Numerical Methods And Analysis For The Schr(o|¨)dinger-Poisson Equations And Their Applications

In this thesis, we study the numerical methods and analysis for the Schr(o|¨)dinger-Poisson equations together with some applications. It consists of three main parts:(1) thecomputation of ground state and dynamics by accurate and efcient algorithms in diferent dimensions;(2) the optimal error estimates of two compact finite diference discretiza-tions;(3) the dimension reduction analysis of the three dimensional (3D) Schr(o|¨)dinger-Poisson equations under anisotropic potentials.In the first part, we first introduce the normalized gradient flow method, time splitting method and semi-implicit finite diference method, and then propose diferentalgorithms to approximate the Poisson potential, which include fast convolution method,sine pseudospectral method, Fourier pseudospectral and finite diference method basedon artificial boundary condition. We point out the inconsistency at0-mode in Fourierpseudospectral approximation which results in a significant loss of accuracy. Finally,we apply the backward Euler sine pseudospectral and time-splitting sine pseudospectralmethods to study the ground state and dynamics of3D Schr(o|¨)dinger-Poisson equations indiferent setups.In the second part, we propose two compact finite diference methods for theSchr(o|¨)dinger-Poisson equations in a bounded domain and establish their optimal errorestimates under proper regularity assumptions. The conservative Crank-Nicolson com-pact finite diference method and the semi-implicit compact finite diference method areboth of order O(h~4+τ~2) in discrete l~2, H~1and l~∞norms with mesh size h and time step τ.In the last part, we present rigorous dimension reduction analysis from the3DSchr(o|¨)dinger-Poisson equations to lower dimensional reduced equations, namely the Surface Adiabatic Model (SAM), Surface Density Model (SDM) and Line Adiabatic Model(LAM). Efcient and accurate numerical schemes are proposed to approximate the efective potentials, which are combined with a backward Euler pseudospectral method andtime splitting pseudospectral method to study the ground state and dynamics respectively. Extensive numerical results are reported to confirm the reduction and convergencerate in terms of ground state and dynamics. In fact, the SDM describes the behavior ofthe2D quantum degenerated electron gas by utilizing the the inversion of square root of Laplacian to describe the electron-electron interaction potential. Finally, we study theground state and dynamics of SDM in diferent setups together with some applications ingraphene.