Ever since joining SE, I have heard many people mention Poisson's equation and the Laplacian. I have also started to encounter these terms more in resources I have been directed to. I am consumed however. Two me it just looks like Poisson's equation is just the divergence of the gradient of some function.
Is there something particularly useful about taking the divergence of the gradient of a function?
Many equations in physics and engineering are either Poisson or related to Poisson equation.
For example numerical methods to solve the Navier Stokes equations often determine the pressure by solving a Poisson equation derived from the condition that the predicted velocities satisfy the continuity equation --see for example Chorin (1967).
The stream vorticity method for resolving 2D flows also involves the solution of a Poisson equation. More specifically, the stream function and the vorticity of the flow are related through a Poisson equation.
More generally, harmonic functions and potential theory occur frequently in physics and engineering in domains such as fluid dynamics, electromagnetics, and heat-transfer.
Chorin, A. J. (1967), "The numerical solution of the Navier-Stokes equations for an incompressible fluid", Bull. Am. Math. Soc. 73: 928–931