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

Gong, Wei, Liu, Wenbin, Tan, Zhiyu, Yan, Ningning (2019) A convergent adaptive finite element method for elliptic Dirichlet boundary control problems. IMA Journal of Numerical Analysis, 39 (4). pp. 1985-2015. ISSN 1464-3642. (doi:10.1093/imanum/dry051)

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https://doi.org/10.1093/imanum/dry051

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.

Item Type: Article
DOI/Identification number: 10.1093/imanum/dry051
Uncontrolled keywords: Applied Mathematics, General Mathematics, Computational Mathematics
Divisions: Faculties > Social Sciences > Kent Business School
SWORD Depositor: JISC Publications Router
Depositing User: Steve Wenbin Liu
Date Deposited: 09 Nov 2018 16:32 UTC
Last Modified: 20 Nov 2019 15:20 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/69216 (The current URI for this page, for reference purposes)
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