Abstract: An alternating direction implicit (ADI) Legendre spectral element method for the two dimensional Schrodinger equation is developed, and the optimal H¹ error estimate for the linear case is given. This method converts the two dimensional computation to one dimension with domain decomposition so that the size of problem and the storage are reduced. The main advantage of the method is its high degree of parallel computation. By choosing appropriate base functions, strip sparse matrices are derived. Numerical experiments show validity of the method. 

Key words: alternating direction implicit (ADI) method, Schrdinger equation, spectral element method