ProximalGalerkin

This repository contains implementations of the proximal Galerkin finite element method and other proximal numerical methods for variational problems with inequality constraints derived in

bibtex @misc{dokken2025latent, title={The latent variable proximal point algorithm for variational problems with constraints}, author={Dokken, {J\o rgen} S. and Farrell, Patrick~E. and Keith, Brendan and Papadopoulos, Ioannis~P.A. and Surowiec, Thomas~M.}, year={2025}, }

Please cite the aforementioned manuscript if using the code in this repository.

Installation instructions

We provide a single docker container ghcr.io/methods-group/proximalgalerkin from Proximal Galerkin Docker container registry that provides an installation of all dependencies used in the examples of this paper.

One can start the image with

bash docker run -ti -v $(pwd):/root/shared --name=proximal-examples ghcr.io/methods-group/proximalgalerkin:v0.3.0

This shares the current directory with the docker container under the location /root/shared. To restart this container at a later instance call

bash docker container start -i proximal-examples

Within this installation you find all examples under /root/LVPP.

MFEM

When wanting to run MFEM-examples, one has to navigate to /root/LVPP/mfem/examples and call make name_of_example to compile the corresponding demo.

The MFEM scripts from this paper are already placed in this location in the container.

FEniCS/Firedrake compatibility

FEniCS and Firedrake are installed in separate virtual environments within the container. At launch, the user gets to use FEniCS by default. To change to Firedrake, call

bash source firedrake-mode

To change back to FEniCS call

bash source dolfinx-mode

Table of Examples and Figures

The following table associates each implementation to the examples and figures in the paper. Further information to run the codes is provided for each specific example can be found in the corresponding folder.

| Figure | Folder | Backend | Problem Type | | :----: | :--------------------------------------------------------------------: | :-------------------------------: | ---------------------------- | | 3b | 01obstacleproblem | FEniCS | Obstacle problem (FEM) | | 3c(i) | 01obstacleproblem | Julia | Obstacle problem (FD) | | 3c(ii) | 01obstacleproblem | MultivariateOrthogonalPolynomials | Obstacle problem (Spectral) | | 4 | 02_signorini | FEniCS | Signorini | | 5 | 03_fracture | Firedrake/FEniCS | Fracture | | 6 | 04_multiphase | FEniCS | Cahn-Hilliard | | 7 | 05obstacletype_qvi | Gridap.jl/FEniCS | Thermoforming QVI | | 8 | 06gradientconstraints | FEniCS | Gradient constraint | | 9 | 07eigenvalueconstraints | Firedrake/FEniCS | Landau–de Gennes | | 10 | 08intersectingconstraints | Firedrake/FEniCS | Intersections of constraints | | 11 | 09_eikonal | MFEM/FEniCS | Eikonal equation | | 12 | 10mongeampere | Firedrake/FEniCS | Monge-Ampere |