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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topZouraris, 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

top- 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.
- G.D. Akrivis, Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal.13 (1993) 115-124.
- 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. Zbl0739.65096
- S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Texts Appl. Math.15, Springer-Verlag, New York (1994). Zbl0804.65101
- H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations. Nonlinear Analysis4 (1980) 677-681. Zbl0451.35023
- T. Cazenave and A. Haraux, Introduction aux problémes d'évolution semi-linéaires. Ellipses, Paris (1990). Zbl0786.35070
- R.Y. Chiao, E. Garmire and C. Townes, Self-trapping of optical beams. Phys. Rev. Lett.13 (1964) 479-482.
- 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. Zbl0685.65110
- 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. Zbl0832.65136
- Y. Jingqi, Time decay of the solutions to a nonlinear Schrödinger equation in an exterior domain in ${\mathbb{R}}^{2}$. Nonlinear Analysis19 (1992) 563-571. Zbl0776.35071
- 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. Zbl0774.65091
- 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. Zbl0896.65068
- H.Y. Lee, Fully discrete methods for the nonlinear Schrödinger equation. Comput. Math. Appl.28 (1994) 9-24. Zbl0808.65133
- H.A. Levine, The role of critical exponents in blowup theorems. SIAM Review32 (1990) 262-288. Zbl0706.35008
- H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity. J. Math. Soc. Japan46 (1994) 557-586. Zbl0829.35121
- A.C. Newell, Solitons in mathematics and mathematical physics. CBMS Appl. Math. Ser.48, SIAM, Philadelphia (1988).
- J.J. Rasmussen and K. Rypdal, Blow-up in nonlinear Schroedinger equations-I: A general review. Physica Scripta33 (1986) 481-497. Zbl1063.35545
- M.P. Robinson and G. Fairweather, Orthogonal spline collocation methods for Schrödinger-type equations in one space variable. Numer. Math.68 (1994) 355-376. Zbl0806.65123
- 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. Zbl1063.35546
- J.M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schroedinger equation. Math. Comp.43 (1984) 21-27. Zbl0555.65061
- W.A. Strauss, Nonlinear wave equations. CBMS Regional Conference Series Math. No. 73, AMS, Providence, RI (1989).
- V.I. Talanov, Self-focusing of wave beams in nonlinear media. JETP Lett.2 (1965) 138-141.
- V. Thomée, Galerkin finite-element methods for parabolic problems. Springer Series Comput. Math.25, Springer-Verlag, Berlin, Heidelberg (1997). Zbl0884.65097
- Y. Tourigny, Optimal H1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation. IMA J. Numer. Anal.11 (1991) 509-523. Zbl0737.65095
- M. Tsutsumi and N. Hayashi, Classical solutions of nonlinear Schrödinger equations in higher dimensions. Math. Z.177 (1981) 217-234. Zbl0438.35028
- V.E. Zakharov, Collapse of Langmuir waves. Sov. Phys. JETP35 (1972) 908-922.

## Citations in EuDML Documents

top## NotesEmbed ?

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