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研究生： 丹明輝
Rizal Dian Azmi
論文名稱： A Nonlinear Multiscale Finite Element Method for Poisson-Boltzmann Equation
指導教授： 黃楓南
Feng-Nan Hwang
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2014
畢業學年度： 103
語文別： 英文
論文頁數： 43
中文關鍵詞： 多尺度 、有限元素法 、Poisson-Boltzmann
外文關鍵詞： Multiscale, Finite Element Method, Poisson-Boltzmann
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    • 在兩個均勻帶電的平行板中，對稱的電解質通常使用泊松-波茲曼方程來做模擬，進而使用非線性泊松-波茲曼方程做修正。而針對此問題，我們以不精確地回溯牛頓迭代法來求解。在離散方法上若採用標準的蓋勒肯法，則需要細密的網格才可達到較為精確的解。因此為了避免網格的需求，我們在較少的網格上採用多重尺度有限元素法來求出更精確的解。在本文中導入多重尺度基底及泡泡函數，並使用此兩者來對全域的多重尺度有限元素方程求解，同時我們也展現了多重尺度在每個牛頓法迭代次數的改變。

    The Poisson–Boltzmann equation (PBE) is used to model the symmetric electrolyte in two parallel uniformly charged plates. A Correction problem formed to approach the
    nonlinear Poisson-Boltzmann equation. The inexact Newton with backtracking method iteration used to solve that correction problem. In the discretization, the standard Galerkin method requires the small grid to achieves accurate result. To avoid this, a multiscale finite element is used to get the better accurate result with a few grid partition. A residual free method is used to derive the multiscale basis and bubble function. This multiscale basis and bubble function used to solved the global solution of multiscale finite element method. It is shown that multiscale basis is changed in each Newton iteration.

    1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Poisson-Boltzmann Equation in Symmetric Electrolyte Problem . . . . . . 3 3 Numerical Method for 1D Poisson Boltzmann Equation . . . . . . . . . . . 5 3.1 Poisson-Boltzmann equation and it’s linearization . . . . . . . . . . . . . 5 3.2 Variational formulation for the corection equation . . . . . . . . . . . . . 6 3.3 Galerkin finite element formulation . . . . . . . . . . . . . . . . . . . . . 7 3.4 Multiscale Finite Element Method (MsFEM) . . . . . . . . . . . . . . . 11 3.5 An Review of The Inexact Newton Method with Backtracking (INB) . . . 13 3.6 Iterated INB with Galerkin FEM and MsFEM for Poisson-Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.1 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Grid Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Appendix: Stiffness matrix and load vector of finite element methods) . . . . . . . 30

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