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Numerical Algorithms
Published in: Numerical Algorithms 4/2021

08-05-2020 | Original Paper

Unconditional optimal error estimates of linearized backward Euler Galerkin FEMs for nonlinear Schrödinger-Helmholtz equations

Authors: Yun-Bo Yang, Yao-Lin Jiang

Published in: Numerical Algorithms | Issue 4/2021

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Abstract

In this paper, we establish unconditionally optimal error estimates for linearized backward Euler Galerkin finite element methods (FEMs) applied to nonlinear Schrödinger-Helmholtz equations. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and derive error estimates in temporal direction. Second, by the above achievements, we obtain the boundedness of the numerical solution in \(L^{\infty }\)-norm. Then, the optimal L2 error estimates for r-order FEMs are derived without any restriction on the time step size. Numerical results in both two- and three-dimensional spaces are provided to illustrate the theoretical predictions and demonstrate the efficiency of the methods.
previous article On the circumcentered-reflection method for the convex feasibility problem
next article A two-grid method for discontinuous Galerkin approximations to nonlinear Sobolev equations

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Literature
1.
Adams, R.A., Fournier, J.J.: Sobolev Spaces. Academic press, New York (2003)MATH
2.
Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math. 59, 31–53 (1991)MathSciNetMATH
3.
Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for general nonlinear Schrödinger equations. SIAM J. Sci. Comput. 33, 1008–1033 (2011)MathSciNetMATH
4.
Bao, W., Cai, Y.: Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 50, 492–521 (2012)MathSciNetMATH
5.
Bao, W., Mauser, N.J., Stimming, H.P.: Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger-Poisson-Xα model. Commun. Math. Sci. 1, 809–828 (2003)MathSciNetMATH
6.
Berezin, F.A., Shubin, M.A.: The Schrödinger Equation. Kluwer Academic Publishers, Dordrecht (1991)MATH
7.
Bohun, S., Illner, R., Lange, H., Zweifel, P.F.: Error estimates for Galerkin approximations to the periodic Schrödinger-Poisson system. Z.MM Z. Angew. Math. Mech. 76, 7–13 (1996)MATH
8.
Borz, A., Decker, E.: Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation. J. Comput. Appl. Math. 193, 65–88 (2006)MathSciNetMATH
9.
Bratsos, A.G.: A modified numerical scheme for the cubic Schrödinger equation. Numer. Methods Part Differ. Equ. 27, 608–620 (2011)MathSciNetMATH
10.
Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (1994)MATH
11.
Cao, Y., Musslimani, Z.H., Titi, E.S.: Nonlinear Schrödinger-Helmholtz equation as numercal regularization of the nonlinear Schrödinger equation. Nonlinearity 21, 879–898 (2008)MathSciNetMATH
12.
Dehghan, M., Taleei, A.: Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method. Numer. Methods Part DifferNumer. Methods Part Differ. Equ. 26, 979–990 (2010)MATH
13.
Douglas, J.J., Ewing, R.E., Wheeler, M.F.: A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Anal. Numer. 17, 249–265 (1983)MathSciNetMATH
14.
Ewing, R.E., Wheeler, M.F.: Galerkin methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 17, 351–365 (1980)MathSciNetMATH
15.
Evans, L.C.: Partial Differential Equations, 2nd edn. AMS, Providence (2010)MATH
16.
Gao, H.: Optimal error estimates of a linearized backward Euler FEM for the Landau-Lifshitz equation. SIAM J. Numer. Anal. 52, 2574–2593 (2014)MathSciNetMATH
17.
Gao, H., Li, B., Sun, W.: Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity. SIAM J. Numer. Anal. 52, 1183–1202 (2014)MathSciNetMATH
18.
Harrison, R., Moroz, I., Tod, K.P.: A numerical study of the Schrödinger-Newton equations. Nonlinearity 16, 101–122 (2003)MathSciNetMATH
19.
Hou, Y., Li, B., Sun, W.: Error estimates of splitting Galerkin methods for heat and sweat transport in textile materials. SIAM J. Numer. Anal. 51, 88–111 (2013)MathSciNetMATH
20.
21.
Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. Part IV: Error analysis for the second order time discretization. SIAM J. Numer. Anal. 2, 353–384 (1990)MATH
22.
Leo, M.D., Rial, D.: Well posedness and smoothing effect of Schrödinger-Poisson equation. J. Math. Phys. 48, 093509 (2007)MathSciNetMATH
23.
Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Inter. J. Numer. Anal. Model. 10, 622–633 (2013)MathSciNetMATH
24.
Li, B.: Mathematical Modeling, Analysis and Computation for Some Complex and Nonlinear Flow Problems. PhD Thesis. City University of Hong Kong, Hong Kong (2012)
25.
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)MathSciNetMATH
26.
Li, B., Gao, H., Sun, W.W.: Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear thermistor equations. SIAM J. Numer. Anal. 52, 933–954 (2014)MathSciNetMATH
27.
Lu, T., Cai, W.: A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger-Poisson equations with discontinuous potentials. J. Comput. Appl. Math. 220, 588–614 (2008)MathSciNetMATH
28.
Lubich, C.: On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77, 2141–2153 (2008)MATH
29.
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)MathSciNetMATH
30.
Masaki, S.: Energy solution to a Schrödinger-Poisson system in the two-dimensional whole space. SIAM J. Math. Anal. 43, 2719–2731 (2011)MathSciNetMATH
31.
Mu, M., Huang, Y.: An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations. SIAM J. Numer. Anal. 35, 1740–1761 (1998)MathSciNetMATH
32.
Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3) 20, 733–737 (1966)MathSciNetMATH
33.
Pathria, D.: Exact solutions for a generalized nonlinear Schrödinger equation. Phys. Scr. 39, 673–679 (1989)
34.
Pelinovsky, D.E., Afanasjev, V.V., Kivshar, Y.S.: Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation. Phys. Rev. E 53, 1940–1953 (1996)
35.
Reichel, B., Leble, S.: On convergence and stability of a numerical scheme of coupled nonlinear Schrödinger equations. Comput. Math. Appl. 55, 745–759 (2008)MathSciNetMATH
36.
Sanz-Serna, J.M.: Methods for the numerical solution of nonlinear Schrödinger equation. Math. Comput. 43, 21–27 (1984)MATH
37.
Stimming, H.P.: The IVP for the Schrödinger-Poisson-Xα equation in one dimension. Math. Models Methods Appl. Sci. 8, 1169–1180 (2005)MathSciNetMATH
38.
Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999)MATH
39.
Sun, W., Wang, J.: Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear Schrödinger system in 3D. J. Comput. Appl. Math. 317, 685–699 (2017)MathSciNetMATH
40.
Sun, Z., Zhao, D.: On the \(L^{\infty }\) convergence of a difference scheme for coupled nonlinear Schrödinger. Comput. Math. Appl. 59, 3286–3300 (2010)MathSciNetMATH
41.
Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)MATH
42.
Tourigny, Y.: Optimal H1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation. IMA J. Numer. Anal. 11, 509–523 (1991)MathSciNetMATH
43.
Wang, J.: A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60, 390–407 (2014)MathSciNetMATH
44.
Zhang, Y., Dong, X.: On the computation of ground state and dynamics of Schrödinger-Poisson-Slater system. J. Comput. Phys. 220, 2660–2676 (2011)MATH
45.
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)MathSciNetMATH
Metadata
Title
Unconditional optimal error estimates of linearized backward Euler Galerkin FEMs for nonlinear Schrödinger-Helmholtz equations
Authors
Yun-Bo Yang
Yao-Lin Jiang
Publication date
08-05-2020
Publisher
Springer US
Published in
Numerical Algorithms / Issue 4/2021
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-020-00942-5

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