# Some mixed boundary conditions in FEniCS

When you first start to use a piece of software, it is good to have some really stupid examples to refer to, which expert users would probably laugh at. For the noob, like me, it is sometimes difficult to figure out how to do even the most basic things. Take the following example:

You have some kind of partial differential equation involving a function $F(z,t)$, in one time variable $0\leq t \leq T$ and one space variable $z \in [0,L]$. The given boundary and initial conditions are $F(0,t) = g(t)$ and $F(z,0) = 0$. How to formulate these conditions in FEniCS?

Note that in the FEniCS implementation we are actually only considering a single variable, since the time differential will be incorporated by applying a finite difference-type calculation over specified time intervals.

First, we need to define the relevant boundaries. For the first one, $F(0,t) = g(t)$, In FEniCS, we define a function that tests whether a point is on the boundary by

def GammaD(x, on_boundary):
return near(x[0],0)

The use of “near” is because we are using floating point numbers, and can’t test for exact equality. Now that we can test for this boundary, we can specify the boundary condition:

g = Expression('...')
bc = DirichletBC(V, g, GammaD)

where $V$ was specified earlier in the code, as usual. In my case, I wanted $F(0,t) = \sin \omega t$, so I set $g(x[0],t) = (1-x[0])*\sin \omega t$. What about the initial condition $F(z,0) = 0$? That is also taken care of by the expression for $g$. But does the restricted boundary condition take this initial condition into account? Since we later specify

u_n = interpolate(u_D, V)

and initially set $t=0$ in $u_D$, it is indeed specified. If we want to add Neumann boundary conditions, we don’t need to do anything more – they are included by default.

Credit where it is due: the post at https://sites.math.rutgers.edu/~falk/math575/Boundary-conditions.html was very helpful, as was http://vefur.simula.no/~hpl/homepage/fenics-tutorial/release-1.0/webm/timedep.html.

As I need to add more and trickier conditions, I’ll update this post.

Not an ad, and I’m not getting paid to put it here. Just a thing that works, and that did wonders for my mother, when all the drugs couldn’t:

# Installing FEniCS to run with Jupyter Notebook

So, I have some PDEs that I need to solve numerically, and came across the FEniCS package. Problem is, the thing just did not want to work on my Mac. I tried several ways, including https://github.com/FEniCS/dolfinx#conda. I even tried to do the same on an Ubuntu installation, with no luck. Although I could import dolfinx successfully in the command line once the environment had been activated, I had no joy importing it in Jupyter Notebook or VSCode.

During the installation, I saw that there were several inconsistencies in my Anaconda installation, so I tried to remove them with

conda install anaconda

This took ages (more than 40 hours), so prepare yourself for a long haul. After this, I installed Notebook and then tried

conda create -n fenicsx-env
conda activate fenicsx-env
conda install -c conda-forge fenics-dolfinx mpich pyvista

Still no luck when trying

python3 -c "import dolfinx"

(In a previous installation this was actually successful, but I still couldn’t import it from Notebook.)

In https://stackoverflow.com/questions/73108638/install-fenics-to-use-it-within-jupyter-lab, I saw that someone had some success by launching the environment in Anaconda Navigator, which I now attempted, and which gave me some errors on start up. To fix this, I tried

conda update anaconda-navigator
conda update navigator-updateranaconda-navigator --reset

which I found at https://stackoverflow.com/questions/46329455/anaconda-an-unexpected-error-ocurred-on-navigator-start-up. Note that this can also take a bloody long time. It also does not seem to work… In the meantime, I was getting desperate and started throwing commands around. I tried (from https://fenicsproject.org/qa/13194/how-to-use-fenics-in-jupyter-by-anaconda/):

conda create -n fenicsproject -c conda-forge python=3.11 jupyter fenics

(this will depend on your version of Python, of course). Now, I activated the environment and tried the import:

conda activate fenicsprojectpython3 -c "import fenics"

This ran without any errors, which at least gave me hope. Next, I opened Notebook and tried

from fenics import *

Success! For the first time, I did not get a “No module…” type error. Now I have to see if everything works as advertised. By the way, what I would have tried next would have been to do the install with Docker (which is actually recommended in the FEniCS manual). I should also mention that I did a pip3 install, which did not work. I will still see if this method is successful on Ubuntu as well, and let you know.