Accessing the basis function of a Finite Element

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If you know what point in the reference element you want to evaluate your function at, you can use dolfinx.fem.Expression, consider the mwe

from mpi4py import MPI

import dolfinx
import numpy as np
import ufl

def eval_trial_function(V, reference_points):
    mesh = V.mesh
    u = ufl.TrialFunction(V)
    expr = dolfinx.fem.Expression(u, reference_points)

    num_cells = mesh.topology.index_map(mesh.topology.dim).size_local
    basis_functions = expr.eval(mesh, np.arange(num_cells, dtype=np.int32)).reshape(num_cells, len(reference_points), -1)
    for i, functions in enumerate(basis_functions):
        for j, point in enumerate(reference_points):    
            print(f"Cell {i}: Point {point} {functions[j, :]}")


mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 1, 1, dolfinx.cpp.mesh.CellType.triangle)
V = dolfinx.fem.functionspace(mesh, ("N1curl", 1))

# Points in reference element
points = np.array([[0.0, 0.0], [1.0, 0.0]])
eval_trial_function(V, points)

yielding:

Cell 0: Point [0. 0.] [ 0.  0.  1.  0.  1. -1.]
Cell 0: Point [1. 0.] [0. 0. 1. 1. 0. 0.]
Cell 1: Point [0. 0.] [-0.  1. -1. -0.  0.  1.]
Cell 1: Point [1. 0.] [-1.  0.  0. -0.  0.  1.]