### Homogeneous Neumann BC not fulfilled?

So I checked the same in the Cahn-Hilliard Demo and it seems that they are not fulfilled there either.

```
from __future__ import print_function
import random
from dolfin import *
set_log_level(WARNING)
# Class representing the intial conditions
class InitialConditions(Expression):
def __init__(self, **kwargs):
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,)
# 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
# Create mesh and build function space
mesh = UnitSquareMesh(48, 48)
P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
ME = FunctionSpace(mesh, P1*P1)
n = FacetNormal(mesh)
# 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(degree=1)
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 = NonlinearVariationalProblem(L, u, J=a)
solver = NonlinearVariationalSolver(problem)
# Output file
file = File("output.pvd", "compressed")
# Step in time
t = 0.0
T = 10*dt
while (t < T):
t += dt
u0.assign(u)
solver.solve()
c, mu = u.split()
c.rename('c','phase')
print(str(assemble(inner(grad(c),n)*ds)))
file << (project(grad(c)), t)
```

I expected that the output should be something close to zero, which is not the case (sth. up to -5.0). If I write the projected gradient into the output file and inspect it in paraview, I would expect that e.g. on the lower boundary the second component should be close to zero, which is also not the case. I also tried different polynomials orders without success.

I am aware that there are errors involved in projecting the gradient and that the BC are only weakly enforced and hence maybe not that close to zero, however I think values in the order of 1 should not be there.

Did I check the conditions not correctly? Is something wrong with the implementation?

### 1 Answer

I've run into this problem (unrelated to FEniCS, actually with a code I implemented with another library and also with something written from scratch) when using the Crank-Nicolson (exactly theta = 0.5) time discretization scheme with Neumann boundary conditions. Off the top of my head, I don't remember what my conclusion was; so I'll have to dig up some notes. For now, what happens when you set theta = 0.51? Also as long as we're at it, maybe try a fully implicit run.

https://fenicsproject.org/olddocs/dolfin/2016.2.0/python/demo/documented/cahn-hilliard/python/demo_cahn-hilliard.py.html

I can't tell you by heart but I think with the boundary 0-Neumann for $c$

cand $\mu$μthis should be well-posed. There is some work by Elliott and Songmu on this.