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In this paper, we approximate the solution of the coupled nonlinear Schr\"odinger equations by using a fully discrete finite element scheme based on the standard Galerkin method in space and implicit midpoint discretization in time. The proposed scheme guarantees the conservation of the total mass and the energy. First, a priori error estimates for the fully discrete Galerkin method is derived. Second, the existence of the approximated solution is proved by virtue of the Brouwer fixed point theorem. Moreover, the uniqueness of the solution is shown. Finally, convergence orders of the fully discrete Crank-Nicolson scheme are discussed. The end of the paper is devoted to some numerical experiments.
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참고문헌 (33)
참고문헌 신청
R. A. Adams / 1975 / Sobolev Spaces / Academic Press
Noureddine Atouani / 2013 / Galerkin finite element method for the Rosenau-RLW equation / Computers & Mathematics with Applications 66(3):289~303
Harish P. Bhatt / 2014 / Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations / Applied Mathematics and Computation 228:271~291
A. Biswas / 2012 / Soliton solutions of the perturbed resonant nonlinear Schrodinger's equation with full nonlinearity by semi-inverse variational principle / Quant. Phys. Lett 1(2):79~86
Salah Boulaaras / 2013 / The finite element approximation of evolutionary Hamilton–Jacobi–Bellman equations with nonlinear source terms / Indagationes Mathematicae 24(1):161~173
Noureddine Atouani / 2013 / Galerkin finite element method for the Rosenau-RLW equation / Computers & Mathematics with Applications 66(3):289~303
Harish P. Bhatt / 2014 / Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations / Applied Mathematics and Computation 228:271~291
A. Biswas / 2012 / Soliton solutions of the perturbed resonant nonlinear Schrodinger's equation with full nonlinearity by semi-inverse variational principle / Quant. Phys. Lett 1(2):79~86
Salah Boulaaras / 2013 / The finite element approximation of evolutionary Hamilton–Jacobi–Bellman equations with nonlinear source terms / Indagationes Mathematicae 24(1):161~173
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