Unconditional optimal error estimates of a linearized mass- and energy- conservation FEM for a coupled nonlinear Schrödinger equations

Abstract

In this paper, we study a linearized finite element method for a coupled nonlinear Schrödinger system (CNLS). We apply a modified leap-frog scheme for time discretization and a Galerkin finite element method for spatial discretization. We prove that this method is conservative for mass and energy in a modified form. By an error splitting technique, we split the error into two parts, the temporal error and the spatial error. First we prove the uniform boundedness in H2-norm for the solutions of the time-discrete system, and obtain the temporal error estimates. Second, we prove the pointwise uniform boundedness of the finite element solution. Then we get the optimal L2-norm error estimates without any time-step restriction. Finally, in order to confirm our theoretical analysis, we provide numerical examples to validate the convergence-order, unconditional stability and conservation for mass and energy.