# Survey of Discrete Laplace-Beltrami (LBO) on Mesh
###### tags: `Laplace`

## 1. 簡介
在傳統的$\mathbb{R}^2$空間中，若我們考慮間距固定為h均勻網格，我們可以透過5-points的有限差分法得到$\triangle u$ 的2階逼近(see [LeVeque FDM] P.59)：
> \begin{split}
> \triangle_5 u(p) := \frac{1}{h^2}\Big[\sum_{j \in R_1(p)}u(p_j)-4u(p) \Big]=\triangle u(p)+\frac{1}{4!}(u_{xxxx}+u_{yyyy})h^2 +o(h^2)
> \end{split}

我們可以知道
> \begin{split}
> \triangle_5 u(p)-\triangle u(p) = O(h^2)
> \end{split}

而且可以透9-points的有限差分及特殊的疊帶手法得到4階逼近。(待補)
現在我們想要將曲面的Laplace-Beltrami Operator進行離散化，而同樣地我們必須考慮離散解對於真實解的收斂行為(隨著網格加密)。

## 2. 曲面$S$上的Laplace-Beltrami Operator(LBO)
### 2-1. 定義
從傳統微分幾何的觀點，我們定義了一個0-form, $u$, 的Laplace-Beltrami Operator如下
> \begin{split}
> \triangle_S u := *d*d u = d^* d u
> \end{split}

其中$d^*: H^p(\Omega) \rightarrow H^{p-1}(\Omega)$ 是對偶外微分(exterior co-differentiate operator)

### 2-2. Cotan-Formular
(see Meyer 2003)
> \begin{split}
> \triangle_M f(p)&=\frac{1}{A_M(p)}\sum_{j \in R_1(p)}\frac{\cot \alpha_{i,j}+\cot \beta_{i,j}}{2}\big[f(p_j)-f(p)\big]
> \end{split}

其中角度...怎麼計算的
*插入圖片

### 2-3. 熱方程下的LBO
(see Belkin et al. 2008 & Xinge Li et al. 2015)
透過曲面上的熱核($H_S^t(x,y)$)及熱方程
> \begin{split}
> \begin{cases}
> \triangle_S u(x,t)= \frac{\partial}{\partial t}\int_{S}H^t_S(x,y)f(y)dy\\
> u(x,0)=f(x)
> \end{cases}
> \end{split}

考慮熱核的漸近展開
> \begin{split}
> H^t_S(x,y) \sim \frac{1}{4 \pi t}e^{\frac{-d_S(x,y)^2}{4t}}
> \end{split}

並定義
> \begin{split}
> F^t_S f(x)=\frac{1}{4 \pi t^2}\int_S e^{\frac{-d_S(x,y)^2}{4t}}\big(f(x)-f(y) \big)d \nu (y)
> \end{split}

## 3. 均勻網格球面
(see [Xu, 2006])
這篇論文中，沿用[Meyer 2003]中的contangent formaular表示離散LBO，計算球面上精確的LBO：
> \begin{split}
> \triangle_S f(p) = (\triangle_S p)^T \nabla f(p) + \triangle f(p) - p^THf(p)p, p \in S
> \end{split}

並估計誤差及在均勻網格的條件下討論其收斂性
> \begin{split}
> \triangle_M f(p) = \triangle_S f(p) + O(r), \ \ \ as\ r \rightarrow 0.
> \end{split}

## 4. 其他可以參考的資料(未讀)
1. D. Liu, G. Xu, Q. Zhang. *A discrete scheme of Laplace-Beltrami operator and its convergence over quadrilateral meshes* Computers and Mathematics with Applications 55 (2008) 1081-1093.

## Reference
>1. Randall J. LeVeque. *Finite Difference Methods for Ordinary and Partial Differential Equations*
>2. M. Meyer, M. Desbrun, P. Schröder, A. H. Barr. *Discrete Differential-Geometry Operators for Triangulated 2-Manifolds.* Visualization and Mathematics January 2003.
>3. M. Belkin, J. Sum, Y. Wang. *Discrete Laplcae Operator on Meshed Surfaces.* Pro ceedings of the twenty-fourth annual symposium on Computational geometry. June 2008 Pages 278-287.
>4. Xinge Li, Guoliang Xu, Yonhjie Jessica Zhang. *Localized discrete Laplace-Beltrami operator over triangular mesh.* Computer Aided Geometry Design 39(2015)67-82
>5. Guoliang Xu, *Discrete Laplace-Beltrami Operator on Sphere and Optimal Spherical Triangulations.* International Hournal of Computational Geometry & Applications Vol. 16, No. 01, pp.75-93(2006)