What about bouncing off this blog post, and use it as an example to illustrate how to decide for appropriate local meshing density?

=> to allow to locally resolve the dependent variable gradients.

One of the strong points of COMSOL is to mesh once the physics has been defined, which allows so nicely to couple all these “physics” together, hence putting back the meshing operation where it belongs: a final “mathematical” discretization of the domains (with obvious similarity to sound sampling theory).

At the point u_init = 0, you say above that one need to estimate the derivative f’(u_init).

From my knowledge this is done (when the full set of equations cannot be analytically derived) by taking difference between mesh nodes, hence you should at least have one mesh node between u_init and your u_solution (even if u_solution is unknown).

For many physics the value of f ‘(u_init) can be estimated:

i.e. in thermal diffusion, for temporal models, the known thermal diffusivity alpha_T [m^2/s] = k/rho/Cp gives a good link between the mesh spatial extend Dz and the related minimum time stepping by respecting Dt > N*Dz^2/alpha_T. Using a N=2 gives a reasonable mesh size to just pass the Nyquist criteria such to evaluate grad(T), hence to avoid negative absolute temperatures (or negative concentration for chemical diffusion) when solving.

I hope you have more time than me to put this together in a clearer and better illustrated way, here on the COMSOL blog

And thanks again for your series of well written texts

Sincerely

Ivar