Unconditionally Optimal Error Estimates For Semi-implicit BDF2-FEM For A Cubic Schr(?)dinger Equation

Nonlinear Schr(?)dinger equation is a basic equation for studying numerical solutions of partial differential equations.In the process of studying the numerical solution of the equation,many numerical methods have been established,for instance:finite difference method,finite element method,finite volume method,geometric volume method,fractional Fourier transform method,Fourier series analysis method and pseudospectral method.Each method has its own unique features.The cubic Schr(?)dinger equation is more representative in the study of nonlinear Schr(?)dinger equations.Compared with the general nonlinear equations,the calculation and derivation of these equations are intuitionistic and concise.Consequently,In this paper,we study optimal error estimates for a BDF2-FEM for a cubic Schr(?)dinger equation.First,we split an error estimate into two parts:the temporal-discretization and the spatial-discretization.Because of the complexity of the non-linear Schr(?)dinger equation,the method of calculating error estimates separately is adopted.The mathematical logic is clear and the calculation is concise.By introducing a temporal-discretization equation,we acquire the uniform boundedness of the solution and the the time-discrete error estimate.Secondly,the finite element method is used to solve the numerical solution of the nonlinear Schr(?)dinger equation,and the error estimate of the full discrete finite element solution is obtained,go a step further,we obtain unconditionally optimal error estimate of the second-order backward difference(BDF2-FEM)semi-implicit scheme for the cubic Schr(?)dinger equation.At the end of the paper,a numerical example is given to verify the theoretical analysis.Under the L~2 norm,the error estimates of linear finite element are in direct proportion to h~2,and the error estimates of quadratic finite element are in direct proportion to h~3.Therefore,the numerical results are accordance with the theoretical analysis.The error estimates of linear finite element and quadratic finite element tend to be constant under L~2 norm when the mesh size is refined successive steps at a fixed time step T.Therefore,the stability of the BDF2-FEM algorithm for the original equation has no compulsory condition for the time step,and this method can be used to analyze other nonlinear parabolic equations.