NGSTrefftz

an add-on to NGSolve for Trefftz methods

Binder Docker Image Version (latest semver) PyPI GitHub Workflow Status status docs

NGSTrefftz provides a framework to implement Trefftz finite element spaces for NGSolve, with several Trefftz spaces already implemented. Additionally, Trefftz-DG on tent-pitched meshes for the acoustic wave equation is implemented using meshes provided by ngstents. Furthermore, the package includes an implementation of the embedded Trefftz method.

Try it out!

The documentation is available here:

docs

You can also try the example notebooks directly in Binder:

Binder

or run them locally using Docker:

git clone https://github.com/PaulSt/NGSTrefftz
cd NGSTrefftz && docker build -t ngstrefftz_jupyter .
docker run -p 8888:8888 ngstrefftz_jupyter

Installation

The recommended installation is via pip:

pip install ngstrefftz

Alternatively, to build NGSTrefftz from source:

git clone --recursive https://github.com/PaulSt/NGSTrefftz
mkdir ./NGSTrefftz/make && cd ./NGSTrefftz/make
cmake ../ && make install

When building from source, the following dependencies are required:

For access to the newest features, the nightly version of NGSolve usually works best.

News

🚀 Jan, 2026: Trefftz Workshop will be held on 7-9 September 2026 in Vienna, Austria

⚠️ Apr, 2025: TrefftzEmbedding has a new interface, please check the documentation for details.

🚀 Jul, 2024: Conforming Trefftz embedding implementation by @johann-cm with examples in ngstSpaceKit

Older news

⚠️ Oct, 2022: With v0.2.0 the git history has undergone a major cleanup, please make sure to clone the repo anew.

🚀 Oct, 2022: New and improved implementation of the embedded Trefftz method via EmbeddedTrefftzFES!

🚀 Aug, 2022: pip-installer available, now using wheels!

🚀 Mar, 2022: NGSTrefftz now has a website!

⚠️ Feb, 2022: If you are using NGSolve nightly releases: NGSolve@eda758d breaks a dependency and NGSolve@3d52ecd produces import issue. Make sure to update ngstrefftz submodules and move to newest ngsolve version, at least NGSolve@5839a09.

🚀 Jan, 2022: NGSTrefftz is now available via pip!

🚀 Nov, 2021: NGSTrefftz now comes in a docker and with binder notebooks!

Publications and citation

If you are using ngstrefftz in your academic work, please consider citing the JOSS paper:

@article{Stocker2022NGSTrefftz,
  author  = {Stocker, Paul},
  title   = {{NGSTrefftz}: Add-on to {NGSolve} for {Trefftz} methods},
  journal = {Journal of Open Source Software},
  year    = {2022},
  volume  = {7},
  number  = {71},
  pages   = {4135},
  doi     = {10.21105/joss.04135}
}

Publications using NGSTrefftz

  • Embedded Trefftz DG method for steady Navier-Stokes flow. Part II: Nonlinear problem
    Paul Stocker, Igor Voulis, Christoph Lehrenfeld, Philip L. Lederer
    arXiv

  • Embedded Trefftz DG method for steady Navier-Stokes flow. Part I: Oseen linearization
    Paul Stocker, Igor Voulis, Christoph Lehrenfeld, Philip L. Lederer
    arXiv

  • Embedded Trefftz DG method for reaction-diffusion problems on anisotropic meshes
    Sergio Gómez, Chiara Perinati, Paul Stocker, Igor Voulis
    arXiv

  • Discontinuous Galerkin Trefftz Methods for Model Reduction of Wave Phenomena
    Tobias Born, Karsten Urban
    PAMM article

  • Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz–Hodge decompositions
    Tim Brüers, Christoph Lehrenfeld, Tim van Beeck, Max Wardetzky
    arXiv

  • A discontinuous Galerkin method for elliptic-hyperbolic equations
    Chiara Perinati, Lise-Marie Imbert-Gérard, Andrea Moiola, Paul Stocker
    arXiv

  • Embedded Trefftz DG method for the Helmholtz equation
    Paul Stocker, Igor Voulis
    arXiv

  • A unified framework for Trefftz-like discretization methods
    Philip L. Lederer, Christoph Lehrenfeld, Paul Stocker, Igor Voulis
    arXiv

  • Inf-sup stable space-time Local Discontinuous Galerkin method for the heat equation
    Sergio Gómez, Chiara Perinati, Paul Stocker
    arXiv

  • Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems
    Lise-Marie Imbert-Gérard, Andrea Moiola, Chiara Perinati, Paul Stocker
    arXiv

  • Trefftz Discontinuous Galerkin discretization for the Stokes problem
    Philip L. Lederer, Christoph Lehrenfeld, Paul Stocker
    arXiv

  • Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems
    Fabian Heimann, Christoph Lehrenfeld, Paul Stocker, Henry von Wahl
    arXiv

  • Embedded Trefftz discontinuous Galerkin methods
    Christoph Lehrenfeld, Paul Stocker
    arXiv

  • A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
    Lise-Marie Imbert-Gérard, Andrea Moiola, Paul Stocker
    arXiv

  • Tent pitching and Trefftz-DG method for the acoustic wave equation
    Ilaria Perugia, Joachim Schöberl, Paul Stocker, Christoph Wintersteiger
    arXiv

  • On the Conforming Trefftz Finite Element Method and Applications
    Johann Carl Meyer, Master’s thesis

  • Space-time Trefftz DG methods for parabolic PDEs
    Constanze Heil, Master’s thesis

  • Embedded Trefftz Trace DG Methods for PDEs on unfitted Surfaces
    Erik Schlesinger, Master’s thesis