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

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

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

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 $\phi$ 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.

Heat equation

As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications:

$$\frac{\partial \phi}{\partial t} = \alpha \Delta \phi$$

Dependencies

numpy
matplotlib