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

Irene Kyza

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

  • Volume: 45, Issue: 4, page 761-778
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

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 45.4 (2011): 761-778. <http://eudml.org/doc/273327>.

@article{Kyza2011,
abstract = {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.},
author = {Kyza, Irene},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {linear Schrödinger equation; Crank-Nicolson method; crank-nicolson reconstruction; a posteriori error analysis; energy techniques; L∞(L2)- and L∞(H1)-norm; Crank-Nicolson reconstruction; - and -norm; numerical examples},
language = {eng},
number = {4},
pages = {761-778},
publisher = {EDP-Sciences},
title = {A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations},
url = {http://eudml.org/doc/273327},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Kyza, Irene
TI - A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2011
PB - EDP-Sciences
VL - 45
IS - 4
SP - 761
EP - 778
AB - 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.
LA - eng
KW - linear Schrödinger equation; Crank-Nicolson method; crank-nicolson reconstruction; a posteriori error analysis; energy techniques; L∞(L2)- and L∞(H1)-norm; Crank-Nicolson reconstruction; - and -norm; numerical examples
UR - http://eudml.org/doc/273327
ER -

References

top
  1. [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. Zbl0744.65085MR1135988
  2. [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. Zbl1101.65094MR2196979
  3. [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. Zbl1188.65108MR2557872
  4. [4] R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains. Bull. Soc. Math. France136 (2008) 27–65. Zbl1157.35100MR2415335
  5. [5] D. Bohm, Quantum Theory. Dover Publications, New York (1979). 
  6. [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. [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). Zbl0755.35001MR1156075
  8. [8] W. Dörfler, A time-and space-adaptive algorithm for the linear time-dependent Schrödinger equation. Numer. Math.73 (1996) 419–448. Zbl0860.65097MR1393174
  9. [9] L.C. Evans, Partial Differential Equations. Second edition, American Mathematical Society, Providence (2002). MR2597943
  10. [10] P. Górka, Convergence of logarithmic quantum mechanics to the linear one. Lett. Math. Phys.81 (2007) 253–264. Zbl1136.81356MR2355491
  11. [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. [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. Zbl0896.65068MR1459390
  13. [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. Zbl0934.65110MR1712169
  14. [14] I. Kyza, A posteriori error estimates for approximations of semilinear parabolic and Schrödinger-type equations. Ph.D. thesis, University of Crete (2009). 
  15. [15] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications 2. Dunod, Paris (1968). Zbl0165.10801MR247244
  16. [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. Zbl1215.65154MR2520298
  17. [17] Ch. Makridakis, Space and time reconstructions in a posteriori analysis of evolution problems. ESAIM: Proc. 21 (2007) 31–44. Zbl1128.65062MR2404052
  18. [18] M.O. Scully and M.S. Zubairy, Quantum Optics. Cambridge University Press (2002). 
  19. [19] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Second edition, Springer-Verlag, Berlin (2006). Zbl0528.65052
  20. [20] R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation. Calcolo40 (2003) 195–212. Zbl1168.65418MR2025602

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.