Laplace-Beltrami

In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami. Wikipedia

$$\Delta f = div\left( \nabla f\right)$$

The Laplace–Beltrami operator, like the Laplacian, is the (Riemannian) divergence of the (Riemannian) gradient.

Cotangent laplacian

In practice, a very simple discretization can actually work quite well—especially on fine triangulations with “nice” elements (e.g., those that satisfy the so-called Delaunay condition). A standard choice is to use the 2-dimensional cotangent formula:

$$\left(L\phi\right)_i := \frac{1}{2}\sum_{ij}\left( \cot \alpha_{ij} + \cot \beta_{ij} \right)\left( \phi_j - \phi_i \right)$$

In other words: the Laplacian of the function u at vertex $i$ is obtained by summing up the difference across all edges $ij$ incident on $i$, weighted by the sum of the cotangents of the two interior angles $\alpha_{ij}$, $\beta_{ij}$ opposite $ij$.

Partial differential equations

It can be used for solving/approximating the poisson equation, diffusion equation, heat equation, wave equation or any other equation which involves the laplacian.

Program

This program is an exaple of the Laplace-Beltrami operator discretization of a scalarfield $\phi$ defined on a manifold $M$ using the cotangent laplacian formula in Python, using the following libraries:

numpy
matplotlib