Advanced Finite Element and Hybrid Methods for Problems in Wave Propagation, Nonlinear Optics, and Chemotaxis

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題名:	Advanced Finite Element and Hybrid Methods for Problems in Wave Propagation, Nonlinear Optics, and Chemotaxis
作者:	盧斯非;IHWANI, IVAN LUTHFI
貢獻者:	數學系
關鍵詞:	One keyword per line;有限元素法;Helmholtz 方程;非線性光學;三次諧波產生;Keller–Segel 趨化模型;One keyword per line;Finite Element Method;Helmholtz Equation;Nonlinear Optics;Third-Harmonic Generation;Keller–Segel Chemotaxis Model
日期:	2025-09-19
上傳時間:	2025-10-17 12:56:20 (UTC+8)
出版者:	國立中央大學
摘要:	本論文針對波傳遞、非線性光學與趨化作用等具挑戰性的偏微分方程，發展先進的有限元素與混合數值方法，並提出四項主要貢獻。首先，我們提出用於二維線性 Helmholtz 問題的具自適應氣泡函數增強之多尺度有限元素法；此法結合多尺度有限元素基底與局部計算的氣泡函數，以提升精度並降低污染誤差。其次，針對非線性 Helmholtz 方程，我們提出一種基於牛頓法的穩定化有限元素法，採用「先線性化、後離散化」策略並結合 Galerkin 最小平方（GLS）穩定化，可穩健模擬 Kerr 介質中的光學雙穩態與光傳播。第三，我們發展用於描述三次諧波產生的耦合非線性 Helmholtz 系統之巢狀多尺度穩定化有限元素法；透過定點迭代、穩定化與多尺度設計，能有效捕捉光子晶體結構中的強非線性多頻交互作用，並達成高轉換效率。最後，我們設計一套適用於 Keller–Segel 趨化模型的有限元素–有限差分數值方案，兼具有限元素在幾何處理上的彈性與有限差分在時間積分上的效率，能準確模擬由非線性交叉擴散與聚集效應所引發的解之爆炸（blow-up）現象。綜合而言，這些方法可為物理、光學與生物建模中的複雜偏微分方程提供精確、穩定且高效的數值解決方案。;This dissertation develops advanced finite element and hybrid methods for challenging partial differential equations in wave propagation, nonlinear optics, and chemotaxis. Four contributions are presented. First, we present the multiscale finite element method with adaptive bubble enrichment for the two-dimensional linear Helmholtz problem. This method combines a set of multiscale finite element basis functions with bubble functions, which are computed locally, to improve accuracy and reduce pollution errors. Second, a Newton-based stabilized finite element method is proposed for the nonlinear Helmholtz equation, employing a linearize–then–discretize strategy with Galerkin least-squares stabilization to robustly simulate optical bistability and light propagation in Kerr media. Third, a nested-multiscale stabilized finite element method is developed for coupled nonlinear Helmholtz systems modeling third-harmonic generation. Employing a fixed-point iteration, stabilization, and a multiscale approach, the method efficiently captures strongly nonlinear multi-frequency interactions in photonic crystal structures and achieves high conversion efficiency. Finally, a finite element–finite difference scheme is designed for the Keller–Segel chemotaxis model. This scheme leverages the geometric flexibility of the finite element method and the time-stepping efficiency of the finite difference method. It is capable of accurately simulating blow-up phenomena arising from nonlinear cross-diffusion and aggregation effects. These methods collectively provide accurate, stable, and efficient solutions for complex partial differential equations arising in physics, optics, and biological modeling.
顯示於類別:	[數學研究所] 博碩士論文

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