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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 $d(x_i,x_j)$.

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    $\begingroup$ 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. $\endgroup$
    – lightxbulb
    Feb 1 at 12:14
  • $\begingroup$ @lightxbulb that was actually the string I was looking to follow, thanks Laplacian matrix expert $\endgroup$
    – Aner
    Feb 2 at 12:49

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In a plane, you have something that resembles the graph Laplacian in the form of the 5-point stencil. You can derive the 5-point stencil as an approximation of the finite element matrix by using a specific quadrature rule (namely, the 2-dimensional generalization of the trapezoidal rule) to compute the integrals that define the finite element method.

I would suspect that you can apply the same kind of principle on manifolds: Subdivide it into triangles or quadrilaterals, set up the finite element integrals, then approximate these integrals by a quadrature formula that only evaluates the integrand at the node points.

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