I used dolfinx to solve the Poisson equation and obtained the solution uh using MPI parallel computing. I want to know how to compute the value of the solution at a specific point in the domain. I tried the following function, but the results were all NaN. Are there any examples or tutorials on how to compute the value of the solution at specific points in an MPI environment?

def insert_value(self, u, points):

nr_points = len(points)

u_values = np.ones((nr_points,))

cells =

points_on_proc =

tree = bb_tree(self.mesh, self.mesh.geometry.dim)

cell_candidates = compute_collisions_points(tree,

points) # Find cells whose bounding-box collide with the the points

colliding_cells = compute_colliding_cells(self.mesh, cell_candidates,

points) # Choose one of the cells that contains the point

```
indices = np.ones((nr_points,), dtype=int) * (-1)
# roots = np.ones((nr_points,), dtype=int)
# Each process creates an array of size 3 with values equal to its rank
sendbuf = np.full(nr_points, 0, dtype=int)
u_values_send = np.full(nr_points, 0, dtype=float)
# Root process initializes the receive buffer to gather all arrays
if self.myrank == 0:
recvbuf = np.empty(self.size * nr_points, dtype=int)
u_values_recv = np.full(nr_points * self.size, 0, dtype=float)
else:
recvbuf = None
u_values_recv = None
count = 0
for i, point in enumerate(points):
if len(colliding_cells.links(i)) > 0:
sendbuf[i] = 1
points_on_proc.append(point)
cells.append(colliding_cells.links(i)[0])
indices[i] = count
count += 1
points_on_proc = np.array(points_on_proc, dtype=np.float64)
u_values_temp = u.eval(points_on_proc, cells)
for i in range(nr_points):
if sendbuf[i] == 1:
u_values_send[i] = u_values_temp[indices[i]]
# Gather the arrays at the root process
self.comm.Gather(sendbuf, recvbuf, root=0)
self.comm.Gather(u_values_send, u_values_recv, root=0)
if self.myrank == 0:
for i in range(nr_points):
for j in range(self.size):
if recvbuf[i + j * nr_points] == 1:
u_values[i] = u_values_recv[i + j * nr_points]
u_values = self.comm.bcast(u_values, root=0)
return u_values
```