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On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation

Georgios E. Zouraris

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 3, page 389-405
  • ISSN: 0764-583X

Abstract

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We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves the L2 norm. We prove optimal order a priori error estimates in the L2 and H1 norms, under mild mesh conditions for two and three space dimensions.

How to cite

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Zouraris, Georgios E.. "On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation." ESAIM: Mathematical Modelling and Numerical Analysis 35.3 (2010): 389-405. <http://eudml.org/doc/197599>.

@article{Zouraris2010,
abstract = { We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves the L2 norm. We prove optimal order a priori error estimates in the L2 and H1 norms, under mild mesh conditions for two and three space dimensions. },
author = {Zouraris, Georgios E.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear Schrödinger equation; two-step time discretization; linearly implicit method; finite element method; L2 and H1 error estimates; optimal order of convergence.; nonlinear Schrödinger equation; convergence; error bounds},
language = {eng},
month = {3},
number = {3},
pages = {389-405},
publisher = {EDP Sciences},
title = {On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation},
url = {http://eudml.org/doc/197599},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Zouraris, Georgios E.
TI - On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 3
SP - 389
EP - 405
AB - We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves the L2 norm. We prove optimal order a priori error estimates in the L2 and H1 norms, under mild mesh conditions for two and three space dimensions.
LA - eng
KW - Nonlinear Schrödinger equation; two-step time discretization; linearly implicit method; finite element method; L2 and H1 error estimates; optimal order of convergence.; nonlinear Schrödinger equation; convergence; error bounds
UR - http://eudml.org/doc/197599
ER -

References

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  1. S.A. Akhamanov, A.P. Sukhonorov and R.V. Khoklov, Self-focusing and self-trapping of intense light beams in a nonlinear medium. Sov. Phys. JETP23 (1966) 1025-1033.  
  2. G.D. Akrivis, Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal.13 (1993) 115-124.  
  3. G.D. Akrivis, V.A. Dougalis and O.A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math.59 (1991) 31-53.  
  4. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Texts Appl. Math.15, Springer-Verlag, New York (1994).  
  5. H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations. Nonlinear Analysis4 (1980) 677-681.  
  6. T. Cazenave and A. Haraux, Introduction aux problémes d'évolution semi-linéaires. Ellipses, Paris (1990).  
  7. R.Y. Chiao, E. Garmire and C. Townes, Self-trapping of optical beams. Phys. Rev. Lett.13 (1964) 479-482.  
  8. A. Cloot, B.M. Herbst and J.A.C. Weideman, A numerical study of the nonlinear Schrödinger equation involving quintic terms. J. Comput. Phys.86 (1990) 127-146.  
  9. Z. Fei, V.M. Pérez-García and L. Vázquez, Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Appl. Math. Comput.71 (1995) 165-177.  
  10. Y. Jingqi, Time decay of the solutions to a nonlinear Schrödinger equation in an exterior domain in . Nonlinear Analysis19 (1992) 563-571.  
  11. O. Karakashian, G.D. Akrivis and V.A. Dougalis, On optimal order error estimates for the nonlinear Schrödinger equation. SIAM J. Numer. Anal.30 (1993) 377-400.  
  12. O. Karakashian and Ch. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: The discontinuous Galerkin method. Math. Comp.67 (1998) 479-499.  
  13. H.Y. Lee, Fully discrete methods for the nonlinear Schrödinger equation. Comput. Math. Appl.28 (1994) 9-24.  
  14. H.A. Levine, The role of critical exponents in blowup theorems. SIAM Review32 (1990) 262-288.  
  15. H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity. J. Math. Soc. Japan46 (1994) 557-586.  
  16. A.C. Newell, Solitons in mathematics and mathematical physics. CBMS Appl. Math. Ser.48, SIAM, Philadelphia (1988).  
  17. J.J. Rasmussen and K. Rypdal, Blow-up in nonlinear Schroedinger equations-I: A general review. Physica Scripta33 (1986) 481-497.  
  18. M.P. Robinson and G. Fairweather, Orthogonal spline collocation methods for Schrödinger-type equations in one space variable. Numer. Math.68 (1994) 355-376.  
  19. K. Rypdal and J.J. Rasmussen, Blow-up in nonlinear Schroedinger equations-II: Similarity structure of the blow-up singularity. Physica Scripta33 (1986) 498-504.  
  20. J.M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schroedinger equation. Math. Comp.43 (1984) 21-27.  
  21. W.A. Strauss, Nonlinear wave equations. CBMS Regional Conference Series Math. No. 73, AMS, Providence, RI (1989).  
  22. V.I. Talanov, Self-focusing of wave beams in nonlinear media. JETP Lett.2 (1965) 138-141.  
  23. V. Thomée, Galerkin finite-element methods for parabolic problems. Springer Series Comput. Math.25, Springer-Verlag, Berlin, Heidelberg (1997).  
  24. Y. Tourigny, Optimal H1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation. IMA J. Numer. Anal.11 (1991) 509-523.  
  25. M. Tsutsumi and N. Hayashi, Classical solutions of nonlinear Schrödinger equations in higher dimensions. Math. Z.177 (1981) 217-234.  
  26. V.E. Zakharov, Collapse of Langmuir waves. Sov. Phys. JETP35 (1972) 908-922.  

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