Subtlety of projection in conservative FEM
I was recently exposed to the subtlety of projection by my friend Wietse (who has agreed to being associated to this first blog post of mine.) 1 Projections are ubiquitous in mathematics and they come in different guises. In this post I want to highlight a very important and subtle distinction between projections and interpolations in the FEM. I have the suspicion that this is one of those topics that is obvious to the initiated, but mysterious to the student. So it is of value to discuss these subtleties.
In the following we let
projections
Given a function
Conservation of mass in finite elements
In flow problems where the flow is modelled by for example the Darcy or the Stokes equations, one is also concerned with conservation of mass of the fluid. Conservation of mass simply means that in a closed system, the amount of fluid exiting the domain equals the amount coming in. This is a physical law, in contrast to the equations mentioned above which are just models in the end. Mathematically this law is written
Thus it is of paramount importance that the equation above is satisfied, and we must therefore take extreme care in how we approximate
If we know
Extra: should it matter to which polynomial space I project?
Short answer is no; not if the function you are testing against lies in both spaces. As an example, let
If
Here is the same post in pdf format.
1 Potential mistakes herein are entirely my own!
2 If we are sampling