# Reference request: graph Laplacian approximation for domains/manifolds

Is there any reference on solving the heat equation on irregular geometries via creating a mesh and using the graph Laplacian instead of FEM techniques? (Convergence to real solution).

That is to say, given a set of points $$\{x_i\}\in\Omega$$, with $$\Omega$$ a domain, what is the Laplacian matrix best approximating the Laplacian operator in $$\Omega$$ when restricting functions $$f\in\mathcal{L}^2(\Omega)$$ to the evaluation $$f(x_i)\in\mathbb{R}^n$$? I imagine each entry $$L_{ij}$$ should be only a function of the Euclidean distance $$d(x_i,x_j)$$.

• I think you're looking for a discrete Laplace-Beltrami operator. The graph Laplacian takes only the topology into account. If you have a triangular mesh you can see: mobile.rodolphe-vaillant.fr/entry/101/… If you have a more general polygonal mesh there was a paper by Marc Alexa and Warzdetsky. If you don't have a mesh then there are some mesh-free ways to go about this. From your question it looks like you just have a point cloud. But then you speak of "irregular geometries" at the beginning so it is unclear what you are given. Feb 1 at 12:14
• @lightxbulb that was actually the string I was looking to follow, thanks Laplacian matrix expert
– Aner
Feb 2 at 12:49