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A posteriori error analysis for the Crank-Nicolson method 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. [email protected]
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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.

Keywords

Linear Schrödinger equationCrank-Nicolson methodCrank-Nicolson reconstructiona posteriori error analysisenergy techniquesL∞(L2)- and L∞(H1)-norm
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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