Because the nonlinear Schrödinger equation is met in many physical problems and is applied widely, The research on well-posedness for its solution and numerical methods has aroused more and more interest. The self-adjoint case has been considered by many authors. For a class of system of nonlinear and non-self-adjoint Schrodinger equations which refers to excitons occurring in one dimensional molecular crystals and in a spiral biomolecules, Guo Boling studied in [6], the pure initial and periodic initial value problems of this system and obtained the existence and uniqueness of its solution. In [7] we discuss the difference solution of this system and obtained its error estimate. In this paper, we shall study the finite element method for the periodic initial value problem of this system. Just as Guo pointed out in [6], since it has a non-self-adjoint term, it not only brings about trouble in mathematics, but also creates more difficulty in numerical analysis.
Our analysis will show that for this system, in theoretically we can obtain the same results as when it has no non-self-adjoint term.