Low-pass filter: create and integrate local supported function over domain

Question

Hi!

I'm currently looking into explicit low-pass filtering for use with dynamic methods in LES (http://en.wikipedia.org/wiki/Filter_%28large_eddy_simulation%29). One method when implementing a generalized top hat box filter is to:

• Define function G(dof) as such: $G(dof_0) = 1$ for $dof_0$ and all dofs on cells that have $dof_0$ as vertex. Zero else.
• Integrate G times the function for filtering over the (local) domain.
• Do this for all dofs in mesh.

Would it be possible to implement this in FEniCS in an "easy" and fast way?

Thanks!

Answer 1

How about letting W be the finite element space that has the dofs you are referring to as all dofs in the mesh. Let f be the "function for filtering". Then do

v = TestFunction(W)
L =  f*v*dx
b = assemble(L)

Now b contains the value for each dof in the mesh.

Comments

Hi Marie!

Thank you for an instructive answer. I am not 100 percent sure that I fully understand your suggestion; but, let's say that G is defined similar to a CG1 function, 1 at a dof, 0 for all other dofs. Then your solution would be perfect! However, G is ( maybe somewhat unclear from my post) not only one in the main dof, but also one at all dofs in elements that contains dof as vertex. If $x_d$ is some dof:

$$G(x_d) = \cases{
1, & \text{for }x_d\text{ and for dofs in cells which have }x_d\text{ as vertex.}\cr
0, & \text{else}}$$

E.g. for this case http://bildr.no/image/UmZHT2xs.jpeg $G$ would be equal to one for all marked dofs, and zero else.

$G$ defined this way is specific for this specific approximation to the filter I found in some paper. It may be possible to utilize CG elements and integrate as you proposed Marie, that may yield some good results.