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

Kyza, Irene

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.

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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