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Convergence of Adaptive Crouzeix–Raviart and Morley FEM for Distributed Optimal Control Problems

  • Asha K. Dond ORCID logo EMAIL logo , Neela Nataraj and Subham Nayak
Published/Copyright: February 10, 2024

Abstract

This article discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by 𝑚-harmonic operators for m = 1 , 2 . A variational discretization approach is employed and the state and adjoint variables are discretized using nonconforming finite elements. Error equivalence results at the continuous and discrete levels lead to a priori and a posteriori error estimates for the optimal control problem. The general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality establishes the quasi-optimality. Numerical results demonstrate the theoretically predicted orders of convergence and the efficiency of the adaptive estimator.

MSC 2020: 65N30; 65N15; 49M25; 49M05

Award Identifier / Grant number: SRG/2020/001027

Award Identifier / Grant number: SPF/2020/000019

Funding statement: Asha K. Dond gratefully acknowledges funding from the Science and Engineering Research Board (SERB), Government of India, through the Start-up Research Grant, Project No. SRG/2020/001027. Neela Nataraj gratefully acknowledges the SERB POWER Fellowship SPF/2020/000019 and also the kind hospitality of IISER Tvm. Asha K. Dond and Subham Nayak acknowledge the research visit to IITB from the SERB POWER Fellowship SPF/2020/000019.

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Received: 2023-03-30
Revised: 2023-11-17
Accepted: 2024-01-15
Published Online: 2024-02-10
Published in Print: 2024-07-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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