Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method

Jérome Droniou, Neela Nataraj, Devika Shylaja

Research output: Contribution to journalArticleResearchpeer-review

1 Citation (Scopus)

Abstract

The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low-order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, non-conforming and mimetic finite difference methods confirm the theoretical rates of convergence.

Original languageEnglish
Pages (from-to)609-637
Number of pages29
JournalComputational Methods in Applied Mathematics
Volume18
Issue number4
DOIs
Publication statusPublished - Oct 2018

Keywords

  • Elliptic Equations
  • Error Estimates
  • Finite Elements
  • GDM
  • Mimetic Finite Differences
  • Neumann Boundary Conditions
  • Optimal Control
  • Super-Convergence

Cite this

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abstract = "The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low-order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, non-conforming and mimetic finite difference methods confirm the theoretical rates of convergence.",
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Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method. / Droniou, Jérome; Nataraj, Neela; Shylaja, Devika.

In: Computational Methods in Applied Mathematics, Vol. 18, No. 4, 10.2018, p. 609-637.

Research output: Contribution to journalArticleResearchpeer-review

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