Abstract
In this paper, a new mixed finite element scheme in space and a linearized backward Euler scheme in time are presented and investigated for the nonlinear Schrödinger equations. By introducing a suitable time-discrete system, both the errors in L2- and H1-norms for the original variable and L2-norm for the flux variable are derived without any time-step restriction, while previous works always required certain conditions between time step and space size. Finally, some numerical results are provided to verify the theoretical analysis.
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Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, New York (2003)
Ablowitz, M.J., Segue, H.: Solitons and the Inverse Scattering Transformation. SIAM, Philadelphia (1981)
Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for general nonlinear schrödinger equations. SIAM J. Sci. Comput. 33, 1008–1033 (2011)
Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear schrödinger equation. Nmer. Math. 59, 31–53 (1911)
Akrivis, G.D.: Finite difference discretization of the cubic schrödinger equation. IMA J. Numer. Analysis. 13(1), 115–124 (1993)
Bratsos, A.G.: A Modified numerical scheme for the cubic schrödinger equation. Numer. Methods Partial Differential Equations 27, 608–620 (2011)
Borzi, A., Decker, E.: Analysis of a leap-frog pseudospectral scheme for the schrödinger equation. J. Comput. Appl. Math. 193, 65–88 (2006)
Ciarlet, P.G.: The finite element method for elliptic problem. North Holland, Amsterdam (1978)
Chang, Q., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear schrödinger equation. J. Comput. Phys. 148(2), 397–415 (1999)
Cai, W.T., Li, J., Chen, Z.X.: Unconditional convergence and optimal error estimates of the Euler semi-implicit scheme for a generalized nonlinear schrödinger equation. Adv. Comput. Math. 42(6), 1311–1330 (2016)
Chen, S.C., Chen, H.R.: New mixed element schemes for second order elliptic problem. Math. Numer. Sin. 32, 213–218 (2010). (in Chinese)
Chen, Z.X.: Finite Element Methods and Their Applications. Springer, Berlin (2005)
Dehghan, M., Taleei, A.: Numerical solution of nonlinear schrödinger equation by using time-space pseudo-spectral method. Numer. Methods Partial Differential Equations 26, 979–990 (2010)
Ebaid, A., Khaled, S.M.: New types of exact solutions for nonlinear schrödinger equation with cubic nonlinearity. J. Comput. Appl. Math. 235, 1984–1992 (2011)
Gao, Y.L., Mei, L.Q.: Implicit-explicit multistep methods for general two-dimensional nonlinear schrödinger equations. Appl. Nmer. Math. 109, 41–60 (2016)
Gao, H.D.: Optimal error analysis of Galerkin FEMs for nonlinear Joule heating equations. J. Sci. Comput. 58, 627–647 (2014)
Jin, J., Wu, X.: Analysis of finite element method for one-dimensional time-dependent schrödinger equation on unbounded domain. J. Comput. Appl. Math. 220, 240–256 (2008)
Hasegawa, A., Kodama, Y.: Solitons in optical communications. Rev. Mod. Phys. 68(2), 423–444 (1996)
Li, L.X., Wang, M.: The (G’/G)-expansion method and travelling wave solutions for a high-order nonlinear schrödinger equation. Appl. Math. Comput. 208(2), 440–445 (2009)
Lin, Q., Liu, X.Q.: Global superconvergence estimates of finite element method for schrödinger equation. J. Comput. Math. 16(6), 521–526 (1998)
Liao, H., Sun, Z., Shi, H.: Error estimate of fourth-order compact scheme for linear schrödinger equations. SIAM J. Numer. Anal. 47, 4381–4401 (2010)
Li, B.K., Fairweather, G., Bialecki, B.: Discrete-time orthogonal spline collocation methods for schrödinger equations in two space variables. SIAM J. Numer. Anal. 35, 453–477 (1998)
Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622–633 (2013)
Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959–1977 (2013)
Newell, A.C.: Solitons in Mathematical and Physics. SIAM, Philadelphia (1985)
Reichel, B., Leble, S.: On convergence and stability of a numerical scheme of coupled nonlinear schrödinger equations. Comput. Math. Appl. 55, 745–759 (2008)
Sun, Z., Zhao, D.: On the \(l_{\infty }\) convergence of a difference scheme for coupled nonlinear schrödinger equations. Comput. Math. Appl. 59, 3286–3300 (2010)
Sun, W.W., Wang, J.L.: Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear schrödinger system in 3D. J. Comput. Appl. Math. 317, 685–699 (2017)
Sanz-Serna, J.M.: Methods for the numerical solution of nonlinear schrödinger equation. Math. Comput. 43, 21–27 (1984)
Shi, D.Y., Wang, J.J.: Unconditional superconvergence analysis of a crank-nicolson galerkin FEM for nonlinear schrödinger equation. J. Sci. Comput. 72(3), 1093–1118 (2017)
Shi, D.Y., Wang, P.L., Zhao, Y.M.: Superconvergence analysis of anisotropic linear triangular finite element for nonlinear schrödinger equation. Appl. Math. Lett. 38, 129–134 (2014)
Shi, D.Y., Liao, X., Wang, L.L.: A nonconforming quadrilateral finite element approximation to nonlinear schrödinger equation. Acta. Math. Sci. 37(3), 584–592 (2017)
Shi, D.Y., Liao, X., Wang, L.L.: Superconvergence analysis of conforming finite element method for nonlinear schrödinger equation. Appl. Math. Comput. 289(20), 298–310 (2016)
Shi, F., Yu, J.P., Li, K.T.: A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair. Int. J. Comput. Math. 88, 2293–2305 (2011)
Tourigny, Y.: Optimal h 1 estimates for two time-discrete Galerkin approximations of a nonlinear schrödinger equation. IMA J. Numer. Anal. 11, 509–523 (1991)
Thomee, V.: Galerkin Finite Element Methods for Paraboloc Problems. Springer, Berlin (1977)
Wang, J.L.: A new error analysis of crank-nicolson galerkin FEMs for a generalized nonlinear schrödinger equation. J. Sci. Comput. 60(2), 390–407 (2014)
Weng, Z.F., Feng, X.L., Huang, P.Z.: A new mixed finite element method based on the Crank-Nicolson scheme for the parabolic problems. Appl. Math. Model. 36, 5068–5079 (2012)
Wu, L.: Two-grid mixed finite-element methods for nonlinear schrödinger equations. Numer. Methods for Partial Differential Equations 28, 63–73 (2012)
Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear schrödinger equations. J. Comput. Phys. 205, 72–77 (2005)
Zhang, H.: Extended Jacobi elliptic function expansion method and its applications. Commun. Nonlinear Sci. Numer. Simul. 12(5), 627–635 (2007)
Zouraris, G.E.: On the convergence of a linear two-step finite element method for the nonlinear schrödinger equation. M2AN Math. Model. Numer. Anal. 35, 389–405 (2001)
Zlamal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10, 229–240 (1973)
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This work is supported by the National Natural Science Foundation of China (Nos. 11671369; 11271340).
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Communicated by: Long Chen
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Shi, D., Yang, H. Unconditionally optimal error estimates of a new mixed FEM for nonlinear Schrödinger equations. Adv Comput Math 45, 3173–3194 (2019). https://doi.org/10.1007/s10444-019-09732-7
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DOI: https://doi.org/10.1007/s10444-019-09732-7
Keywords
- Nonlinear Schrödinger equtions
- New mixed finite element scheme
- Unconditionally optimal error estimates