Hello there! I’m studying the heat equation from this website: The heat equation — FEniCSx tutorial.

```
import matplotlib.pyplot as plt
import pyvista
import ufl
import numpy as np
from petsc4py import PETSc
from mpi4py import MPI
from dolfinx import fem, mesh, io, plot
from dolfinx.fem.petsc import assemble_vector, assemble_matrix, create_vector, apply_lifting, set_bc
# Define temporal parameters
t = 0 # Start time
T = 1.0 # Final time
num_steps = 50
dt = T / num_steps # time step size
# Define mesh
nx, ny = 50, 50
domain = mesh.create_rectangle(MPI.COMM_WORLD, [np.array([-2, -2]), np.array([2, 2])],
[nx, ny], mesh.CellType.triangle)
V = fem.FunctionSpace(domain, ("Lagrange", 1))
# Create initial condition
def initial_condition(x, a=5):
return np.exp(-a * (x[0]**2 + x[1]**2))
u_n = fem.Function(V)
u_n.name = "u_n"
u_n.interpolate(initial_condition)
# Create boundary condition
fdim = domain.topology.dim - 1
boundary_facets = mesh.locate_entities_boundary(
domain, fdim, lambda x: np.full(x.shape[1], True, dtype=bool))
bc = fem.dirichletbc(PETSc.ScalarType(0), fem.locate_dofs_topological(V, fdim, boundary_facets), V)
xdmf = io.XDMFFile(domain.comm, "diffusion.xdmf", "w")
xdmf.write_mesh(domain)
# Define solution variable, and interpolate initial solution for visualization in Paraview
uh = fem.Function(V)
uh.name = "uh"
uh.interpolate(initial_condition)
xdmf.write_function(uh, t)
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
f = fem.Constant(domain, PETSc.ScalarType(0))
a = u * v * ufl.dx + dt * ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx
L = (u_n + dt * f) * v * ufl.dx
bilinear_form = fem.form(a)
linear_form = fem.form(L)
A = assemble_matrix(bilinear_form, bcs=[bc])
A.assemble()
b = create_vector(linear_form)
solver = PETSc.KSP().create(domain.comm)
solver.setOperators(A)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
import matplotlib as mpl
pyvista.start_xvfb()
grid = pyvista.UnstructuredGrid(*plot.vtk_mesh(V))
plotter = pyvista.Plotter()
plotter.open_gif("u_time.gif", fps=10)
grid.point_data["uh"] = uh.x.array
warped = grid.warp_by_scalar("uh", factor=1)
viridis = mpl.colormaps.get_cmap("viridis").resampled(25)
sargs = dict(title_font_size=25, label_font_size=20, fmt="%.2e", color="black",
position_x=0.1, position_y=0.8, width=0.8, height=0.1)
renderer = plotter.add_mesh(warped, show_edges=True, lighting=False,
cmap=viridis, scalar_bar_args=sargs,
clim=[0, max(uh.x.array)])
for i in range(num_steps):
t += dt
# Update the right hand side reusing the initial vector
with b.localForm() as loc_b:
loc_b.set(0)
assemble_vector(b, linear_form)
# Apply Dirichlet boundary condition to the vector
apply_lifting(b, [bilinear_form], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
# Solve linear problem
solver.solve(b, uh.vector)
uh.x.scatter_forward()
# Update solution at previous time step (u_n)
u_n.x.array[:] = uh.x.array
# Write solution to file
xdmf.write_function(uh, t)
# Update plot
warped = grid.warp_by_scalar("uh", factor=1)
plotter.update_coordinates(warped.points.copy(), render=False)
plotter.update_scalars(uh.x.array, render=False)
plotter.write_frame()
plotter.close()
xdmf.close()
```

But this error:

```
HDF5-DIAG: Error detected in HDF5 (1.10.7) MPI-process 0:
#000: ../../../src/H5F.c line 366 in H5Fcreate(): unable to create file
major: File accessibility
minor: Unable to open file
#001: ../../../src/H5Fint.c line 1713 in H5F_open(): unable to lock the file
major: File accessibility
minor: Unable to lock file
#002: ../../../src/H5FD.c line 1675 in H5FD_lock(): driver lock request failed
major: Virtual File Layer
minor: Unable to lock file
#003: ../../../src/H5FDsec2.c line 990 in H5FD__sec2_lock(): unable to lock file, errno = 11, error message = 'Resource temporarily unavailable'
major: Virtual File Layer
minor: Unable to lock file
Traceback (most recent call last):
File "/home/jandui/Documents/MATH code/AAAAAAAAAAAAa/heat_updated.py", line 40, in <module>
xdmf = io.XDMFFile(domain.comm, "diffusion.xdmf", "w")
RuntimeError: Failed to create HDF5 file.
```

My questions are:

Why is this error about xmdf appearing? I don’t know what I should install now; most of the libraries come with fenicsx.

Also, how can I create multiple heat points with specific shapes at the square domain with different functions and constants? (I mean, maybe put two heat sourcers (one at center in a circular domain and another one in the left corner with sin or exponential function, just an example)

Thanks for your attention.