A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method

Karakashian, Ohannes; Makridakis, Charalambos

doi:10.1090/S0025-5718-98-00946-6

A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method

Authors: Ohannes Karakashian and Charalambos Makridakis
Journal: Math. Comp. 67 (1998), 479-499
MSC (1991): Primary 65M60, 65M12
DOI: https://doi.org/10.1090/S0025-5718-98-00946-6
MathSciNet review: 1459390
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Abstract: The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrödinger equation is analyzed in this paper. We show the existence of the resulting approximations and prove optimal order error estimates in $L^{\infty }(L^{2} ) .$ These estimates are valid under weak restrictions on the space-time mesh.

Georgios D. Akrivis, Vassilios A. Dougalis, and Ohannes A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math. 59 (1991), no. 1, 31–53. MR 1103752, DOI https://doi.org/10.1007/BF01385769
I. Babuška and J. Osborn, Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer. Math. 34 (1980), no. 1, 41–62. MR 560793, DOI https://doi.org/10.1007/BF01463997
L. Bales and I. Lasiecka, Continuous finite elements in space and time for the nonhomogeneous wave equation, Comput. Math. Appl. 27 (1994), no. 3, 91–102. MR 1255008, DOI https://doi.org/10.1016/0898-1221%2894%2990048-5
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258
Felix E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Proc. Sympos. Appl. Math., Vol. XVII, Amer. Math. Soc., Providence, R.I., 1965, pp. 24–49. MR 0197933

R.Y. Chiao, E. Garmire and C. Townes, Self-trapping of optical beams, Phys. Rev. Lett. 73 (1964), 479–482.

M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI https://doi.org/10.1090/S0025-5718-1987-0878688-2

Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174

Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR 0448814

K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monographs, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR 774402

Willy Dörfler, A time- and space-adaptive algorithm for the linear time-dependent Schrödinger equation, Numer. Math. 73 (1996), no. 4, 419–448. MR 1393174, DOI https://doi.org/10.1007/s002110050199

Todd Dupont, Mesh modification for evolution equations, Math. Comp. 39 (1982), no. 159, 85–107. MR 658215, DOI https://doi.org/10.1090/S0025-5718-1982-0658215-0

Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43–77. MR 1083324, DOI https://doi.org/10.1137/0728003

Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in $L_\infty L_2$ and $L_\infty L_\infty $, SIAM J. Numer. Anal. 32 (1995), no. 3, 706–740. MR 1335652, DOI https://doi.org/10.1137/0732033

Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal. 32 (1995), no. 6, 1729–1749. MR 1360457, DOI https://doi.org/10.1137/0732078

Kenneth Eriksson, Claes Johnson, and Vidar Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 4, 611–643 (English, with French summary). MR 826227, DOI https://doi.org/10.1051/m2an/1985190406111

Donald A. French and Todd E. Peterson, A continuous space-time finite element method for the wave equation, Math. Comp. 65 (1996), no. 214, 491–506. MR 1325867, DOI https://doi.org/10.1090/S0025-5718-96-00685-0

E. Hairer and G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems. MR 1111480

Thomas J. R. Hughes and Gregory M. Hulbert, Space-time finite element methods for elastodynamics: formulations and error estimates, Comput. Methods Appl. Mech. Engrg. 66 (1988), no. 3, 339–363. MR 928689, DOI https://doi.org/10.1016/0045-7825%2888%2990006-0

Pierre Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain, SIAM J. Numer. Anal. 15 (1978), no. 5, 912–928. MR 507554, DOI https://doi.org/10.1137/0715059

Claes Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 107 (1993), no. 1-2, 117–129. MR 1241479, DOI https://doi.org/10.1016/0045-7825%2893%2990170-3

Ohannes Karakashian, Georgios D. Akrivis, and Vassilios A. Dougalis, On optimal order error estimates for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 30 (1993), no. 2, 377–400. MR 1211396, DOI https://doi.org/10.1137/0730018

O. Karakashian and Ch. Makridakis, A space-time finite element mathod for the nonlinear Schrödinger equation: The discontinuous Galerkin method, Preprint no. 96-9, Dept. of Math., University of Crete, Heraklion, Greece (1996).

O. Karakashian and Ch. Makridakis, A space-time finite element mathod for the nonlinear Schrödinger equation. II: The continuous Galerkin method, Preprint no. 96-11, Dept. of Math., University of Crete, Heraklion, Greece (1996).

P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. Publication No. 33. MR 0658142

Alan C. Newell, Solitons in mathematics and physics, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 48, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR 847245

J. Juul Rasmussen and K. Rypdal, Blow-up in nonlinear Schroedinger equations. I. A general review, Phys. Scripta 33 (1986), no. 6, 481–497. MR 870142, DOI https://doi.org/10.1088/0031-8949/33/6/001

A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp. 64 (1995), no. 211, 907–928. MR 1297478, DOI https://doi.org/10.1090/S0025-5718-1995-1297478-7

Walter A. Strauss, The nonlinear Schrödinger equation, Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977) North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam-New York, 1978, pp. 452–465. MR 519654

V.I. Talanov, Self-focusing of wave beams in nonlinear media, JETP Lett. 2 (1965), 138–141.

Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045

G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954