New a posteriori L ∞ ( L 2 ) and L 2 ( L 2 ) -error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

Zuliang Lu

Applications of Mathematics (2016)

  • Volume: 61, Issue: 2, page 135-163
  • ISSN: 0862-7940

Abstract

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We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L ∞ ( J ; L 2 ( Ω ) ) -norm and L 2 ( J ; L 2 ( Ω ) ) -norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.

How to cite

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Lu, Zuliang. "New a posteriori $L^{\infty }(L^2) $ and $L^2(L^2)$-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems." Applications of Mathematics 61.2 (2016): 135-163. <http://eudml.org/doc/276761>.

@article{Lu2016,
abstract = {We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in $L^\{\infty \}(J;L^2(\Omega )) $-norm and $L^2(J;L^2(\Omega ))$-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.},
author = {Lu, Zuliang},
journal = {Applications of Mathematics},
keywords = {a posteriori error estimate; general optimal control problem; nonlinear parabolic equation; mixed finite element method; a posteriori error estimate; general optimal control problem; nonlinear parabolic equation; mixed finite element method},
language = {eng},
number = {2},
pages = {135-163},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {New a posteriori $L^\{\infty \}(L^2) $ and $L^2(L^2)$-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems},
url = {http://eudml.org/doc/276761},
volume = {61},
year = {2016},
}

TY - JOUR
AU - Lu, Zuliang
TI - New a posteriori $L^{\infty }(L^2) $ and $L^2(L^2)$-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems
JO - Applications of Mathematics
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 2
SP - 135
EP - 163
AB - We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in $L^{\infty }(J;L^2(\Omega )) $-norm and $L^2(J;L^2(\Omega ))$-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.
LA - eng
KW - a posteriori error estimate; general optimal control problem; nonlinear parabolic equation; mixed finite element method; a posteriori error estimate; general optimal control problem; nonlinear parabolic equation; mixed finite element method
UR - http://eudml.org/doc/276761
ER -

References

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