Nedelec basis functions of different meshes

General
1

Hi,

  1. I am trying to reach the first-order Nedelec function for any edge of the element. I defined a basis function on any element, for example a master triangular element. Now, I want to access the basis function of any edge. For example, I want to access the basis function of the hypotenuse edge for the master triangle that I defined in the code. How can I do that?
    For master triangle:
    V_coarse = FunctionSpace(mesh_coarse, (“Nedelec 1st kind H(curl)”, 1))
    u_coarse = TrialFunction(V_coarse)
    v_coarse = TestFunction(V_coarse)

  2. I defined the Nedelec basis functions for the master triangle in the example. I also defined Nedelec basis functions for a different geometry. I want to obtain the inner product of the basis functions of my new element (9-sided finer geometry) with the basis function that I found corresponding to any side of the master triangle (obtained result from my first question).

x = np.array([[0.0, 0.0], # Vertex 0
[1.0, 0.0], # Vertex 1
[0.0, 1.0], # Vertex 2
[0.5, 0.0], # Vertex 3 (midpoint of edge 0-1)
[0.5, 0.5], # Vertex 4 (midpoint of edge 1-2)
[0.0, 0.5]]) # Vertex 5 (midpoint of edge 2-0)

cells = np.array([[0, 3, 5], # Lower left triangle
[3, 1, 4], # Lower right triangle
[3, 4, 5], # Middle triangle
[5, 4, 2]], # Top triangle
dtype=np.int32)

mesh_fine = dolfinx.mesh.create_mesh(MPI.COMM_WORLD, cells, x, domain)

After multiplying these vectors, I want to do the integration in new smaller triangles (inner product calculation on the finer geometry). I couldn’t do this with built-in functions, the functions in similar questions I found on this website seem cannot be applicable for Fenicsx, I wonder how I can do this. Shortly, I would like to ask how I can calculate the inner product of Nedelec basis functions of different meshes in any geometry.

  1. Finally, how can I reach the basis function or norm of the basis function at any point in the cell? For example, for the master triangle, how can I reach basis function norm value or corresponding total basis function at the midpoint or any point of the triangle?

Thank you very much,

2

Please make a reproducible example.
Your snippets doesn’t make it clear what you want to achieve.
It Seems like you want to do the following:

  1. given a cell in one mesh, evaluate the basis functions at a certain point in this mesh.
  2. combine basis functions on this grid with the basis on another grid (evaluated in the same point in physical space?)
3

Thank you for your answer.

My questions are:

  1. Given a cell in one mesh, evaluate the basis functions at a certain point in this mesh.

  2. I would like to calculate the inner product of two trial functions belonging to different meshes. For example, I tried to calculate the inner product of two trial functions on the domain of the second trial function. But this approach doesn’t work. I wanted to ask how I can do this.

a = ufl.inner(u_fine, v_coarse) * ufl.dx(mesh_fine)

u_fine and v_coarse are Nedelec trial functions assigned for different meshes. I want to calculate their inner product on finer mesh.