Spectral Element Method For Large-Scale Computation Of Potential Well Problems

The motion of particles in a potential obeys a classical mathematical physical equation-the Schrodinger equation,which has important applications in many fields,such as quantum me-chanics,quantum chemistry,and condensed matter physics.Eigenfunctions of the Schrodinger equation with a disordered potential will be highly localized,and this phenomenon is called Anderson localization.The disordered potential can be simulated by the piecewise constant po-tential.However,the finite element method will face great difficulties as the dimension and the complexity of the piecewise constant potential increase when solving the eigenvalue problem and the corresponding source problem.In this thesis we develop a spectral element method to solve the difficulties above,verify the Anderson localization phenomenon,and successfully develop a theory and algorithm to predict the eigenvalues and eigenfunctions by solving the source problem.Taking advantage of the efficient spectral element method and its fast algorithm,we can optimize large-scale matrices of the multidimensional discrete problem block-by-block to create an efficient way that allows numerical experiments to be achieved with high precision and a low cost.With the efficient numerical method we successfully solve the two kinds of problems in the three-dimensional case.Then,we can predict the eigenvalues and eigenfunctions and develop the relevant empirical theory and algorithm.In conclusion,we develop a spectral element method for the eigenvalue problem and source problem with a large-scale piecewise constant potential well,obtain an efficient nu-merical method,and successfully complete the numerical experiments in the three-dimensional case.Results from the numerical experiments verify the Anderson localization phenomenon and confirm the reliability of the theory which predicts the eigenvalues and eigenfunctions.