Optimal L2 error estimates of two structure-preserving finite element methods for Schrödinger-Boussinesq equations

Abstract

This paper is concerned with the construction and analysis of two structure-preserving finite element approximation schemes for solving one and two dimensional coupled Schrödinger-Boussinesq (SBq) equations. Firstly, two finite element approximation schemes are developed and the total mass and energy preserving properties of the proposed schemes are demonstrated in the discrete sense. Secondly, by use of the innovative cut-off function method, an auxiliary fully-discrete system is established, which leads to the unique solvability with the Brouwer's fixed-pointed theorem and the optimal $L^2$ error estimates for the proposed nonlinear scheme. Finally, linearized iterative algorithms are showed rigorously and extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical methods.