A New Error Analysis of Crank–Nicolson Galerkin FEMs for a Generalized Nonlinear Schrödinger Equation

Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation

“…It seems that these results have never be seen in the existing literature. How to remove this restriction as [12] and extend our results to nonconforming finite element case still remain open.…”

“…The NLSE is one of the most important equations in various areas of mathematics and physics, such as nonlinear optics, optical pulses, plasma physics and water waves [1] . Numerical methods for the NLSE have been investigated extensively, see [2][3][4][5][6] for finite difference method, [7][8][9][10][11][12] for finite element method (FEM), [13,14] for discontinuous Galerkin method, and [15][16][17][18][19] for meshless method. However, all of these studies only concentrated on the convergence analysis.…”

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

“…Such a technique has been widely used in error analysis for many different nonlinear PDEs, see for example, [12][13][14][15][16] for nonlinear parabolic equation, [17] for nonlinear Schrödinger equation, [18,19] for Navier-Stokes equations. Such splitting techniques were also used to other equations [22][23][24][25]. Then the spatial error reduces to the unconditional boundedness of numerical solution in maximum norm.…”

Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations

“…A semi-implicit scheme for the nonlinear term results in a linear system with a variable coefficient matrix of time, and an explicit treatment for the nonlinear term gives a constant matrix. Some authors have exploited IMEX methods to study NLS equation [36,38]. Wang [36] studied the linearized Crank-Nicolson scheme for nonlinear Schrödinger equation and obtained the optimal L 2 error estimate without any time-step restrictions.…”

“…Some authors have exploited IMEX methods to study NLS equation [36,38]. Wang [36] studied the linearized Crank-Nicolson scheme for nonlinear Schrödinger equation and obtained the optimal L 2 error estimate without any time-step restrictions. Xu and Zhang [38] presented the linearized ADI schemes for solving two-dimensional cubic nonlinear Schrödinger equations.…”