Dirichlet Boundary ◽  
Mixed Finite Element ◽  
Lagrangian Multiplier ◽  
Finite Element Discretization ◽  
Dirichlet Boundary Control ◽  
Element Discretization ◽  
Polygonal Domains ◽  
Constrained Problems

AbstractWe consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in {H^{1/2}(\Gamma)}. To avoid computing the latter norm numerically, we realize it using the {H^{1}(\Omega)} norm of the harmonic extension of the control. We propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier. In the case of convex polygonal domains, optimal error estimates in the {H^{1}} and {L_{2}} norm are proven. We also consider and analyze the case of control constrained problems.

Improved error estimates for semidiscrete finite element solutions of parabolic Dirichlet boundary control problems

2019 ◽  
Vol 40 (4) ◽  
pp. 2898-2939 ◽  
Author(s):  
Wei Gong ◽  
Buyang Li
Keyword(s):  
Finite Element ◽  
Boundary Control ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Finite Element Discretization ◽  
Dirichlet Boundary Control ◽  
Element Discretization ◽  
Boundary Control Problem ◽  
Boundary Control Problems

Abstract The parabolic Dirichlet boundary control problem and its finite element discretization are considered in convex polygonal and polyhedral domains. We improve the existing results on the regularity of the solutions by establishing and utilizing the maximal $L^p$-regularity of parabolic equations under inhomogeneous Dirichlet boundary conditions. Based on the proved regularity of the solutions, we prove ${\mathcal O}(h^{1-1/q_0-\epsilon })$ convergence for the semidiscrete finite element solutions for some $q_0>2$, with $q_0$ depending on the maximal interior angle at the corners and edges of the domain and $\epsilon$ being a positive number that can be arbitrarily small.

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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