Gradient of basic functions evaluated in each point of the mesh

General
1

For the \phi_i's, the basis functions obtained by the approximation of a P1 (linear) finite element space related to the discretization of a cylinder \Omega,

The setup is as follows:

r=15
large=70
begin=[-15, 0, 0]
ending=[45, 0, 0]
cylinder=Cylinder(Point(begin),Point(ending), r ,r)
mesh_cilinder = generate_mesh(cylinder, 10)
Space = FunctionSpace(mesh_cilinder,'P', 1)
basis_function=TrialFunction(Space)

I am trying to compute the gradients of the basis functions evaluated at each vertex of the mesh. Specifically, I want to compute the following matrices: \big[\nabla_x \phi_i(x_j)\big]_{i,j}, \big[\nabla_y \phi_i(x_j)\big]_{i,j}, \big[\nabla_z \phi_i(x_j)\big]_{i,j}, where x_j are the coordinates of the mesh vertices.

My approach was to create an expression for each basis function and then interpolate it over the entire function space:

coordinates_nodes=mesh_cilinder.coordinates()
cells_nodes=mesh_cilinder.cells()
Space = FunctionSpace(mesh_cilinder,'P', 1)
basis_function=TrialFunction(Space)

basis_function_list=[ ]
integral_basis=[ ]
n=Space.dim()


for i in range(n):
    # Create a function that represents the i-th basis function   
    x0=coordinates_nodes[i][0]
    x1=coordinates_nodes[i][1]
    x2=coordinates_nodes[i][2]
    phi_node = interpolate(Expression('(x[0] == x0 && x[1] == x1 && x[2] == x2) ? 1 : 0', degree=3, x0=x0,x1=x1,x2=x2),Space)
    basis_function_list.append(phi_node)

and after that compute the gradient and evaluate in the vertex of the mesh, for example for phi_0

u0=interpolate(basis_function_list[0],Space)
gradf   = grad(u0)
u,v,w   = gradf
project(u).vector().get_local()
project(v).vector().get_local()
project(w).vector().get_local()

However, the values I obtained for the gradients are not what I expected, since they are not constant, however the derivatives of the basis functions should be piecewise constant since I am using P1.

Could you please help me understand the correct way to compute the gradients of the basis functions and evaluate them at the mesh vertices for computing the matrices?

2

Please read https://fenicsproject.discourse.group/t/read-before-posting-how-do-i-get-my-question-answered/21: on top of all written there, it’s very difficult to read your code, because you haven’t formatted it properly with “```”.

From what I can gather, the space in which you want to apply interpolate is a DG 0 space (and not the P 1 space you define), because, as you say, the gradient of a P 1 function is piecewise constant function.

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3

Thanks, for your answer it worked, however it takes a lot of time when the size of discretization is small. might you have another idea for computing that matrices efficiently?

4

Switch to DOLFINx and use dolfinx.fem.Expression to compute it, as you can interpolate the gradient onto a compatible DG-0 or DG-1 space and easily extract the values.

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