Two-grid Finite Element Method For Time-independent Nonlinear Schr¨odinger Equation

This paper is targeted on solving initial and boundary value problem ofnonlinear time-independent Schr(¨|o)dinger equation,exploring numerical method offinite element discretization based on a coarse grid and a fine one,structuring cal-culation format and carrying out theoretical analysis and numerical experiments.The principle of the two grid method based on finite element includes twosteps.Firstly,obtain the finite element approximation of the original problem onthe coarse grid.Secondly, get the finite element approximating solution of a sig-nificantly simplified problem on the fine grid and this solution has the sameconvergence order as the one by standard finite element method.Here,the sim-plification means the decoupling of the coupled problem and the linearity of thenonlinear problem.In this paper,we structure the two grid algorithm of the prob-lem in,give the error estimates of the solution obtained by two grid method andget the conclusion that the numerical solution by two grid method has the sameconvergence rate as the solution by standard finite element method(in the senseof‖·‖₁).Computation can be reduced greatly by two grid finite element methodbecause the coarse grid can be chosen very coarse.