Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

Konstantinos Chrysafinos

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 1, page 189-206
  • ISSN: 0764-583X

Abstract

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A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at arbitrary time points under minimal regularity assumptions, and a discrete compactness argument for discontinuous Galerkin schemes (see Walkington [SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4).

How to cite

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Chrysafinos, Konstantinos. "Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's." ESAIM: Mathematical Modelling and Numerical Analysis 44.1 (2010): 189-206. <http://eudml.org/doc/250854>.

@article{Chrysafinos2010,
abstract = { A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at arbitrary time points under minimal regularity assumptions, and a discrete compactness argument for discontinuous Galerkin schemes (see Walkington [SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4). },
author = {Chrysafinos, Konstantinos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Discontinuous Galerkin approximations; distributed controls; stability estimates; semi-linear parabolic PDE's.; distributed controls; semi-linear parabolic PDE's; optimal control; discontinuous Galerkin finite element method; convergence},
language = {eng},
month = {3},
number = {1},
pages = {189-206},
publisher = {EDP Sciences},
title = {Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's},
url = {http://eudml.org/doc/250854},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Chrysafinos, Konstantinos
TI - Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 1
SP - 189
EP - 206
AB - A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at arbitrary time points under minimal regularity assumptions, and a discrete compactness argument for discontinuous Galerkin schemes (see Walkington [SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4).
LA - eng
KW - Discontinuous Galerkin approximations; distributed controls; stability estimates; semi-linear parabolic PDE's.; distributed controls; semi-linear parabolic PDE's; optimal control; discontinuous Galerkin finite element method; convergence
UR - http://eudml.org/doc/250854
ER -

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