Point interpolation of shape function derivatives

[SouzaEM]

Is it possible to calculate the values of the shape function derivatives using vertex meshes? As suggested in #3782 and from the FWI example (https://www.firedrakeproject.org/demos/full_waveform_inversion.py.html), it is possible to obtain the shape function values as follows:

from firedrake import *

mesh = UnitSquareMesh(2, 2)

x_v = [np.r_[0.25, 0.25]]
mesh_v = VertexOnlyMesh(mesh, x_v)

V_v = FunctionSpace(mesh_v, "DG", 0)
phi_v = assemble(TestFunction(V_v)*dx)

V = FunctionSpace(mesh, "CG", 1)
phi = Cofunction(V.dual()).interpolate(phi_v)
print(phi.dat.data)

However, as the function spaces defined on vertex meshes may only be DG0, I was not able to assemble the gradient of the test functions.

[Ig-dolci]

Do you want a Cofunction that is a result of the assembly of shape function gradient in DG0? Then interpolate it into V space? This does not look like the right thing to do once DG0 has dimension 0 and a constant function in DG0 does not have a well-defined gradient. Or I am wrong and it is possible...

[SouzaEM]

I want the values of the derivatives of the shape function of the "parent" function space (CG1 in the example above).

As I far as I got from studying the vertex mesh, the code above gives the shape function values at point (0.25, 0.25).

[Ig-dolci]

This code interpolates a Cofunction (phi_v in V_v.dual()) onto V.dual(). Are you asking for the derivative of phi in V.dual()? If so, which derivative?

[SouzaEM]

I'm asking for the spatial derivatives of the shape functions ($\frac{\partial\phi_i}{\partial x_j}$) evaluated at (0.25, 0.25).

By running the code above, the output is [0. 0.5 0.5 0. 0. 0. 0. 0. 0. ], which is $\phi_i$(0.25, 0.25) for all the shape functions of CG1 over this mesh.

[wence-]

Interpolate grad(phi) into a VectorFunctionSpace of appropriate dimension, no?

[SouzaEM]

Do you suggest something like this?

from firedrake import *

mesh = UnitSquareMesh(2, 2)

V = FunctionSpace(mesh, "CG", 1)
v = TestFunction(V)

from firedrake.__future__ import interpolate

W = VectorFunctionSpace(mesh, "DG", 0)
w = assemble(interpolate(grad(v), W))

Then how do I integrate this code with point interpolation? I need the shape function derivatives in few points of the domain.