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

@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

docker run -ti -v $(pwd):/root/shared --name=proximal-examples ghcr.io/methods-group/proximalgalerkin:v0.2.0-alpha

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

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

source firedrake-mode

To change back to FEniCS call

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
2b	01_obstacle_problem	FEniCS	Obstacle problem (FEM)
2c(i)	01_obstacle_problem	Julia	Obstacle problem (FD)
2c(ii)	01_obstacle_problem	MultivariateOrthogonalPolynomials	Obstacle problem (Spectral)
3	02_signorini	FEniCS	Signorini
4	03_fracture	Firedrake/FEniCS	Fracture
5	04_multiphase	FEniCS	Cahn-Hilliard
6	05_obstacle_type_qvi	Gridap.jl/FEniCS	Thermoforming QVI
7	06_gradient_constraints	FEniCS	Gradient constraint
8	07_eigenvalue_constraints	Firedrake/FEniCS	Landau–de Gennes
9	08_intersecting_constraints	Firedrake/FEniCS	Intersections of constraints
10	09_equality_constraints	FEniCS	Harmonic map
11	11_eikonal	MFEM/FEniCS	Eikonal equation
12	12_monge_ampere	Firedrake/FEniCS	Monge-Ampere

Example 10: Linear Equality Constraints

Note that there is no numerical example for this setting because the derived variational formulation is equivalent to the standard Lagrange multiplier formulation for this class of problems.