This repository contains software and instructions to reproduce the numerical experiments in the paper

A hybridized Nitsche method for sign-changing elliptic PDEs

• authors: Erik Burman(1), Alexandre Ern(2) and Janosch Preuss(1)
• (1): University College London
• (2): CERMICS and INRIA Paris

We describe two options to setup the software for running the experiments.

• downloading a docker image from Zenodo or Docker Hub which contains all dependencies and tools to run the application,
• or installing everything manually on your own local machine.

We recommend the first option as it is simple and will produce exactly the results given in the paper (as the complete compute environment has been fixed). The second option provides higher flexibility but may be more complicated. It is mainly geared towards users who want to further develop the code. Please contact [email protected] if problems occur.

The instructions for running the image are geared towards users who have access to a Unix like environment with a bash shell. Windows users may use Linux subsystems or tools like Git BASH or MobaXterm to run these commands.

• Please install the docker platform for your distribution as described here.
• After installation the Docker daemon has to be started. This can either be done on boot or manually. In most Linux distributions the command for the latter is either sudo systemctl start docker or sudo service docker start.
• Pull the docker image using the command docker pull janosch2888/sign-changing-repro:v1.
• Run the image with sudo docker run -it janosch2888/sign-changing-repro:v1 bash.
• Proceed further as described in How to reproduce.

• For this option the first two steps are the same as above.
• The image can be downloaded here.
• Assuming that sign-changing-repro.tar is the filename of the downloaded image, please load the image with sudo docker load < sign-changing-repro.tar.
• Run the image with sudo docker run -it janosch2888/sign-changing-repro:v1 bash.
• Proceed further as described in How to reproduce.

We need to install NGSolve and a small extension called ngs_refsol manually. For reference: The code has been developed using commit 819b0d3da731bb078204fa54293be0d9feb45842 of the former and commit f3c5d52cae6a8f24a488d94337178956ace07abc of the latter library. Installation instructions for NGSolve using package managers are available here and instructions to build from source are here. Once NGSolve has been installed we can install ngs_refsol as follows:

git clone https://gitlab.gwdg.de/learned_infinite_elements/ngs_refsol.git 
cd ngs_refsol
python3 setup.py install --user

For compiling the figures you will also need a recent latex distribution installed on your machine. Now we are ready to clone the repository using

git clone https://github.com/UCL/sign-changing-repro.git 

and proceed as described in How to reproduce.

The python scripts for runnings the numerical experiments are located in the folder scripts. To run an experiment we change to this folder and run the corresponding file. After execution has finished the produced data will be available in the folder data. For the purpose of comparison, the folder data_ref contains a copy of the data which has been used for the plots in the paper. The data in both folders should be identical.

To generate the plots as shown in the article from the data just produced we change to the folder plots and compile the corresponding latex file. Below we decribe the above process for each of the figures in the article in detail. For viewing the generated pdf file, say figure.pdf, the figure has to be copied to the host machine. This can be done by executing the following commands in a new terminal window (not the one in which docker is run):

CONTAINER_ID=$(sudo docker ps -alq)
sudo docker cp $CONTAINER_ID:/home/app/sign-changing-repro/plots/figure.pdf \
/path/on/host/machine/figure.pdf

Here, /path/on/host/machine/ has to be adapted according to the file structure on the host machine. The file figure.pdf can then be found at the designated path on the host machine and inspected with a common pdf viewer. (The command above assumes that the reproduction image is the latest docker image to be started on the machine). Alternatively, if a recent latex distribution is available on the host machine it is also possible to copy data and tex files to the latter and compile the figures there.

Change to directory scripts. Run

python3 symmetric_cavity-easy.py  

Afterwards, new data files of the form Cavity-k__i__-unstructured-easy.dat will be available in the folder data. Here, i in [1,2,3] describes the finite element order k as defined in the paper. The data in the files is structured in the follwing columns:

• h: proportional to the width of the mesh.
• h1nat: contains the H^1-error for the Galerkin stabilization not shown in the paper.
• hybridstab: contains the H^1-error for the new method proposed in this paper.

This will gerate the data for the left plot. Now to produce the data for the right plot we run

python3 symmetric_cavity-high-contrast.py 

Afterwards, the data will be available in the file Cavity-k__i__-unstructured-high-contrast.dat which have the same structure as above. Then, to generate Figure 2. switch to the folder plots and run

lualatex -pdf Cavity-easy.tex 

Change to directory scripts. Run

python3 symmetric_cavity.py 

Afterwards, the data for the unstructed meshes will be available in the file Cavity-k__i__-unstructured.dat (to be found in the data folder) and the data on the symmetric meshes in the file Cavity-k1-symmetric.dat. As above, i in [1,2,3] denotes the polynomial degree.

Then, to generate Figure 3. switch to the folder plots and run

lualatex -pdf Cavity-near-critical-contrast.tex

Change to directory scripts. Run

python3 SolveMetaMaterial.py 

This will generate all the data.

• The vtk data for the plot without(!) claok (Figure 4 (A)) is available in NoCloak-order3.vtu in the folder numexp.
• The vtk data for the plot with claok (Figure 4 (B)) is available in MetaMaterial-order3.vtu in the folder numexp.
• The data for the convergence plots in Figure 5 is available in the files MetaMaterial-k__i__.dat where i in [2,3,4] denotes the polynomial order of the FEM. These data files contain the following columns
  • h is the mesh width.
  • Galerkin-inner is the H1-error for the Galerkin method in subdomain \Omega_i.
  • Galerkin-outer is the H1-error for the Galerkin method in subdomain \Omega_e.
  • Hybridstab-inner is the H1-error for the stabilized method in subdomain \Omega_i.
  • Hybridstab-outer is the H1-error for the stabilized method in subdomain \Omega_e.

To generate the Figure 5, switch to the folder plots and run

lualatex -pdf MetaMaterial-conv.tex

Change to directory scripts. Run

python3 unsymmetric-cavity.py 

Two data files will be created:

• The file Cavity-nonsymmetric-k2-unstructured-critical.dat contains in the column H1 the H1-error and in the column IF the error in the H^1/2-norm on the interface.
• The file Cavity-nonsymmetric-k2-unstructured-critical-log.dat contains these columns as well, but the column fh gives additionally the logarithmic scaling of h (the x-axis in the right plot) and the column ref contains the data for the gray reference line. The data for the vtk plot is available in the file Cavity-nonsymmetric-k2-unstructured-critical.vtu in the folder numexp.

To generate Figure 6, switch to the folder plots and run

lualatex -pdf Unsymmetric-cavity-critical-k2.tex