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I was looking over Langtangen's book on finite volume methods, and he mentions in section 5.5.1 that the second order or Laplacian term in a PDE can be written either as:

$$ \nabla^2 u \quad \text{or} \quad \nabla \cdot \nabla u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} $$

Both of these notations are analytically equivalent, but the $\nabla \cdot \nabla u$ approach seems to focus on this idea of the divergence of the gradient.

I was just trying to understand the physical intuition behind defining the Laplacian as the divergence of the gradient? I understand that the Laplacian operator represents diffusion and that it looks to reduce the deviation between a point and its neighboring points. But I was not clear on how this idea relates to the ideas of divergence--which relates to flux, and gradient--which describes how a scalar valued function is changing at a point given its variables.

RobPratt
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krishnab
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  • Have you seen the continuity equation? This becomes the heat equation when the flux is proportional to the negative gradient. – Ian Oct 03 '22 at 21:22
  • @Ian Oh yes, i believe I have seen that before in the context of fluid dynamics. I will have to try and find some notes on that. – krishnab Oct 03 '22 at 21:35
  • @TrystwithFreedom thanks for the suggestion, but I don't think that post answers my question. Indeed, I mention some of the intuition behind the laplacian in my original post, so the laplacian itself was never really the question--it was more about the divergence of the gradient produces the laplacian. – krishnab Oct 04 '22 at 15:07

1 Answers1

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The Laplacian operator measures the deviation between a point and the average of its neighboring points.

A neighboring point where the value is lower will have its gradient pointing towards you. A neighboring point where the value is higher will have its gradient pointing away from you.

So the Laplacian is just the measure of average outgoingness of the gradient. But that's exactly what the divergence operator does.

enter image description here

For instance, if you think of the vector field above as the gradient of a function, the first one represents a local maximum (negative Laplacian), the second one represents a local minimum (positive Laplacian), the last one a region where the function is linear (in particular it is harmonic, zero Laplacian)

justt
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    yes this is very helpful. I get what you mean now. Thanks for the instructive diagrams as well. You answered exactly what I needed, so the gradient points towards higher values from a point--whether that is inward (towards me) or outward, and the divergence resolves whether the gradients near a point average inward or outward. Thanks again. – krishnab Oct 04 '22 at 15:15