Hello FEniCSxers,
I have a general question about finite elements: are there rigorous ways to compare different terms appearing in a PDE or in a weak form?
For example, say we have a heat equation of the form \nabla \cdot (\kappa \nabla T)+q_1(T)+q_2(T)=0, i.e. there is the usual conduction term and additional 2 source terms. What are ways to compare these terms? We can of course compute them and plot them over their mesh/domain, but that would only give a visual clue, not a quantitative answer.
We could compute their L2 norm, or Linfinity norm. Would those be more appropriate?
In which other ways could we compare those terms?
In this specific case, if for example we find that q_1(T) dominates, we could conclude that it is the main term that affects T, the variable we’re solving for. (This motivates the question I ask).
In your context non-dimensionalisation may not be trivial, so you may want to investigate relative coefficients. Analogous to, for example, relative permittivity or relative permeability.
There are much more advanced techniques beyond my skill which you can probably find in an introductory analysis book.
Thanks a lot nate for your reply.
I am not quite sure I fully understand the difference between non dimensionalization and relative coefficients. I think I have tried a way to have relative coefficients, i.e. I found a mathematical inequality that the coefficients in front of the different terms in the PDE have to satisfy for some terms to be as big as \nabla \cdot (\kappa \nabla T), these inequalities come from a non dimensional relationship (very similar to the adimensional thermoelectric figure of merit zT=TS^2/(\kappa \rho)). I guess I could rewrite the PDE in terms of these relationships, i.e. only in terms of z and non-dimensionalized variables. But I didn’t do it yet.
What annoys me a bit is that these are “approximations”, where the geometry cancels out, and I suspect that things can be surprisingly more complicated than this. This is why I wanted to use FEM to inspect the magnitude of the different terms. Maybe I could just evaluate those terms at mesh nodes points and plot some histogram of their value, and their magnitude (or L2 norm). A histogram would be possibly more relevant than a heat map, but there would be a bias in how the points where these terms are evaluated are selected, i.e. non uniformly randomly over the domain. So I am not sure it’s a good metric. Heck, I don’t pretend to reinvent the wheel since I am not qualified enough.
As regards to your last comment, I am surprised that you mentioned analysis. I guess I’ll have to search for “analysis PDE coefficients” or something like this.