The gradient discretization method for optimal control problems, with superconvergence for nonconforming finite elements and mixed-hybrid mimetic finite differences

Jérome Droniou, Neela Nataraj, Devika Shylaja

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)

Abstract

In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretization method. Gradient schemes are defined for the optimality system of the control problem. Error es- timates for state, adjoint, and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution are discussed. These supercon- vergence results are shown to apply to nonconforming P1 finite elements and to the mixed-hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, nonconforming and mixed-hybrid mimetic finite difference schemes.

Original languageEnglish
Pages (from-to)3640-3672
Number of pages33
JournalSIAM Journal on Control and Optimization
Volume55
Issue number6
DOIs
Publication statusPublished - 2017

Keywords

  • Elliptic equations
  • Error estimates
  • Gradient discretization method
  • Gradient schemes
  • Mimetic finite differences
  • Nonconforming P1 finite elements
  • Numerical schemes
  • Optimal control
  • Super- convergence

Cite this

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title = "The gradient discretization method for optimal control problems, with superconvergence for nonconforming finite elements and mixed-hybrid mimetic finite differences",
abstract = "In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretization method. Gradient schemes are defined for the optimality system of the control problem. Error es- timates for state, adjoint, and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution are discussed. These supercon- vergence results are shown to apply to nonconforming P1 finite elements and to the mixed-hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, nonconforming and mixed-hybrid mimetic finite difference schemes.",
keywords = "Elliptic equations, Error estimates, Gradient discretization method, Gradient schemes, Mimetic finite differences, Nonconforming P1 finite elements, Numerical schemes, Optimal control, Super- convergence",
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The gradient discretization method for optimal control problems, with superconvergence for nonconforming finite elements and mixed-hybrid mimetic finite differences. / Droniou, Jérome; Nataraj, Neela; Shylaja, Devika.

In: SIAM Journal on Control and Optimization, Vol. 55, No. 6, 2017, p. 3640-3672.

Research output: Contribution to journalArticleResearchpeer-review

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KW - Error estimates

KW - Gradient discretization method

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