### You get what you mesh for

In finite element modeling, in fact, in all the numerical modeling, the resolution of the results depends on the space where the problem is projected. As to PDE’s numerical solution, the projection is from space with infinite dimension to space with finite dimension. So no matter it’s finite element or finite volume or finite difference, the size of the mesh is what you can see at the finest scale. Beyond that, it’s either physical *modeling* (such as turbulence subscale modeling) or some other educated *guess*.

Here is an example.

This plot shows a finite element solution of Navier-Stokes equation, in a water waves problem. The free surface is *tracked* using Level Set method. Taking closer look reveals the following not-so-smooth “water surface”:

The resolution of the free surface is restricted by the size of the element. For any mesh, examing close enough reveals similar wiggles. To put it another way, in the numerical simulation, free surface is not really a *surface*, it’s more like a *layer* with finite thickness. If I plot the centroid of the elements that reside on the free surface, it looks like this:

This is what a non-polished numerical result should look like. Smoothing such as with cubic splines sweepts everything under the table, including the message: “you get what you mesh for”.