A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations*

Irene Kyza

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 4, page 761-778
  • ISSN: 0764-583X

Abstract

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We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L∞(L2)- and the L∞(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds that are of optimal order in the L∞(L2)-norm, but of suboptimal order in the L∞(H1)-norm. The optimality in the case of L∞(H1)-norm is recovered by using an auxiliary initial- and boundary-value problem.

How to cite

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Kyza, Irene. "A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations*." ESAIM: Mathematical Modelling and Numerical Analysis 45.4 (2011): 761-778. <http://eudml.org/doc/197399>.

@article{Kyza2011,
abstract = { We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L∞(L2)- and the L∞(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds that are of optimal order in the L∞(L2)-norm, but of suboptimal order in the L∞(H1)-norm. The optimality in the case of L∞(H1)-norm is recovered by using an auxiliary initial- and boundary-value problem. },
author = {Kyza, Irene},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Linear Schrödinger equation; Crank-Nicolson method; Crank-Nicolson reconstruction; a posteriori error analysis; energy techniques; L∞(L2)- and L∞(H1)-norm; linear Schrödinger equation; Crank-Nicolson reconstruction; a posteriori error analysis; energy techniques; - and -norm; numerical examples},
language = {eng},
month = {2},
number = {4},
pages = {761-778},
publisher = {EDP Sciences},
title = {A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations*},
url = {http://eudml.org/doc/197399},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Kyza, Irene
TI - A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations*
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/2//
PB - EDP Sciences
VL - 45
IS - 4
SP - 761
EP - 778
AB - We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L∞(L2)- and the L∞(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds that are of optimal order in the L∞(L2)-norm, but of suboptimal order in the L∞(H1)-norm. The optimality in the case of L∞(H1)-norm is recovered by using an auxiliary initial- and boundary-value problem.
LA - eng
KW - Linear Schrödinger equation; Crank-Nicolson method; Crank-Nicolson reconstruction; a posteriori error analysis; energy techniques; L∞(L2)- and L∞(H1)-norm; linear Schrödinger equation; Crank-Nicolson reconstruction; a posteriori error analysis; energy techniques; - and -norm; numerical examples
UR - http://eudml.org/doc/197399
ER -

References

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