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.
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.
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)