On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation
Zouraris, Georgios E.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001), p. 389-405 / Harvested from Numdam

We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit 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.

Publié le : 2001-01-01
Classification:  65M12,  65M60
@article{M2AN_2001__35_3_389_0,
     author = {Zouraris, Georgios E.},
     title = {On the convergence of a linear two-step finite element method for the nonlinear Schr\"odinger equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {35},
     year = {2001},
     pages = {389-405},
     mrnumber = {1837077},
     zbl = {0991.65088},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2001__35_3_389_0}
}
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 - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) pp. 389-405. http://gdmltest.u-ga.fr/item/M2AN_2001__35_3_389_0/

[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. JETP 23 (1966) 1025-1033.

[2] G.D. Akrivis, Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal. 13 (1993) 115-124. | Zbl 0762.65070

[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. | Zbl 0739.65096

[4] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Texts Appl. Math. 15, Springer-Verlag, New York (1994). | MR 1278258 | Zbl 0804.65101

[5] H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations. Nonlinear Analysis 4 (1980) 677-681. | Zbl 0451.35023

[6] T. Cazenave and A. Haraux, Introduction aux problémes d'évolution semi-linéaires. Ellipses, Paris (1990). | Zbl 0786.35070

[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. | Zbl 0685.65110

[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. | Zbl 0832.65136

[10] Y. Jingqi, Time decay of the solutions to a nonlinear Schrödinger equation in an exterior domain in 2 . Nonlinear Analysis 19 (1992) 563-571. | Zbl 0776.35071

[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. | Zbl 0774.65091

[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. | Zbl 0896.65068

[13] H.Y. Lee, Fully discrete methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 28 (1994) 9-24. | Zbl 0808.65133

[14] H.A. Levine, The role of critical exponents in blowup theorems. SIAM Review 32 (1990) 262-288. | Zbl 0706.35008

[15] H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity. J. Math. Soc. Japan 46 (1994) 557-586. | Zbl 0829.35121

[16] A.C. Newell, Solitons in mathematics and mathematical physics. CBMS Appl. Math. Ser. 48, SIAM, Philadelphia (1988). | Zbl 0565.35003

[17] J.J. Rasmussen and K. Rypdal, Blow-up in nonlinear Schroedinger equations-I: A general review. Physica Scripta 33 (1986) 481-497. | Zbl 1063.35545

[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. | Zbl 0806.65123

[19] K. Rypdal and J.J. Rasmussen, Blow-up in nonlinear Schroedinger equations-II: Similarity structure of the blow-up singularity. Physica Scripta 33 (1986) 498-504. | Zbl 1063.35546

[20] J.M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schroedinger equation. Math. Comp. 43 (1984) 21-27. | Zbl 0555.65061

[21] W.A. Strauss, Nonlinear wave equations. CBMS Regional Conference Series Math. No. 73, AMS, Providence, RI (1989). | MR 1032250 | Zbl 0714.35003

[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). | Zbl 0528.65052

[24] Y. Tourigny, Optimal H 1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation. IMA J. Numer. Anal. 11 (1991) 509-523. | Zbl 0737.65095

[25] M. Tsutsumi and N. Hayashi, Classical solutions of nonlinear Schrödinger equations in higher dimensions. Math. Z. 177 (1981) 217-234. | Zbl 0438.35028

[26] V.E. Zakharov, Collapse of Langmuir waves. Sov. Phys. JETP 35 (1972) 908-922.