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標題: 使用立方次厄米特有限單法延續法解旋轉玻色-愛因斯坦凝聚
A Cubic Hermite Finite Element-Continuation Method for Numerical Solutions of Rotating Bose-Einstein Condensates
作者: 施因澤
關鍵字: 數學類
摘要: 我們提出一年的研究計畫,主要是提出使用立方次厄米特有限單元-延續法來解旋轉玻色-愛因斯坦凝聚。我們將內容分為兩部分,第一部分處理以及分析有限單法離散化偏微分方程式。第二部分利用延續法處理非線性的薛丁格方程組以其控制分支的解答曲線。在第一部分裡,我們先使用泛函分析的技巧研究與此特徵值問題相對應的變分問題,並且化簡為有限維的問題來研究此PDE 解的性質。然後我們提出一個Adini 有限單元的算則,用以離散化薛丁格特徵值問題來解薛丁格特徵值問題的前幾個最小特徵對偶。在使用矩形元素逼近時,我們能得到四次的收斂速率。在此部分我們已部份完成使用有限單元離散特徵值問題的計算工作。在本計畫的第二部分,我們將使用數值延續法來追蹤非線性的薛丁格方程組正解,我們主要研究第一個特徵向量的分支特性以及它的超流體密度等位圖。我們將與張勝麟,簡澄陞於2007 發表的文章做比較。該文章是目前惟一以數值方法繪出超流體密度等位圖。
This is one year proposal and the research focuses on a cubic Hermite finite elementcontinuationmethod for numerical solutions of the rotating Bose-Einstein condensate (BEC).There are two main topics for this project. First, we process and analyze the finite elementmethod for differential equations, and secondly we implement the continuation algorithm fortwo coupled nonlinear Schrdinger eigenvalue problems and trace the solution curve and itsfirst branch.In this first part, we will show that the existence of the solution of nonlinear Schrdingereigenvalue problem and its associated variational formulation in the Hilbert space 10 H . Nextwe will use the finite element approximation of the variational formulation in the finitedimensional subspace with Hermite basis functions. Here we propose the Adini finite elementto discretize the Schrdinger eigenvalue problem and solve the first eigen-pairs. We will showthat the Adini finite element method for nonlinear Schrdinger eigenvalue problem convergesin 4th order.In Part two, we will use the continuation method to trace the solution for the nonlinearSchrdinger eigenvalue problem. We will study first solution branch and the contour itssuperfluid densities. We will make a comparison with the first known literature presented byChang and Chien in 2007 that discretized the BEC system by the centered difference method.
其他識別: NSC98-2115-M005-005
Appears in Collections:應用數學系所



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