在這篇文章裡，我們使用有限元素法來解定義在不規則定義域上的Neumann 問題。我們從定義在曲線上的偏微分方程出發，並且使用與曲線邊界不吻合的均勻笛卡爾網格。在這個方法中，方程式的邊界條件會被一個定義域內的節點上的函數值的線性組合所取代。我們使用線性以及二次的有限元素法在與邊界不吻合的網格上，而無論定義域的邊界是如何割過元素，方程式離散化之後產生的線性系統都會是穩定的。我們也從能量範數的角度給出誤差分析，並且展示幾個數值範例。

In this thesis, we consider a finite element method to solve partial differential equations on a irregular domain with Neumann boundary condition. In particular, we start with partial differential equations defined on curves. Instead of curve fitted meshes, we use Cartesian grids that do not fit the curved boundary. The boundary condition is replaced by a linear constraint on inner nodes. We apply both piecewise linear and quadratic element on unfitted meshes. The resulted linear system is stable whenever the boundary cut though the elements. We give error estimate on energy norm and show some numerical examples.

1 Introduction 2
2 The technique 3
2.1 A PDE on a curve . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 A General Neumann problem . . . . . . . . . . . . . . . . . . . . 6
3 Error estimate 7
4 Numerical example 9
5 Conclusion 12

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