How to calculate Facet average

Question

What's the best way to compute the average value of a function on each facet of the mesh?

I suppose one way is to explicitly call assemble in a loop with the domain restricted to each facet in turn.. but maybe there's a better way?

Øyvind Evju · Answer 1 · Feb 25, 2014

Best answer

Could you use Crouzeix-Raviart elements?

CR = FunctionSpace(mesh, "CR", 1)
facet_avg = interpolate(function, CR)

answered Feb 25, 2014 by Øyvind Evju FEniCS Expert (17,700 points)
selected Feb 25, 2014 by Nikolaus Rath

That's clever. I didn't think of using a different finite element for this.

commented Feb 25, 2014 by Nikolaus Rath FEniCS User (2,100 points)

I was looking for something else and I saw this answer. The average value of a function it is not always equal to its value at the midpoint ... that is valid only for constants and linear polynomials.

commented Sep 5, 2014 by fotini.karakatsani FEniCS Novice (500 points)

Yes, that's why you have to use Crouzeix-Raviart elements. For CR, the value on each face is constant and thus equal to the average of the interpolated function.

commented Sep 5, 2014 by Nikolaus Rath FEniCS User (2,100 points)

Maybe I do not understand how the interpolant is computed in FEniCS, but interpolation at a point means that we take the value of the function at that point.
The average value of a function g on a facet F is
1/facet_area(F) \int_F g(s) ds.
How can that be equal to the value g(midpoint of F) which is exactly what you get when you interpolate g by using CR elements.

commented Sep 5, 2014 by fotini.karakatsani FEniCS Novice (500 points)

I was just about to write that interpolate minimizes the mean square deviation between the two functions, but then I realized that I may have confused this project. So your point may be valid.. Maybe the proper way to do this is to project into the CR space. Hm.

commented Sep 8, 2014 by Nikolaus Rath FEniCS User (2,100 points)

How to calculate Facet average

1 Answer