A Two-Grid Finite-Element Method for the Nonlinear Schrödinger Equation

Abstract

In this paper, some two-grid finite element schemes are constructed for solving the nonlinear Schrödinger equation. With these schemes, the solution of the original problem is reduced to the solution of the same problem on a much coarser grid together with the solutions of two linear problems on a fine grid. We have shown, both theoretically and numerically, that our schemes are efficient and achieve asymptotically optimal accuracy.Mathematics subject classification: 65N30, 65N55

A Two-Grid Finite-Volume Method for the Schrödinger Equation

“…For any real-valued and complex-valued function, the inner product (•,•) and standard Sobolev norms ( • m,p , • m,∞ ) have been defined in the same way as in [44].…”

Low order nonconforming finite element method for time-dependent nonlinear Schrödinger equation

“…Up to now, the two-grid method was deeply researched for different problems [20][21][22][23][24][25]. Especially, the two-grid method was used to solve the linear Schrödinger equation (LSE) and NLSE in [26][27][28][29][30]. However, this method is rarely considered for nonconforming elements.…”

Two‐grid method for two‐dimensional nonlinear Schrödinger equation by finite element method

“…In recent years, there has been tremendous interest in developing finite difference schemes and finite element methods for two‐dimensional Schrödinger equations . Wang studied the split‐step finite difference method for various cases of NLS equations, such as cubic NLS equations, coupled NLS equations with constant coefficients and Gross‐Pitaevskii equations in 1D, 2D, and 3D in .…”

“…Dawson and Chen proposed a two‐grid method for quasilinear reaction diffusion equations. Jin et al successfully extended the two‐grid finite element method to solve coupled partial differential equations, such as the linear Schrödinger equation, where the equations on the fine grid are decoupled so that the computational complexity of solving the Schrödinger equation is comparable to solving two decoupled Poisson equations on the same fine grid. Chien et al proposed two‐grid discretization schemes with two‐loop continuation algorithms for nonlinear Schrödinger equations, where the centered difference approximations, the six‐node triangular elements and the Adini elements are used to discretize the partial differential equations.…”

“…Numerical experiments using these schemes have been shown to be efficient; however, no rigorous error analysis has been conducted. Zhang et al recently extended the approach given in to time‐dependent linear Schrödinger equations, where the error convergence rate is of the order of in the ‐norm. Therefore, it is natural to develop a two‐grid scheme for time‐dependent nonlinear Schrödinger equations −1.3 that produces a better error analysis result.…”