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

Published online by Cambridge University Press:  21 February 2011

Irene Kyza
Irene Kyza*
Affiliation:
Department of Mathematics, Mathematics Building, University of Maryland, College Park, 20742-4015 MD, USA. kyza@math.umd.edu

Abstract

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We prove a posteriori error estimates of optimal order for linearSchrödinger-type equations in the L ∞(L 2)- and theL ∞(H 1)-norm. We discretize only in time by theCrank-Nicolson method. The direct use of the reconstructiontechnique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds thatare of optimal order in the L ∞(L 2)-norm, but ofsuboptimal order in the L ∞(H 1)-norm. The optimality inthe case of L ∞(H 1)-norm is recovered by using anauxiliary initial- and boundary-value problem.

Keywords

Linear Schrödinger equationCrank-Nicolson methodCrank-Nicolson reconstructiona posteriori error analysisenergytechniquesL ∞(L 2)- and L ∞(H 1)-norm

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 4 , July 2011, pp. 761 - 778
Copyright
© EDP Sciences, SMAI, 2011

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