The Finite Elements Method convergence survival guide

Ok, people, let’s face it. Many of us had no idea about the maths behind Finite Elements Method (FEM) simulations when we initially started using them. And most of us have felt that horrible feeling when we repeatedly saw the message ‘No convergence’ instead of a nice output. What is horrible about it is that, unless you or your group develops the software, you probably have no idea where to start looking for fixing the problem.

I heard from a lot of people in my field (electronics) that they have felt discouraged with FEM and did not even want to touch it. But I’m here to tell you that there is hope! I decided to write this guide after almost 10 years of using this type of software (on and off), first for nanoelectonics and then for semiconducting quantum technologies. It might be that the issue is actually more important in the latter, because of the differences in magnitude (or the scale of things) between classical and quantum effects (yes, not everything is quantum in quantum TCAD). But this is a user’s and not developer’s guide, so I will also keep it math-and code-free.

3D and 2D reciprocal lattice vectors (Python example)

When I started using Density Functional Theory and, even more, when trying to re-create its results in a meaningful piece of material, like a nanoribbon or a nanowire, a lot of the problems came during geometry creation. This is natural, as DFT is a theory to derive quantities like eigenenergies, in a crystal lattice with periodicity. But when trying to switch to a model that is finite in one or more directions, you have to start a sort of mix and match procedure.

During this time (and assuming you are not the one who writes the software), visualizing vectors is maybe the less useful thing to do. Still, there are cases where you need it in order to get a better understanding of things, like visualizing Weyl points.