Unconditional stability and convergence of Crank–Nicolson Galerkin FEMs for a nonlinear Schrödinger–Helmholtz system

Wang, Jilu

doi:10.1007/s00211-017-0944-0

Cited by 22 publications

(13 citation statements)

References 40 publications

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“…Comparing to previous error analysis with conditional stability in [34][35][36], optimal error estimates were obtained unconditionally in [37,38]. Recently, this new technique has been used to analyze linearized FEMs for nonlinear Schrödinger type equation [24][25][26][27][28] and many other PDEs [39][40][41][42][43][44]. To our best knowledge, this new technique is mainly carried out for nonlinear parabolic type of equations and also possibly coupled with elliptic type of equations.…”

Section: Introductionmentioning

confidence: 95%

“…In the last several decades, numerical simulations of both the nonlinear Schrödinger equation and the nonlinear Schrödinger-type equation have been studied extensively. For examples, finite difference methods [15][16][17][18][19][20][21][22][23], finite element methods [24][25][26][27][28][29][30][31][32] and Fourier spectral method [33]. In the field of finite difference methods, an implicit nonconservative difference scheme had been developed in [16] for solving nonlinear Schrödinger equation, the method needs lots of algebraic operators.…”

Section: Introductionmentioning

confidence: 99%

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A linearized energy–conservative finite element method for the nonlinear Schrödinger equation with wave operator

Cai

Pan

2019

Applied Numerical Mathematics

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In this paper, we propose a linearized finite element method (FEM) for solving the cubic nonlinear Schrödinger equation with wave operator. In this method, a modified leap-frog scheme is applied for time discretization and a Galerkin finite element method is applied for spatial discretization. We prove that the proposed method keeps the energy conservation in the given discrete norm. Comparing with non-conservative schemes, our algorithm keeps higher stability. Meanwhile, an optimal error estimate for the proposed scheme is given by an error splitting technique. That is, we split the error into two parts, one from temporal discretization and the other from spatial discretization. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and obtain error estimates in temporal direction. With the help of the preliminary temporal estimates, we then prove the pointwise uniform boundedness of the finite element solution, and obtain the optimal L 2 -norm error estimates in the sense that the time step size is not related to spatial mesh size. Finally, numerical examples are provided to validate the convergence-order, unconditional stability and energy conservation.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

A linearized energy–conservative finite element method for the nonlinear Schrödinger equation with wave operator

Cai

Pan

2019

Applied Numerical Mathematics

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“…, then the semi-discrete Fourier spectral-Galerkin approximation for solving (10) and (11) is to find u N ∈ S N such that…”

Section: Fourier-galerkin Schemementioning

confidence: 99%

“…Also, see the other related works on the numerical schemes for Schrödinger equations. [4][5][6][7][8][9][10] By defining some new integral over mechanical path, Laskin 11,12 and Laskin and Zaslavsky 13 obtained the spatial fractional derivative operator; on this basis, spatial fractional quantum mechanics has been developed. Guo and Xu 14 and Cheng 15 introduced some applications of space fractional equation in physics.…”

Section: Introductionmentioning

confidence: 99%

“…Schrödinger equations were regarded as a very important equation in quantum mechanics; it is widely used in plasma physics, chemistry, material science, and so on. Also, see the other related works on the numerical schemes for Schrödinger equations . By defining some new integral over mechanical path, Laskin and Laskin and Zaslavsky obtained the spatial fractional derivative operator; on this basis, spatial fractional quantum mechanics has been developed.…”

Section: Introductionmentioning

confidence: 99%

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An efficient numerical approach to solve Schrödinger equations with space fractional derivative

Zhang

Lin

Wang

2019

Math Methods in App Sciences

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We design and analyze an efficient numerical approach to solve the coupled Schrödinger equations with space‐fractional derivative. The numerical scheme is based on leap‐frog in time direction and Fourier method in spatial direction. The advantage of the numerical scheme is that only a linear equation needs to be solved for each time step size, and we proved that the energy and mass of space‐fractional coupled Schrödinger equations (SFCSEs) are conserved in the case of full‐discrete scheme. Moreover, we also analyze the error estimate of the numerical scheme, and numerical solutions converge with the order scriptOfalse(△t2+N−sfalse) in L2 norm. Numerical examples are illustrated to verify the theoretical results.

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L^p error estimate of nonlinear Schrödinger equation using a two‐grid finite element method

2023

Numerical Methods Partial

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A two-grid finite element scheme is presented for nonlinear Schrödinger equation. H 1 superconvergence error estimates of finite element solutions based on L ∞ bound of finite element solutions is proved. The optimal order error estimations of the two-grid solution in the L p and H 1 norm are given without any time-step size conditions. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse, with no loss in the order of accuracy.

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Unconditional stability and convergence of Crank–Nicolson Galerkin FEMs for a nonlinear Schrödinger–Helmholtz system

Cited by 22 publications

References 40 publications

A linearized energy–conservative finite element method for the nonlinear Schrödinger equation with wave operator

A linearized energy–conservative finite element method for the nonlinear Schrödinger equation with wave operator

An efficient numerical approach to solve Schrödinger equations with space fractional derivative

L^p error estimate of nonlinear Schrödinger equation using a two‐grid finite element method

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Unconditional stability and convergence of Crank–Nicolson Galerkin FEMs for a nonlinear Schrödinger–Helmholtz system

Cited by 22 publications

References 40 publications

A linearized energy–conservative finite element method for the nonlinear Schrödinger equation with wave operator

A linearized energy–conservative finite element method for the nonlinear Schrödinger equation with wave operator

An efficient numerical approach to solve Schrödinger equations with space fractional derivative

Lp error estimate of nonlinear Schrödinger equation using a two‐grid finite element method

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L^p error estimate of nonlinear Schrödinger equation using a two‐grid finite element method