Structure preserving FEM for the perturbed wave equation of quantum mechanics

The construction and analysis of structure-preserving finite element method (FEM) for computing the perturbed wave equation of quantum mechanics are demonstrated. Firstly, a new fully discrete system is built and proved conservative in the sense of the energy. Meanwhile, the boundedness of the numerical solution is derived. Secondly, the existence and uniqueness of the solution are obtained with the help of the Brouwer fixed-point theorem and some special splitting technique. Thirdly, we provide a comprehensive superclose analysis, offering the global superconvergent result. Finally, numerical results are presented to illustrate the theoretical analysis.