fxdgm

A nonlinear, mixed finite element solver for the DGM electrolyte model

Paper

Jan Habscheid, Satyvir Singh, Lambert Theisen, Stefanie Braun, Manuel Torrilhon, A finite element solver for a thermodynamically consistent electrolyte model, Computer Physics Communications, Volume 319, 2026, 109916.
https://doi.org/10.1016/j.cpc.2025.109916

BibTeX

@article{HABSCHEID2026109916,
title = {A finite element solver for a thermodynamically consistent electrolyte model},
journal = {Computer Physics Communications},
volume = {319},
pages = {109916},
year = {2026},
issn = {0010-4655},
doi = {https://doi.org/10.1016/j.cpc.2025.109916},
url = {https://www.sciencedirect.com/science/article/pii/S0010465525004175},
author = {Jan Habscheid and Satyvir Singh and Lambert Theisen and Stefanie Braun and Manuel Torrilhon},
keywords = {Electrochemistry, Electrical double layer, Thermodynamics, Electrolyte models, FEniCS, Finite element method},
abstract = {In this study, we present a finite element solver for a thermodynamically consistent electrolyte model that accurately captures multicomponent ionic transport by incorporating key physical phenomena such as steric effects, solvation, and pressure coupling. The model is rooted in the principles of non-equilibrium thermodynamics and strictly enforces mass conservation, charge neutrality, and entropy production. It extends beyond classical frameworks like the Nernst–Planck system by employing modified partial mass balances, the electrostatic Poisson equation, and a momentum balance expressed in terms of electrostatic potential, atomic fractions, and pressure, thereby enhancing numerical stability and physical consistency. Implemented using the FEniCSx platform, the solver efficiently handles one- and two-dimensional problems with varied boundary conditions and demonstrates excellent convergence behavior and robustness. Validation against benchmark problems confirms its improved physical fidelity, particularly in regimes characterized by high ionic concentrations and strong electrochemical gradients. Simulation results reveal critical electrolyte phenomena, including electric double layer formation, rectification behavior, and the effects of solvation number, Debye length, and compressibility. The solver’s modular variational formulation facilitates its extension to complex electrochemical systems involving multiple ionic species with asymmetric valences. We publicly provide the documented and validated solver framework.}
}

Physical Background

The system, which is solved, refers to the original work, Overcoming the shortcomings of the Nernst–Planck model, from Wolfgang Dreyer, Clemens Guhlke and Rüdiger Müller in 2013.
This paper introduces a new, generalized Nernst-Planck model, which is thermodynamically consistent, as the classical Nernst-Planck model fails to predict the correct ion-concentrations close to the boundaries. The open-source package FEniCSx was used for the numerical implementation.

Main Features

• Solving steady DGM model in dimensionless units
  • for a ternary electrolyte (cations, anions, neutral solvent)
  • for an electrolyte of N arbitrary species
• Local mesh refinement for one-dimensional domains towards the electrode
• Testcases for the one-dimensional case or the two-dimensional electrolytic diode
• Solutions for the Double-Layer Capacity, both numerical and analytical
• Numerical Convergence with relaxation parameter for newtons method
• Two-dimensional testcases for the example of the electrolytic diode

Installation

First, install the FEniCSx and the necessary compiler via conda and after the fxdgm package with pip. The necessary dependencies can be installed with

conda install -c conda-forge fenics-dolfinx=0.9.0 mpich=4.3.0 pyvista=0.43.10 c-compiler=1.9.0 cxx-compiler=1.9.0 fortran-compiler=1.9.0 -y
git clone https://git.rwth-aachen.de/JanHab/fxdgm.git
cd fxdgm
pip install .

It is also possible to install the FEniCSx backend in a different manner. See the FEniCSx documentation for this. Although this installation method should work, it was not tested for the purpose of this package.

Alternative installation using Docker

Alternatively, the FEniCSx backend and the fxdgm package can be installed at once using Docker.

docker compose build
docker compose run solver

Testing

For testing clone the repository, install pytest and run the tests with

pip install pytest==8.3.3
python -m pytest

Usage

Find the package source code in fxdgm. This implements the nonlinear electrolyte model.

Furthermore, some physical examples are provided in the examples. In the subfolder ReproducableCode is the code, to execute the calculations with some first visualizations. The subfolder Data stores the data for all the simulations in a *.npz file, which can be read with numpy np.load(file.npz). Visualizations creates the necessary figures from the thesis and stores them either in *.svg or *.pdf format in "Figures".

Contact

• Jan Habscheid:
  • Jan.Habscheid@rwth-aachen.de
• Dr. Satyvir Singh
  • ACoM - Applied and Computational Mathematics
  • RWTH Aachen University
  • singh@acom.rwth-aachen.de
• Dr. Lambert Theisen
  • ACoM - Applied and Computational Mathematics
  • RWTH Aachen University
  • theisen@acom.rwth-aachen.de
• Dr. Stefanie Braun
  • ACoM - Applied and Computational Mathematics
  • RWTH Aachen University
  • braun@acom.rwth-aachen.de
• Prof. Dr. Manuel Torrilhon
  • ACoM - Applied and Computational Mathematics
  • RWTH Aachen University
  • mt@acom.rwth-aachen.de