NGSTrefftz

an add-on to NGSolve for Trefftz methods

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:

You can also try the example notebooks directly in 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:

• cmake >= 3.1
• gcc >= 9 or clang >= 10
• lapack >= 3.9
• ngsolve >= 6.2

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
  
• Embedded Trefftz DG method for steady Navier-Stokes flow. Part I: Oseen linearization
  Paul Stocker, Igor Voulis, Christoph Lehrenfeld, Philip L. Lederer
  
• Embedded Trefftz DG method for reaction-diffusion problems on anisotropic meshes
  Sergio Gómez, Chiara Perinati, Paul Stocker, Igor Voulis
  
• 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
  
• A discontinuous Galerkin method for elliptic-hyperbolic equations
  Chiara Perinati, Lise-Marie Imbert-Gérard, Andrea Moiola, Paul Stocker
  
• Embedded Trefftz DG method for the Helmholtz equation
  Paul Stocker, Igor Voulis
  
• A unified framework for Trefftz-like discretization methods
  Philip L. Lederer, Christoph Lehrenfeld, Paul Stocker, Igor Voulis
  
• Inf-sup stable space-time Local Discontinuous Galerkin method for the heat equation
  Sergio Gómez, Chiara Perinati, Paul Stocker
  
• Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems
  Lise-Marie Imbert-Gérard, Andrea Moiola, Chiara Perinati, Paul Stocker
  
• Trefftz Discontinuous Galerkin discretization for the Stokes problem
  Philip L. Lederer, Christoph Lehrenfeld, Paul Stocker
  
• Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems
  Fabian Heimann, Christoph Lehrenfeld, Paul Stocker, Henry von Wahl
  
• Embedded Trefftz discontinuous Galerkin methods
  Christoph Lehrenfeld, Paul Stocker
  
• A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
  Lise-Marie Imbert-Gérard, Andrea Moiola, Paul Stocker
  
• Tent pitching and Trefftz-DG method for the acoustic wave equation
  Ilaria Perugia, Joachim Schöberl, Paul Stocker, Christoph Wintersteiger
  
• 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