Free access
Issue
ESAIM: M2AN
Volume 45, Number 4, July-August 2011
Page(s) 761 - 778
DOI http://dx.doi.org/10.1051/m2an/2010101
Published online 21 February 2011
  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. [MathSciNet]
  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.
  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. [CrossRef] [MathSciNet]
  4. R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains. Bull. Soc. Math. France 136 (2008) 27–65. [MathSciNet]
  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).
  8. W. Dörfler, A time-and space-adaptive algorithm for the linear time-dependent Schrödinger equation. Numer. Math. 73 (1996) 419–448. [CrossRef] [MathSciNet]
  9. L.C. Evans, Partial Differential Equations. Second edition, American Mathematical Society, Providence (2002).
  10. P. Górka, Convergence of logarithmic quantum mechanics to the linear one. Lett. Math. Phys. 81 (2007) 253–264. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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).
  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. [CrossRef] [MathSciNet]
  17. Ch. Makridakis, Space and time reconstructions in a posteriori analysis of evolution problems. ESAIM: Proc. 21 (2007) 31–44. [CrossRef]
  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).
  20. R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003) 195–212. [CrossRef] [MathSciNet]

Recommended for you