Close Search Advanced
Journals
Resources
About Us
Open Access

Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints

Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints
Read Now
Download (PDF)

Year:    2012

Author:    M. Hintermüller, Michael Hinze, Ronald H.W. Hoppe

Journal of Computational Mathematics, Vol. 30 (2012), Iss. 2 : pp. 101–123

Abstract

Adaptive finite element methods for optimization problems for second order linear elliptic partial differential equations subject to pointwise constraints on the $\mathcal{ℓ}^2$-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.

Submit Article

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1109-m3522

Journal of Computational Mathematics, Vol. 30 (2012), Iss. 2 : pp. 101–123

Published online:    2012-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    23

Keywords:    Adaptive finite element method A posteriori errors Dualization Low regularity Pointwise gradient constraints State constraints Weak solutions.

Author Details

M. Hintermüller

Michael Hinze

Ronald H.W. Hoppe

  1. Preconditioned Solution of State Gradient Constrained Elliptic Optimal Control Problems

    Herzog, Roland | Mach, Susann

    SIAM Journal on Numerical Analysis, Vol. 54 (2016), Iss. 2 P.688

    https://doi.org/10.1137/130948045 [Citations: 3]

  2. Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

    Zhou, Zhaojie | Yu, Xiaoming | Yan, Ningning

    Numerical Methods for Partial Differential Equations, Vol. 30 (2014), Iss. 1 P.339

    https://doi.org/10.1002/num.21815 [Citations: 20]

  3. A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization

    Keil, Tim | Mechelli, Luca | Ohlberger, Mario | Schindler, Felix | Volkwein, Stefan

    ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 55 (2021), Iss. 3 P.1239

    https://doi.org/10.1051/m2an/2021019 [Citations: 12]

  4. Optimal Control of Elliptic Equations with Pointwise Constraints on the Gradient of the State in Nonsmooth Polygonal Domains

    Wollner, W.

    SIAM Journal on Control and Optimization, Vol. 50 (2012), Iss. 4 P.2117

    https://doi.org/10.1137/110836419 [Citations: 7]
  5. Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems

    Non-conforming Localized Model Reduction with Online Enrichment: Towards Optimal Complexity in PDE Constrained Optimization

    Ohlberger, Mario | Schindler, Felix

    2017

    https://doi.org/10.1007/978-3-319-57394-6_38 [Citations: 4]
  6. Optimization with PDE Constraints

    Adaptive Finite Elements for Optimally Controlled Elliptic Variational Inequalities of Obstacle Type

    Gaevskaya, A. | Hintermüller, M. | Hoppe, R. H. W. | Löbhard, C.

    2014

    https://doi.org/10.1007/978-3-319-08025-3_4 [Citations: 5]
  7. Geometric Partial Differential Equations - Part II

    Optimal control of geometric partial differential equations

    Hintermüller, Michael | Keil, Tobias

    2021

    https://doi.org/10.1016/bs.hna.2020.10.003 [Citations: 0]
  8. Trends in PDE Constrained Optimization

    A-Priori Error Bounds for Finite Element Approximation of Elliptic Optimal Control Problems with Gradient Constraints

    Deckelnick, Klaus | Hinze, Michael

    2014

    https://doi.org/10.1007/978-3-319-05083-6_23 [Citations: 0]
  9. Transport Processes at Fluidic Interfaces

    Fully Adaptive and Integrated Numerical Methods for the Simulation and Control of Variable Density Multiphase Flows Governed by Diffuse Interface Models

    Hintermüller, Michael | Hinze, Michael | Kahle, Christian | Keil, Tobias

    2017

    https://doi.org/10.1007/978-3-319-56602-3_13 [Citations: 0]

  10. Error Control Based Model Reduction for Parameter Optimization of Elliptic Homogenization Problems

    Ohlberger, Mario | Schaefer, Michael

    IFAC Proceedings Volumes, Vol. 46 (2013), Iss. 26 P.251

    https://doi.org/10.3182/20130925-3-FR-4043.00053 [Citations: 4]

  11. Finite element approximation and iterative method solution of elliptic control problem with constraints to gradient of state

    Dautov, R. | Lapin, A.

    Lobachevskii Journal of Mathematics, Vol. 36 (2015), Iss. 1 P.65

    https://doi.org/10.1134/S1995080215010059 [Citations: 2]