Galerkin finite element method for nonlinear fractional Schrödinger equations

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

[1]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[2]  Jie Xin,et al.  The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition , 2011, Comput. Math. Appl..

[3]  Xuan Zhao,et al.  A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation , 2014, SIAM J. Sci. Comput..

[4]  W. Wyss.  The fractional diffusion equation , 1986 .

[5]  Leilei Wei,et al.  Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation , 2012 .

[6]  G. Akrivis,et al.  On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation , 1991 .

[7]  Jie Xin,et al.  Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation , 2008, Appl. Math. Comput..

[8]  M. Meerschaert,et al.  Finite difference approximations for two-sided space-fractional partial differential equations , 2006 .

[9]  K. Aruna,et al.  Approximate Solutions of Non-linear Fractional Schrodinger Equation Via Differential Transform Method and Modified Differential Transform Method , 2013 .

[10]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[11]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[12]  Yu-Hong Ran,et al.  On HSS-like iteration method for the space fractional coupled nonlinear Schrödinger equations , 2015, Appl. Math. Comput..

[13]  L. Caffarelli,et al.  An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.

[14]  Jiye Yang,et al.  Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations , 2014, J. Comput. Phys..

[15]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[16]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[17]  Chengming Huang,et al.  An energy conservative difference scheme for the nonlinear fractional Schrödinger equations , 2015, J. Comput. Phys..

[18]  I. Turner,et al.  Numerical methods for fractional partial differential equations with Riesz space fractional derivatives , 2010 .

[19]  Fawang Liu,et al.  Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation , 2008, Appl. Math. Comput..

[20]  Xinwei Yu,et al.  Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms , 2012 .

[21]  Fawang Liu,et al.  Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation , 2009, Appl. Math. Comput..

[22]  B. Ross,et al.  Fractional Calculus and Its Applications , 1975 .

[23]  Wei Yang,et al.  Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative , 2013, J. Comput. Phys..

[24]  Charalambos Makridakis,et al.  A Space-Time Finite Element Method for the Nonlinear Schrödinger Equation: The Continuous Galerkin Method , 1999 .

[25]  Jiye Yang,et al.  Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations , 2015, J. Comput. Phys..

[26]  Wei Yang,et al.  A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations , 2014, J. Comput. Phys..

[27]  Charalambos Makridakis,et al.  A space-time finite element method for the nonlinear Schröinger equation: the discontinuous Galerkin method , 1998, Math. Comput..

[28]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[29]  Jiye Yang,et al.  Finite element multigrid method for multi-term time fractional advection diffusion equations , 2015, Int. J. Model. Simul. Sci. Comput..

[30]  Tingchun Wang,et al.  Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions , 2013, J. Comput. Phys..

[31]  Yangquan Chen,et al.  Computers and Mathematics with Applications Numerical Approximation of Nonlinear Fractional Differential Equations with Subdiffusion and Superdiffusion , 2022 .

[32]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[33]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..

[34]  Najeeb Alam Khan,et al.  Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method , 2012 .

[35]  Georgios Akrivis,et al.  Finite difference discretization of the cubic Schrödinger equation , 1993 .

[36]  Da Xu,et al.  Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation , 2014, J. Comput. Phys..

[37]  Mostafa Abbaszadeh,et al.  The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics , 2013 .

[38]  Fawang Liu,et al.  Galerkin finite element approximation of symmetric space-fractional partial differential equations , 2010, Appl. Math. Comput..

[39]  Xianjuan Li,et al.  A Space-Time Spectral Method for the Time Fractional Diffusion Equation , 2009, SIAM J. Numer. Anal..

[40]  Chengming Huang,et al.  A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation , 2014, Numerical Algorithms.

[41]  J. P. Roop.  Variational Solution of the Fractional Advection Dispersion Equation , 2004 .

[42]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .