A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 4, p. 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 : https://doi.org/10.1051/m2an/2010101
Classification:  65M15,  35Q41
@article{M2AN_2011__45_4_761_0,
     author = {Kyza, Irene},
     title = {A posteriori error analysis for the Crank-Nicolson method for linear Schr\"odinger equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {4},
     year = {2011},
     pages = {761-778},
     doi = {10.1051/m2an/2010101},
     zbl = {1269.65088},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_4_761_0}
}
Kyza, Irene. A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 4, pp. 761-778. doi : 10.1051/m2an/2010101. http://www.numdam.org/item/M2AN_2011__45_4_761_0/

[1] G.D. Akrivis and V.A. Dougalis, On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation. RAIRO Modél. Math. Anal. Numér. 25 (1991) 643-670. | Numdam | MR 1135988 | Zbl 0744.65085

[2] G. Akrivis, Ch. Makridakis and R.H. Nochetto, A posteriori error estimates for the Crank-Nicolson method for parabolic equations. Math. Comput. 75 (2006) 511-531. | MR 2196979 | Zbl 1101.65094

[3] G. Akrivis, Ch. Makridakis and R.H. Nochetto, Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods. Numer. Math. 114 (2009) 133-160. | MR 2557872 | Zbl 1188.65108

[4] R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains. Bull. Soc. Math. France 136 (2008) 27-65. | Numdam | MR 2415335 | Zbl 1157.35100

[5] D. Bohm, Quantum Theory. Dover Publications, New York (1979).

[6] A. Brocéhn, Galerkin methods for approximation of solutions of second order partial differential equations of Schrödinger type. Ph.D. thesis, University of Göteborg (1980).

[7] 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). | MR 1156075 | Zbl 0755.35001

[8] W. Dörfler, A time-and space-adaptive algorithm for the linear time-dependent Schrödinger equation. Numer. Math. 73 (1996) 419-448. | MR 1393174 | Zbl 0860.65097

[9] L.C. Evans, Partial Differential Equations. Second edition, American Mathematical Society, Providence (2002). | MR 2597943

[10] P. Górka, Convergence of logarithmic quantum mechanics to the linear one. Lett. Math. Phys. 81 (2007) 253-264. | MR 2355491 | Zbl 1136.81356

[11] Th. Katsaounis and I. Kyza, A posteriori error estimates in the L∞(L2)-norm for Crank-Nicolson fully discrete approximations for linear Schrödinger equations. Preprint.

[12] O. Karakashian and Ch. Makridakis, A space-time finite element method for the nonlinear Schrodinger equation: the discontinuous Galerkin method. Math. Comput. 67 (1998) 479-499. | MR 1459390 | Zbl 0896.65068

[13] O. Karakashian and Ch. Makridakis, A space-time finite element method for the nonlinear Schrodinger equation: the continuous Galerkin method. SIAM J. Numer. Anal. 36 (1999) 1779-1807. | MR 1712169 | Zbl 0934.65110

[14] I. Kyza, A posteriori error estimates for approximations of semilinear parabolic and Schrödinger-type equations. Ph.D. thesis, University of Crete (2009).

[15] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications 2. Dunod, Paris (1968). | MR 247244 | Zbl 0165.10801

[16] A. Lozinski, M. Picasso and V. Prachittham, An anisotropic error estimator for the Crank-Nicolson method: Application to a parabolic problem. SIAM J. Sci. Comput. 31 (2009) 2757-2783. | MR 2520298 | Zbl 1215.65154

[17] Ch. Makridakis, Space and time reconstructions in a posteriori analysis of evolution problems. ESAIM: Proc. 21 (2007) 31-44. | MR 2404052 | Zbl 1128.65062

[18] M.O. Scully and M.S. Zubairy, Quantum Optics. Cambridge University Press (2002).

[19] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Second edition, Springer-Verlag, Berlin (2006). | Zbl 0528.65052

[20] R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003) 195-212. | MR 2025602 | Zbl 1168.65418