Convergence of a Strang splitting finite element discretization for the Schrödinger–Poisson equation

Auzinger, Winfried; Kassebacher, Thomas; Koch, Othmar; Thalhammer, Mechthild

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1245-1278.

Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schrödinger equations. In particular, the Schrödinger–Poisson equation under homogeneous Dirichlet boundary conditions on a finite domain is considered. A rigorous stability and error analysis is carried out for the second-order Strang splitting method and conforming polynomial finite element discretizations. For sufficiently regular solutions the classical orders of convergence are retained, that is, second-order convergence in time and polynomial convergence in space is proven. The established convergence result is confirmed and complemented by numerical illustrations.

Classification : 65J15, 65L05, 65M60, 65M12, 65M15
Mots clés : Nonlinear Schrödinger equations, operator splitting methods, finite element discretization, stability, local error, convergence

R.A. Adams, Sobolev Spaces. Academic Press, Orlando, Fla. (1975). | MR | Zbl

X. Antoine, W. Bao and Ch. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations. Comput. Phys. Commun. 184 (2013). | DOI | MR | Zbl

X. Antoine, C. Besse and P. Klein, Numerical solution of time-dependent nonlinear Schrödinger equations using domain truncation techniques coupled with relaxation scheme. Laser Physics 21 (2011) 1–12. | DOI

W. Auzinger, H. Hofstätter, O. Koch and M. Thalhammer, Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part III: The nonlinear case. J. Comput. Appl. Math. 273 (2014) 182–204. | DOI | MR | Zbl

W. Auzinger and O. Koch, Coefficients of various splitting methods. Available at: http://www.asc.tuwien.ac.at/˜winfried/splitting/.

G. Bao, G. Hu and D. Liu, An $h$-adaptive finite element solver for the calculations of the electronic structures. J. Comput. Phys. 231 (2012) 4967–4979. | DOI | Zbl

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Models 6 (2013) 1–135. | DOI | MR | Zbl

W. Bao, S. Jiang, Q. Tang and Y. Zhang, Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT. J. Comput. Phys. 296 (2015) 72–89. | DOI | MR | Zbl

S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition. Springer Verlag, New York (2002). | MR | Zbl

F. Brezzi and P. Markowitsch, The three-dimensional Wigner–Poisson problem: Existence, uniqueness and approximation. Math. Methods Appl. Sci. 14 (1991) 35–61. | DOI | MR | Zbl

R. Carles, On Fourier time-splitting methods for nonlinear Schrödinger equations in the semiclassical limit. SIAM J. Numer. Anal. 51 (2013) 3232–3258. | DOI | MR | Zbl

G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations. Springer, Berlin, Heidelberg, New York (2002). | MR | Zbl

J. Fang, X. Gao and A. Zhou, A Kohn–Sham equation solver based on hexahedral finite elements. J. Comput. Phys. 231 (2012) 3166–3180. | DOI | MR | Zbl

J. Garcke and M. Griebel, On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. J. Comput. Phys. 165 (2000) 694–716. | DOI | MR | Zbl

L. Gauckler, Convergence of a split-step Hermite method for the Gross–Pitaevskii equation. IMA J. Numer. Anal. 31 (2011) 396–415. | DOI | MR | Zbl

W. Hackbusch, Elliptic Differential Equations: Theory and Numerical Treatment. Springer Verlag, Berlin, Heidelberg, New York (1992). | MR | Zbl

E. Hairer, Ch. Lubich and G. Wanner, Geometric Numerical Integration. Springer, Verlag, Berlin, Heidelberg, New York (2002). | MR | Zbl

G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities. Cambridge Univ. Press, Cambridge (1934). | MR | Zbl

D. Hochstuhl, C.M. Hinz and M. Bonitz, Time-dependent multiconfiguration methods for the numerical simulation of photoionization processes of many-electron atoms. Eur. Phys. J. Special Topics 223 (2014) 177–336. | DOI

R. Illner, P.F. Zweifel and H. Lange, Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner–Poisson and Schrödinger–Poisson systems. Math. Methods Appl. Sci. 17 (1994) 349–376. | DOI | MR | Zbl

O. Karakashian and Ch. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method. Math. Comp. 67 (1998) 479–499. | DOI | MR | Zbl

T. Katsaounis and I. Kyza, A posteriori error control and adaptivity for Crank–Nicolson finite element approximations for the linear Schrödinger equation. Numer. Math. 129 (2015) 55–90. | DOI | MR | Zbl

O. Koch and Ch. Lubich, Analysis and time integration of the multi-configuration time-dependent Hartree-Fock equations in electron dynamics. ASC Report 4/2008, Inst. for Anal. and Sci. Comput., Vienna Univ. of Technology (2008).

O. Koch and Ch. Lubich, Variational splitting time integration of the MCTDHF equations in electron dynamics. IMA J. Numer. Anal. 31 (2011) 379–395. | DOI | MR | Zbl

O. Koch, Ch. Neuhauser and M. Thalhammer, Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics. ESAIM: M2AN 47 (2013) 1265–1284. | DOI | Numdam | MR | Zbl

K. Kormann, A time-space adaptive method for the Schrödinger equation. Tach. Rep. 23 (2012).

Ch. Lubich, On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77 (2008) 2141–2153. | DOI | MR | Zbl

R. Mclachlan and R. Quispel, Splitting methods. Acta Numer. 11 (2002) 341–434. | DOI | MR | Zbl

M. Miklavčič, Applied Functional Analysis and Partial Differential Equations. World Scientific, Singapore (1998). | MR | Zbl

P. Motamarri, M. Nowak, K. Leiter, J. Knap and V. Gavini, Higher-order adaptive finite-element methods for Kohn–Sham density functional theory. J. Comput. Phys. 231 (2012) 6596–6621. | DOI | Zbl

M.A. Olshanskii and E.E. Tyrtyshnikov, Iterative Methods for Linear Systems. SIAM, Philadelphia, PA, USA (2014). | MR | Zbl

Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29 (1992) 209–228. | DOI | MR | Zbl

Y. Saad, Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, PA, USA, 2nd edition (2003). | MR | Zbl

V. Schauer, Finite element based electronic structure calculations. Universität Stuttgart, Inst. f. Mechanik (Bauwesen), Lehrstuhl I (2014).

W.E. Schiesser, The Numerical Method of Lines. Academic Press, San Diego (1991). | MR | Zbl

R. Sidje, Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Software 24 (1998) 130–156. | DOI | Zbl

H. Yu and A. Bandrauk, Three-dimensional Cartesian finite element method for the time dependent Schrödinger equation of molecules in laser fields. J. Chem. Phys. 102 (1995).

Y. Zhou, Y. Saad, M. Tiago and J. Chelikowsky, Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration. Phys. Rev. E 74 (2006) 066704. | DOI | Zbl