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Unconditional stability and convergence of Crank-Nicolson Galerkin FEMs for a nonlinear Schrödinger-Helmholtz system. (English) Zbl 1402.65119
Two linearized Crank-Nicolson Galerkin finite element methods (FEM) are presented for the complex nonlinear Schrödinger equations defined in a bounded and convex polygon or polyhedron in two and three dimensions, respectively. Homogeneous boundary conditions are considered and for simplification the Schrödinger-Helmholtz system is analyzed. Spatial and temporal error analyses are performed. Unconditionally optimal $$L^2$$ error estimates are proven in both two and three space dimensions based on the uniform boundedness of the numerical solution in the $$L^\infty$$ norm. A time-discrete system is introduced using a splitting technique.
Finally, three numerical examples are presented, which illustrate the theoretical convergence and stability results in 2d and 3d. The software FreeFem++ is used with uniform triangular grids on square and cubic domains.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
FreeFem++
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##### References:
 [1] Antoine, X; Besse, C; Klein, P, Absorbing boundary conditions for general nonlinear Schrödinger equations, SIAM J. Sci. Comput., 33, 1008-1033, (2011) · Zbl 1231.35223 [2] Akrivis, GD; Dougalis, VA; Karakashian, OA, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59, 31-53, (1991) · Zbl 0739.65096 [3] 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) · Zbl 1246.35188 [4] Bao, W; Mauser, NJ; Stimming, HP, Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger-Poisson-X$$α$$ model, Commun. Math. Sci., 1, 809-828, (2003) · Zbl 1160.81497 [5] Berezin, F.A., Shubin, M.A.: The Schrödinger Equation. Kluwer Academic Publishers, Dordrecht (1991) · Zbl 0749.35001 [6] Bohun, S; Illner, R; Lange, H; Zweifel, PF, Error estimates for Galerkin approximations to the periodic Schrödinger-Poisson system, ZAMM$$·$$Z, Angew. Math. Mech., 76, 7-13, (1996) · Zbl 0864.65062 [7] Borz, A; Decker, E, Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation, J. Comput. Appl. Math., 193, 65-88, (2006) · Zbl 1118.65107 [8] Bratsos, AG, A modified numerical scheme for the cubic Schrödinger equation, Numer. Methods Part Differ. Equ., 27, 608-620, (2011) · Zbl 1216.65101 [9] Cannon, JR; Lin, Y, Nonclassical $$H^1$$ projection and Galerkin methods for nonlinear parabolic integro-differential equations, Calcolo, 25, 187-201, (1988) · Zbl 0685.65124 [10] Cao, Y; Musslimani, ZH; Titi, ES, Nonlinear Schrödinger-Helmholtz equation as numercal regularization of the nonlinear Schrödinger equation, Nonlinearity, 21, 879-898, (2008) · Zbl 1143.35365 [11] Chang, Q; Jia, E; Sun, W, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148, 397-415, (1999) · Zbl 0923.65059 [12] Chen, Z; Hoffmann, K-H, Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity, Adv. Math. Sci. Appl., 5, 363-389, (1995) · Zbl 0846.65051 [13] Dehghan, M; Taleei, A, Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method, Numer. Methods Part Differ. Equ., 26, 979-990, (2010) · Zbl 1195.65137 [14] Douglas, J; Ewing, RE; Wheeler, MF, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO Anal. Numer., 17, 249-265, (1983) · Zbl 0526.76094 [15] Dupont, T; Fairweather, G; Johnson, JP, Three-level Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 11, 392-410, (1974) · Zbl 0313.65107 [16] Harrison, R; Moroz, I; Tod, KP, A numerical study of the Schrödinger-Newton equations, Nonlinearity, 16, 101-122, (2003) · Zbl 1040.81554 [17] He, Y, The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comput., 77, 2097-2124, (2008) · Zbl 1198.65222 [18] Hou, Y; Li, B; Sun, W, Error analysis of splitting Galerkin methods for heat and sweat transport in textile materials, SIAM J. Numer. Anal., 51, 88-111, (2013) · Zbl 1439.74421 [19] 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) · Zbl 1155.65078 [20] Leo, MD; Rial, D, Well posedness and smoothing effect of Schrödinger-Poisson equation, J. Math. Phys., 48, 093509, (2007) · Zbl 1152.81652 [21] Li, B.: Mathematical modeling, analysis and computation for some complex and nonlinear flow problems, Ph.D. thesis, City University of Hong Kong, Hong Kong (2012) [22] 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) · Zbl 1281.65122 [23] 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) · Zbl 1311.76067 [24] Li, BK; 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) · Zbl 0911.65095 [25] 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) · Zbl 1208.65130 [26] López-Marcos, J.C., Sanz-Serna, J.M.: A definition of stability for nonlinear problems. In: Numerical Treatment of Differential Equations. Teubner-Texte zur Mathematik, Band 104, Leipzig, pp. 216-226 (1988) · Zbl 1218.65115 [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) · Zbl 1146.65072 [28] Lubich, C, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comput., 77, 2141-2153, (2008) · Zbl 1198.65186 [29] Masaki, S, Energy solution to a Schrödinger-Poisson system in the two-dimensional whole space, SIAM J. Math. Anal., 43, 2719-2731, (2011) · Zbl 1233.35179 [30] Mu, M; Huang, Y, An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations, SIAM J. Numer. Anal., 35, 1740-1761, (1998) · Zbl 0914.65129 [31] Pathria, D, Exact solutions for a generalized nonlinear Schrödinger equation, Phys. Scr., 39, 673-679, (1989) [32] Pelinovsky, DE; Afanasjev, VV; Kivshar, YS, Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation, Phys. Rev. E, 53, 1940-1953, (1996) [33] Reichel, B; Leble, S, On convergence and stability of a numerical scheme of coupled nonlinear Schrödinger equations, Comput. Math. Appl., 55, 745-759, (2008) · Zbl 1142.65074 [34] Sanz-Serna, JM, Methods for the numerical solution of nonlinear Schrödinger equation, Math. Comput., 43, 21-27, (1984) · Zbl 0555.65061 [35] Schürmann, HW, Traveling-wave solutions of the cubic-quintic nonlinear Schrödinger equation, Phys. Rev. E, 54, 4312-4320, (1996) [36] Stimming, HP, The IVP for the Schrödinger-Poisson-X$$α$$ equation in one dimension, Math. Models Methods Appl. Sci., 8, 1169-1180, (2005) · Zbl 1087.35086 [37] Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999) · Zbl 0928.35157 [38] 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) · Zbl 1357.65148 [39] Sun, Z; Zhao, D, On the $$L_{∞ }$$ convergence of a difference scheme for coupled nonlinear Schrödinger equations, Comput. Math. Appl., 59, 3286-3300, (2010) · Zbl 1198.65173 [40] Temam, R.: Naiver-Stokes Equations: Theory and Numerical Analysis. North-Holland Publishing Company, Amsterdam (1979) [41] Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997) · Zbl 0884.65097 [42] 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) · Zbl 0737.65095 [43] Wang, T; Guo, B; Zhang, L, New conservative difference schemes for a coupled nonlinear Schrödinger system, Appl. Math. Comput., 217, 1604-1619, (2010) · Zbl 1205.65242 [44] Wu, H; Ma, H; Li, H, Optimal error estimates of the Chebyshev-Legendre spectral method for solving the generalized Burgers equation, SIAM J. Numer. Anal., 41, 659-672, (2003) · Zbl 1050.65083 [45] Zhang, Y; Dong, X, On the computation of ground state and dynamics of Schrödinger-Poisson-Slater system, J. Comput. Phys., 220, 2660-2676, (2011) · Zbl 1218.65115 [46] Zlámal, M, Curved elements in the finite element method. $$\text{I}^*$$, SIAM J. Numer. Anal., 10, 229-240, (1973) · Zbl 0285.65067 [47] Zouraris, GE, 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) · Zbl 0991.65088
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