在這一篇論文中使用快速傅立葉轉換(FFT)、固定點迭代和牛頓法，在橢圓域上數值求解電荷守恆的泊松-玻茲曼方程(CCPB)，此方程包含線性算子與非線性函式。首先，我們使用有限差分法把已橢圓座標參數化的線性算子數值離散，並且使用快速傅立葉轉換得到線性部分的數值解，非線性的部分藉由固定點迭代與牛頓法求解。數值解是二階收斂的，除此之外，運算時間與未知總點數幾乎成正比。

In this thesis, we solve the Charge Conserving Poisson-Boltzmann equation(CCPB) numerically in elliptic domain by using fast Fourier transform(FFT), fixed point iterations and Newton method. The equation consists of a linear operator and a nonlinear function. First, we use the finite difference method to obtain the numerical discretization of the linear operator with elliptic coordinate parameterization and use fast Fourier transform to obtain the solution of the linear part. The non-linear part is solved by the fixed point iteration and Newton method. The convergence of the solution is second order. In addition, the computation time is almost proportional to the number of total unknowns.

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