$\phi$-FEM for the heat equation

Repository containing the codes used in preprint.
Authors : Michel Duprez, Vanessa Lleras, Alexei Lozinski and Killian Vuillemot.

Python codes

Implementations of $\phi$-FEM and standard FEM to solve the Heat equation in different case :

• Folder circle : solve the heat equation on a circle, using the manufactured solution $u = \cos\left(\frac{1}{2} \pi (x^2+y^2)\right) \exp(x) \sin(t)$.

  • convergence.py : code to compare the errors of $\phi$-FEM and a standard FEM on the given problem;
  • sigma_values.py : code to emphasize the influence of $\sigma$ on the error;
  • degree_phi.py : code to emphasize the influence of $l$, the degree of interpolation of the level-set function, on the error;
  • plot_meshes.py : code to plot the domain, an example of mesh used for $\phi$-FEM and an example of conforming mesh used for standard FEM.

• Folder popcorn : solve the heat equation on a popcorn with $f=\exp(-\frac{(x-0.2)^2 + (y-0.3)^2 + (z+0.1)^2}{2\times 0.3^2})$, $u = 0$ on $\Gamma \times (0,T)$ and $u^0 = 0$ in $\Omega$.

  • main.py : code to compare the errors of $\phi$-FEM and a standard FEM on the given problem.
  • plot_meshes : code to plot the popcorn domain, an example of mesh used for $\phi$-FEM and an example of conforming mesh used for standard FEM.

Required packages : FEniCS, matplotlib, numpy, sympy, vedo and pygalmesh. You can run FEniCS using a Docker container with the command

docker run -ti -p 127.0.0.1:8000:8000 -v $(pwd):/home/fenics/shared -w /home/fenics/shared quay.io/fenicsproject/stable:current

or by installing (on ubuntu) with

sudo apt-get install software-properties-common
sudo add-apt-repository ppa:fenics-packages/fenics
sudo apt-get update
sudo apt-get install fenics

Then, just install the other packages (and dependencies) with

pip install matplotlib sympy
pip install numpy==1.23.5
pip install vedo==2022.4.2
sudo apt install libcgal-dev libeigen3-dev
pip install pygalmesh