Hello everyone, I hope everyone is having a good day. I was trying to solve the Cahn-Hilliard equation (code that I found on FeniCs webpage), and found an error. I think this has to do with some outdated versions, since I’ve seen multiples ways to formulate mixed elements, but none seem to work on the code provided.

Here’s the code found on the web:

```
import random
from dolfin import *
# Class representing the intial conditions
class InitialConditions(Expression):
def __init__(self):
random.seed(2 + MPI.rank(mpi_comm_world()))
def eval(self, values, x):
values[0] = 0.63 + 0.02*(0.5 - random.random())
values[1] = 0.0
def value_shape(self):
return (2,)
# Class for interfacing with the Newton solver
class CahnHilliardEquation(NonlinearProblem):
def __init__(self, a, L):
NonlinearProblem.__init__(self)
self.L = L
self.a = a
def F(self, b, x):
assemble(self.L, tensor=b)
def J(self, A, x):
assemble(self.a, tensor=A)
# Model parameters
lmbda = 1.0e-02 # surface parameter
dt = 5.0e-06 # time step
theta = 0.5 # time stepping family, e.g. theta=1 -> backward Euler, theta=0.5 -> Crank-Nicolson
# Form compiler options
parameters["form_compiler"]["optimize"] = True
parameters["form_compiler"]["cpp_optimize"] = True
parameters["form_compiler"]["representation"] = "quadrature"
# Create mesh and define function spaces
mesh = UnitSquareMesh(96, 96)
V = FunctionSpace(mesh, "Lagrange", 1)
ME = V*V
# Define trial and test functions
du = TrialFunction(ME)
q, v = TestFunctions(ME)
# Define functions
u = Function(ME) # current solution
u0 = Function(ME) # solution from previous converged step
# Split mixed functions
dc, dmu = split(du)
c, mu = split(u)
c0, mu0 = split(u0)
# Create intial conditions and interpolate
u_init = InitialConditions()
u.interpolate(u_init)
u0.interpolate(u_init)
# Compute the chemical potential df/dc
c = variable(c)
f = 100*c**2*(1-c)**2
dfdc = diff(f, c)
# mu_(n+theta)
mu_mid = (1.0-theta)*mu0 + theta*mu
# Weak statement of the equations
L0 = c*q*dx - c0*q*dx + dt*dot(grad(mu_mid), grad(q))*dx
L1 = mu*v*dx - dfdc*v*dx - lmbda*dot(grad(c), grad(v))*dx
L = L0 + L1
# Compute directional derivative about u in the direction of du (Jacobian)
a = derivative(L, u, du)
# Create nonlinear problem and Newton solver
problem = CahnHilliardEquation(a, L)
solver = NewtonSolver()
solver.parameters["linear_solver"] = "lu"
solver.parameters["convergence_criterion"] = "incremental"
solver.parameters["relative_tolerance"] = 1e-6
# Output file
file = File("output.pvd", "compressed")
# Step in time
t = 0.0
T = 50*dt
while (t < T):
t += dt
u0.vector()[:] = u.vector()
solver.solve(problem, u.vector())
file << (u.split()[0], t)
plot(u.split()[0])
```

And the error it raises goes like: TypeError: unsupported operand type(s) for *: ‘FunctionSpace’ and ‘FunctionSpace’.

I also tried to modify the formulation for:

ME = MixedElement([V, V])

But got no results as well