Volume 45, Number 4, July-August 2011
|Page(s)||761 - 778|
|Published online||21 February 2011|
A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations*
Department of Mathematics,
University of Maryland,
College Park, 20742-4015 MD, USA. firstname.lastname@example.org
Revised: 16 August 2010
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.
Mathematics Subject Classification: 65M15 / 35Q41
Key words: Linear Schrödinger equation / Crank-Nicolson method / Crank-Nicolson reconstruction / a posteriori error analysis / energy techniques / L∞(L2)- and L∞(H1)-norm
© EDP Sciences, SMAI, 2011
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