立方Schr?dinger方程的半隐格式BDF2-FEM无条件最优误差估计
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摘要
研究了立方Schr?dinger方程的二阶向后差分有限元方法(BDF2-FEM)的无条件最优误差估计.首先,将误差分为时间误差和空间误差两部分.通过引入时间离散方程,得到时间离散方程解的一致有界性,并给出时间误差估计.从而得到该方程在半隐格式下BDF2-FEM无条件最优误差估计.最后,用数值算例验证了理论分析.
The optimal error estimates of the semi-implicit BDF2-FEM were studied for cubic Schr?dinger equations. First, an error estimate was divided into 2 parts: the temporal-discretization and the spatial-discretization. Through introduction of a temporal-discretization equation, the uniform boundedness of the solution and the temporal error estimate were obtained. The unconditionally optimal error estimates of the 2 nd-order backward difference(BDF2-FEM) semi-implicit scheme for cubic Schr?dinger equations were given. Finally, numerical examples verify the theoretical analysis.
The optimal error estimates of the semi-implicit BDF2-FEM were studied for cubic Schr?dinger equations. First, an error estimate was divided into 2 parts: the temporal-discretization and the spatial-discretization. Through introduction of a temporal-discretization equation, the uniform boundedness of the solution and the temporal error estimate were obtained. The unconditionally optimal error estimates of the 2 nd-order backward difference(BDF2-FEM) semi-implicit scheme for cubic Schr?dinger equations were given. Finally, numerical examples verify the theoretical analysis.
引文
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[2] EBAID A,KHALED S M.New types of exact solutions for nonlinear Schr?dinger equation with cubic nonlinearity[J].Journal of Computational and Applied Mathematics,2011,235(8):1984-1992.
[3] LI B Y,SUN W W.Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations[J].International Journal of Numerical Analysis and Modeling,2013,10(3):622-633.
[4] LI B Y,SUN W W.Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear thermistor equations[J].SIAM Journal on Numerical Analysis,2012,52(2):933-954.
[5] LAMBERT J D.Numerical Methods for Ordinary Differential Systems:the Initial Value Problem[J].New York:John Wiley & Sons Inc,1991.
[6] BAKER G,DOUGALIS V,KARAKASHIAN O.On a higher accurate fully discrete Galerkin approximation to the Navier-Stokes equations[J].Mathematics of Computation,1982,39(160):339-375.
[7] CAI W,LI J,CHEN Z.Unconditional convergence and optimal error estimates of the Euler semi-implicit scheme for a generalized nonlinear Schr?dinger equation[J].Advances in Computational Mathematics,2016,42(6):1311-1330.
[8] CAI W,LI J,CHEN Z.Unconditional optimal error estimates for BDF2-FEM for a nonlinear Schr?dinger equation[J].Journal of Computational and Applied Mathematics,2018,331:23-41.
[9] DUPONT T.Three-level Galerkin methods for parabolic equations[J].SIAM Journal on Numerical Analysis,1974,11(2):392-410.
[10] 姜礼尚,庞之垣.有限元方法及其理论[M].北京:人民教育出版社,1979.(JIANG Lishang,PANG Zhiyuan.Finite Element Method and Its Theory[M].Beijing:People’s Education Press,1979.(in Chinese))
[11] BREZZI F,RAPPAZ J,RAVIART P A.Finite Dimensional Approximation of Nonlinear Problems[M].New York:Springer-Verlag,1980.
[12] AKRIVIS G,LARSSON S.Linearly implicit finite element methods for the time-dependent Joule heating problem[J].Bit Numerical Mathematics,2005,45(3):429-442.
[13] JENSEN M,MALQVIST A.Finite element convergence for the Joule heating problem with mixed boundary conditions[J].Bit Numerical Mathematics,2013,53(2):475-496.
[14] BULUT H,PANDIR Y,DEMIRAY S T.Exact solutions of nonlinear Schr?dinger equation with dual power-law nonlinearity by extended trial equation method[J].Waves Random Complex Media,2014,24(4):439-451.
[15] HEYWOOD J G,RANNACHER R.Finite element approximation of the nonstationary Navier-Stokes problem IV:error analysis for second-order time discretization[J].SIAM Journal on Numerical Analysis,1984,27(2):353-384.
[16] FEIT M D,FLECK J A,STEIGER A.Solution of the Schr?dinger equation by a spectral method II:vibrational energy levels of triatomic molecules[J].Journal of Computational Physics,1983,78(1):301-308.
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