A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 4, pp. 761-778.

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.

DOI : 10.1051/m2an/2010101
Classification : 65M15, 35Q41
Mots-clés : linear Schrödinger equation, Crank-Nicolson method, crank-nicolson reconstruction, a posteriori error analysis, energy techniques, L∞(L2)- and L∞(H1)-norm 
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     author = {Kyza, Irene},
     title = {\protect\emph{A posteriori} error analysis for the {Crank-Nicolson} method for linear {Schr\"odinger} equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {761--778},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {4},
     year = {2011},
     doi = {10.1051/m2an/2010101},
     zbl = {1269.65088},
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     url = {http://www.numdam.org/articles/10.1051/m2an/2010101/}
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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 , Tome 45 (2011) no. 4, pp. 761-778. doi : 10.1051/m2an/2010101. http://www.numdam.org/articles/10.1051/m2an/2010101/

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