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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
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Abstract

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
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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References

Akrivis, G.D. and Dougalis, V.A., On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation. RAIRO Modél. Math. Anal. Numér. 25 (1991) 643–670. CrossRef
Akrivis, G., Makridakis, Ch. and Nochetto, R.H., A posteriori error estimates for the Crank-Nicolson method for parabolic equations. Math. Comput. 75 (2006) 511–531. CrossRef
Akrivis, G., Makridakis, Ch. and Nochetto, R.H., Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods. Numer. Math. 114 (2009) 133–160. CrossRef
Anton, R., Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains. Bull. Soc. Math. France 136 (2008) 27–65. CrossRef
D. Bohm, Quantum Theory. Dover Publications, New York (1979).
A. Brocéhn, Galerkin methods for approximation of solutions of second order partial differential equations of Schrödinger type. Ph.D. thesis, University of Göteborg (1980).
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology 5, Evolution Problems I. Second edition, Springer-Verlag, Berlin (2000).
Dörfler, W., A time-and space-adaptive algorithm for the linear time-dependent Schrödinger equation. Numer. Math. 73 (1996) 419–448.
L.C. Evans, Partial Differential Equations. Second edition, American Mathematical Society, Providence (2002).
Górka, P., Convergence of logarithmic quantum mechanics to the linear one. Lett. Math. Phys. 81 (2007) 253–264. CrossRef
Th. Katsaounis and I. Kyza, A posteriori error estimates in the L∞(L 2)-norm for Crank-Nicolson fully discrete approximations for linear Schrödinger equations. Preprint.
Karakashian, O. and Makridakis, Ch., A space-time finite element method for the nonlinear Schrodinger equation: the discontinuous Galerkin method. Math. Comput. 67 (1998) 479–499. CrossRef
Karakashian, O. and Makridakis, Ch., A space-time finite element method for the nonlinear Schrodinger equation: the continuous Galerkin method. SIAM J. Numer. Anal. 36 (1999) 1779–1807. CrossRef
I. Kyza, A posteriori error estimates for approximations of semilinear parabolic and Schrödinger-type equations. Ph.D. thesis, University of Crete (2009).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications 2. Dunod, Paris (1968).
Lozinski, A., Picasso, M. and Prachittham, V., An anisotropic error estimator for the Crank-Nicolson method: Application to a parabolic problem. SIAM J. Sci. Comput. 31 (2009) 2757–2783. CrossRef
Makridakis, Ch., Space and time reconstructions in a posteriori analysis of evolution problems. ESAIM: Proc. 21 (2007) 31–44. CrossRef
M.O. Scully and M.S. Zubairy, Quantum Optics. Cambridge University Press (2002).
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Second edition, Springer-Verlag, Berlin (2006).
Verfürth, R., A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003) 195–212. CrossRef