$\varphi$-FEM for the heat equation

This repository contains the code used in Duprez, Michel, and Lleras, Vanessa, and Lozinski, Alexei, and Vuillemot, Killian. $\varphi$-FEM for the heat equation: optimal convergence on unfitted meshes in space (preprint).

This repository is for reproducibility purposes only

It is "frozen in time" and not maintained. To use our latest $\varphi$-FEM code please refer to the phiFEM repository.

Generate the results

The results can be generated using a container image.
The image is based on ghcr.io/scientificcomputing/fenics-gmsh:2024-05-30. It contains also the following python libraries and their dependencies:

• dataclasses,
• pygalmesh,
• vedo.

Prerequisites

• Git,
• Docker/podman.

Install the image and launch the container

1. Clone this repository in a dedicated directory:

   mkdir heat-equation-phifem/
   git clone https://github.com/PhiFEM/publication_Heat-Equation_fenics.git heat-equation-phifem

2. Download the image from the docker.io registry, in the main directory:

   export CONTAINER_ENGINE=docker
   cd heat-equation-phifem
   sudo -E bash pull-image.sh

3. Launch the container:

   sudo -E bash run-image.sh

Example of usage

From the main directory heat-equation-phifem, launch the convergence script for the circle test case:

cd circle/
python3 convergence.py

Python codes

Implementations of $\varphi$-FEM and standard FEM to solve the heat equation in different cases :

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

Issues and support

Please use the issue tracker to report any issues.

Authors (alphabetical)

Michel Duprez, Inria Nancy Grand-Est
Vanessa Lleras, Université de Montpellier
Alexei Lozinski, Université de Franche-Comté
Killian Vuillemot, Université de Montpellier