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

Published online by Cambridge University Press:  15 April 2002

Georgios E. Zouraris
Georgios E. Zouraris*
Affiliation:
Department of Numerical Analysis and Computing Science (NADA), Royal Institute of Technology (KTH), 10044 Stockholm, Sweden. : Centre de Recherche en Mathématiques de la Décision (CEREMADE), UMR CNRS 7534, Université de Paris IX-Dauphine, Place du Maréchal de Lattre-de-Tassigny, 75775 Paris Cedex 16, France.
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Abstract

We discretize the nonlinear Schrödinger equation,with Dirichlet boundary conditions, by a linearlyimplicit two-step finite element method which conserves the L 2 norm. We prove optimal order a priori error estimates in the L 2 and H 1 norms, under mild mesh conditions for two and three space dimensions.

Keywords

Nonlinear Schrödinger equationtwo-step time discretizationlinearly implicit methodfinite element methodL 2 and H 1 error estimatesoptimal order of convergence.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 3 , May 2001 , pp. 389 - 405
Copyright
© EDP Sciences, SMAI, 2001

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References

Akhamanov, S.A., Sukhonorov, A.P. and Khoklov, R.V., Self-focusing and self-trapping of intense light beams in a nonlinear medium. Sov. Phys. JETP 23 (1966) 1025-1033.
Akrivis, G.D., Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal. 13 (1993) 115-124. CrossRef
Akrivis, G.D., Dougalis, V.A. and Karakashian, O.A., On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math. 59 (1991) 31-53. CrossRef
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Texts Appl. Math. 15, Springer-Verlag, New York (1994).
Brezis, H. and Gallouet, T., Nonlinear Schrödinger evolution equations. Nonlinear Analysis 4 (1980) 677-681. CrossRef
T. Cazenave and A. Haraux, Introduction aux problémes d'évolution semi-linéaires. Ellipses, Paris (1990).
Chiao, R.Y., Garmire, E. and Townes, C., Self-trapping of optical beams. Phys. Rev. Lett. 13 (1964) 479-482. CrossRef
Cloot, A., Herbst, B.M. and Weideman, J.A.C., A numerical study of the nonlinear Schrödinger equation involving quintic terms. J. Comput. Phys. 86 (1990) 127-146. CrossRef
Fei, Z., Pérez-García, V.M. and Vázquez, L., Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Appl. Math. Comput. 71 (1995) 165-177.
Jingqi, Y., Time decay of the solutions to a nonlinear Schrödinger equation in an exterior domain in $\mathbb R^2$ . Nonlinear Analysis 19 (1992) 563-571. CrossRef
Karakashian, O., Akrivis, G.D. and Dougalis, V.A., On optimal order error estimates for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 30 (1993) 377-400. CrossRef
Karakashian, O. and Makridakis, Ch., A space-time finite element method for the nonlinear Schrödinger equation: The discontinuous Galerkin method. Math. Comp. 67 (1998) 479-499. CrossRef
Lee, H.Y., Fully discrete methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 28 (1994) 9-24. CrossRef
Levine, H.A., The role of critical exponents in blowup theorems. SIAM Review 32 (1990) 262-288. CrossRef
Nawa, H., Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity. J. Math. Soc. Japan 46 (1994) 557-586. CrossRef
A.C. Newell, Solitons in mathematics and mathematical physics. CBMS Appl. Math. Ser. 48, SIAM, Philadelphia (1988).
Rasmussen, J.J. and Rypdal, K., Blow-up in nonlinear Schroedinger equations-I: A general review. Physica Scripta 33 (1986) 481-497. CrossRef
Robinson, M.P. and Fairweather, G., Orthogonal spline collocation methods for Schrödinger-type equations in one space variable. Numer. Math. 68 (1994) 355-376. CrossRef
Rypdal, K. and Rasmussen, J.J., Blow-up in nonlinear Schroedinger equations-II: Similarity structure of the blow-up singularity. Physica Scripta 33 (1986) 498-504. CrossRef
Sanz-Serna, J.M., Methods for the numerical solution of the nonlinear Schroedinger equation. Math. Comp. 43 (1984) 21-27. CrossRef
W.A. Strauss, Nonlinear wave equations. CBMS Regional Conference Series Math. No. 73, AMS, Providence, RI (1989).
Talanov, V.I., 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).
Tourigny, Y., Optimal H1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation. IMA J. Numer. Anal. 11 (1991) 509-523. CrossRef
Tsutsumi, M. and Hayashi, N., Classical solutions of nonlinear Schrödinger equations in higher dimensions. Math. Z. 177 (1981) 217-234. CrossRef
Zakharov, V.E., Collapse of Langmuir waves. Sov. Phys. JETP 35 (1972) 908-922.