Abstract
An error estimate is presented for the Newton iterative Crank–Nicolson finite element method for the nonlinear Schrödinger equation, fully discretized by quadrature, without restriction on the grid ratio between temporal step size and spatial mesh size. It is shown that the Newton iterative solution converges double exponentially with respect to the number of iterations to the solution of the implicit Crank–Nicolson method uniformly for all time levels, with optimal convergence in both space and time.
Funding source: Natural Science Foundation of Guangdong Province
Award Identifier / Grant number: 2018A0303100016
Funding source: Research Grants Council, University Grants Committee
Award Identifier / Grant number: 15301818
Award Identifier / Grant number: 14306921
Award Identifier / Grant number: 14306719
Funding source: Hong Kong Polytechnic University
Award Identifier / Grant number: P0031035
Funding statement: The work of the first author was supported by Natural Science Foundation of Guangdong province, China (2018A0303100016). The work of the second author was partially supported by Hong Kong RGC General Research Fund (project 15301818) and an internal grant of the university (Project ID: P0031035, Work Programme: ZZKQ). The work of the third author was substantially supported by Hong Kong RGC General Research Fund (projects 14306921 and 14306719).
Appendix: Proof of Lemma 4.1
(i) By using the definition in (4.1) and Hölder’s inequality, we have
(ii) By the definition of the discrete Laplacian operator, we have
where the second to last inequality is the standard inverse inequality for finite element functions.
Since the inequality above holds for all
Since
This proves the second result of Lemma 4.1.∎
References
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure Appl. Math. (Amsterdam) 140, Elsevier, Amsterdam, 2003. Search in Google Scholar
[2] G. D. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal. 13 (1993), no. 1, 115–124. 10.1093/imanum/13.1.115Search in Google Scholar
[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), no. 1, 31–53. 10.1007/BF01385769Search in Google Scholar
[4] M. Amrein and T. P. Wihler, Fully adaptive Newton–Galerkin methods for semilinear elliptic partial differential equations, SIAM J. Sci. Comput. 37 (2015), no. 4, A1637–A1657. 10.1137/140983537Search in Google Scholar
[5] X. Antoine, W. Bao and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations, Comput. Phys. Commun. 184 (2013), no. 12, 2621–2633. 10.1016/j.cpc.2013.07.012Search in Google Scholar
[6] W. Bao and Y. Cai, Optimal error estimates of finite difference methods for the Gross–Pitaevskii equation with angular momentum rotation, Math. Comp. 82 (2013), no. 281, 99–128. 10.1090/S0025-5718-2012-02617-2Search in Google Scholar
[7] W. Bao, Q. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation, J. Comput. Phys. 235 (2013), 423–445. 10.1016/j.jcp.2012.10.054Search in Google Scholar
[8] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar
[9] M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys. 44 (1981), no. 2, 277–288. 10.1016/0021-9991(81)90052-8Search in Google Scholar
[10] P. Deuflhard, Newton Methods for Nonlinear Problems, Springer Ser. Comput. Math. 35, Springer, Berlin, 2004. Search in Google Scholar
[11] L. C. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, 2010. Search in Google Scholar
[12] X. Feng, B. Li and S. Ma, High-order mass- and energy-conserving SAV-Gauss collocation finite element methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 59 (2021), no. 3, 1566–1591. 10.1137/20M1344998Search in Google Scholar
[13] X. Feng, H. Liu and S. Ma, Mass- and energy-conserved numerical schemes for nonlinear Schrödinger equations, Commun. Comput. Phys. 26 (2019), no. 5, 1365–1396. 10.4208/cicp.2019.js60.05Search in Google Scholar
[14] Y. Gong, Q. Wang, Y. Wang and J. Cai, A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation, J. Comput. Phys. 328 (2017), 354–370. 10.1016/j.jcp.2016.10.022Search in Google Scholar
[15] L. Grafakos, Classical Fourier Analysis, 2nd ed., Grad. Texts in Math. 249, Springer, New York, 2008. 10.1007/978-0-387-09432-8Search in Google Scholar
[16] L. Grafakos and S. Oh, The Kato–Ponce inequality, Comm. Partial Differential Equations 39 (2014), no. 6, 1128–1157. 10.1080/03605302.2013.822885Search in Google Scholar
[17] P. Henning and D. Peterseim, Crank–Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials, Math. Models Methods Appl. Sci. 27 (2017), no. 11, 2147–2184. 10.1142/S0218202517500415Search in Google Scholar
[18] O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: The discontinuous Galerkin method, Math. Comp. 67 (1998), no. 222, 479–499. 10.1090/S0025-5718-98-00946-6Search in Google Scholar
[19] O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: The continuous Galerkin method, SIAM J. Numer. Anal. 36 (1999), no. 6, 1779–1807. 10.1137/S0036142997330111Search in Google Scholar
[20]
B. Li, S. Ma and N. Wang,
Second-order convergence of the linearly extrapolated Crank–Nicolson method for the Navier–Stokes equations with
[21] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University, Cambridge, 2000. Search in Google Scholar
[22] J. M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schroedinger equation, Math. Comp. 43 (1984), no. 167, 21–27. 10.1090/S0025-5718-1984-0744922-XSearch in Google Scholar
[23] J. Shen, J. Xu and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev. 61 (2019), no. 3, 474–506. 10.1137/17M1150153Search in Google Scholar
[24] J. Wang, A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput. 60 (2014), no. 2, 390–407. 10.1007/s10915-013-9799-4Search in Google Scholar
[25] T. Wang, B. Guo and Q. Xu, Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J. Comput. Phys. 243 (2013), 382–399. 10.1016/j.jcp.2013.03.007Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
