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Journal of Scientific Computing

Erschienen in:

18.02.2017

Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear Schrödinger Equation

verfasst von: Dongyang Shi, Junjun Wang

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2017

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Abstract

A linearized Crank–Nicolson Galerkin finite element method with bilinear element for nonlinear Schrödinger equation is studied. By splitting the error into two parts which are called the temporal error and the spatial error, the unconditional superconvergence result is deduced. On one hand, the regularity for a time-discrete system is presented based on the proof of the temporal error. On the other hand, the classical Ritz projection is applied to get the spatial error with order \(O(h^2)\) in \(L^2\)-norm, which plays an important role in getting rid of the restriction of \(\tau \). Then the superclose estimates of order \(O(h^2+\tau ^2)\) in \(H^1\)-norm is arrived at based on the relationship between the Ritz projection and the interpolated operator. At the same time, global superconvergence property is arrived at by the interpolated postprocessing technique. At last, three numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and \(\tau \) is the time step.

Vorheriger Artikel Regularization of Singularities in the Weighted Summation of Dirac-Delta Functions for the Spectral Solution of Hyperbolic Conservation Laws

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Akrivis, G.D.: Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal. 13(1), 115–124 (1993)MathSciNetMATHCrossRef

Karakashian, O., Makridakis, C.: A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method. SIAM J. Numer. Anal. 36(6), 1779–1807 (1999)MathSciNetMATHCrossRef

Shi, D.Y., Liao, X., Wang, L.L.: Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation. Appl. Math. Comput. 289, 298–310 (2016)MathSciNet

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)MathSciNetMATHCrossRef

Dehghan, M., Mirzaei, D.: Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method. Int. J. Numer. Methods Eng. 76, 501–520 (2008)MATHCrossRef

Wu, L.: Two-grid mixed finite-element methods for nonlinear Schrödinger equations. Numer. Methods Part. Diff. Equ. 28(1), 63–73 (2012)MATHCrossRef

Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205, 72–77 (2005)MathSciNetMATHCrossRef

Karakashian, O., Makridakis, C.: A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method. Math. Comp. 67(222), 479–499 (1998)MathSciNetMATHCrossRef

Chang, Q.S., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 148(2), 397–415 (1999)MathSciNetMATHCrossRef

10.

Dehghan, M., Taleei, A.: A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput. Phys. Comm. 181, 43–51 (2010)MathSciNetMATHCrossRef

11.

Bao, W.Z., Cai, Y.Y.: Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 50, 492–521 (2012)MathSciNetMATHCrossRef

12.

Zhang, L.M., Chang, Q.S.: A conservative numerical scheme for a class of nonlinear Schrödinger equation with wave operator. Appl. Math. Comput. 220(1–2), 240–256 (2008)

13.

Lai, X., Yuan, Y.: Galerkin alternating-direction method for a kind of three-dimensional nonlinear hyperbolic problems. Comput. Math. Appl. 57(3), 384–403 (2009)MathSciNetMATHCrossRef

14.

Chen, Y., Huang, Y.: The full-discrete mixed finite element methods for nonlinear hyperbolic equations. Commun. Nonlinear Sci. Numer. Simul. 3(3), 152–155 (1998)MathSciNetMATHCrossRef

15.

Rachford Jr., H.H.: Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations. SIAM J. Numer. Anal. 10(6), 1010–1026 (1973)MathSciNetMATHCrossRef

16.

Luskin, M.: A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions. SIAM J. Numer. Anal. 16(2), 284–299 (1979)MathSciNetMATHCrossRef

17.

Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics. Springer, Sweden (2000)

18.

Shi, D.Y., Wang, J.J., Yan, F.N.: Superconvergence analysis for nonlinear parabolic equation with \(EQ^{rot}_1\) nonconforming finite element. Comput. Appl. Math. (2016). doi:10.1007/s40314-016-0344-6

19.

Shi, D.Y., Tang, Q.L., Gong, W.: A low order characteristic-nonconforming finite element method for nonlinear Sobolev equation with convection-dominated term. Math. Comput. Simulat. 114, 25–36 (2015)MathSciNetCrossRef

20.

Gu, H.M.: Characteristic finite element methods for nonlinear Sobolev equations. Appl. Math. Comput. 102(1), 51–62 (1999)MathSciNet

21.

Liu, B.Y.: The analysis of a finite element method with streamline diffusion for the compressible Navier–Stokes equations. SIAM J. Numer. Anal. 38(1), 1–16 (2000)MathSciNetMATHCrossRef

22.

He, Y.N., Sun, W.W.: Stabilized finite element method based on the Crank–Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations. Math. Comput. 76(257), 115–136 (2006)MathSciNetMATHCrossRef

23.

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)MathSciNetMATHCrossRef

24.

Wang, J.L., Si, Z.Y., Sun, W.W.: A new error analysis of characteristics-mixed FEMs for miscible displacement in porous media. SIAM J. Numer. Anal. 52(6), 3000–3020 (2013)MathSciNetMATHCrossRef

25.

Li, B.Y., Sun, W.W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51(4), 1959–1977 (2013)MathSciNetMATHCrossRef

26.

Gao, H.D.: Optimal error analysis of Galerkin FEMs for nonlinear Joule Heating equations. J. Sci. Comput. 58(3), 627–647 (2014)MathSciNetMATHCrossRef

27.

Li, B.Y., Gao, H.D., Sun, W.W.: Unconditionally optimal error estiamtes of a Crank–Nicolson Galerkin method for the nonlinear thermistor equations. SIAM J. Numer. Anal. 52(2), 933–954 (2014)MathSciNetMATHCrossRef

28.

Gao, H.D.: Unconditional optimal error estiamtes of BDF-Galerkin FEMs for nonlinear thermistor equations. J. Sci. Comput. 66(2), 504–527 (2016)MathSciNetMATHCrossRef

29.

Si, Z.Y., Wang, J.L., Sun, W.W.: Unconditional stability and error estimates of modified characteristics FEMs for the Navier-Stokes equations. Numer. Math. 134(1), 139–161 (2016)MathSciNetMATHCrossRef

30.

Shi, D.Y., Wang, J.J., Yan, F.N.: Unconditional Superconvergence analysis for nonlinear parabolic equation with \(EQ^{rot}_1\) nonconforming finite element. J. Sci. Comput. (2016). doi:10.1007/s10915-016-0243-4

31.

Shi, D.Y., Yan, F.N., Wang, J.J.: Unconditional superconvergence analysis of a new mixed finite element method for nonlinear Sobolev equation. Appl. Math. Comput. 274(1), 182–194 (2016)MathSciNet

32.

Gottlieb, S., Wang, C.: Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers equation. J. Sci. Comput. 53, 102–128 (2012)MathSciNetMATHCrossRef

33.

Cheng, K., Wang, C.: Long time stability of high order multiStep numerical schemes for two-dimensional incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 54(5), 3123–3144 (2016)MathSciNetMATHCrossRef

34.

Shi, D.Y., Wang, F.L., Fan, M.Z., Zhao, Y.M.: A new approach of the lowest order anisotropic mixed finite element high accuracy analysis for nonlinear Sine–Gordon equaitons. Math. Numer. Sin. 37(2), 148–161 (2015)MathSciNetMATH

35.

Wang, H.: Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl. Math. Comput. 170(1), 17–35 (2005)MathSciNetMATH

Titel: Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear Schrödinger Equation
verfasst von: Dongyang Shi
Junjun Wang
Publikationsdatum: 18.02.2017
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2017
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0390-2

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