Hyperboloidal compactification in NGSolve

An executable introduction to finite-element Helmholtz and wave problems with infinity as a computational boundary.

Published book: https://markuswess.github.io/hypFEM

Local build

The notebooks require Python 3.12 and NGSolve. A tested environment can be created with

conda create -n hypfem python=3.12
conda activate hypfem
python -m pip install -r requirements.txt

Build with the execution cache:

make

For review, publication, or continuous integration, use a clean build that executes every notebook from the current source:

make full

The HTML entry point is _build/html/index.html. The clean build is the reproducibility check: a normal cached build can legitimately reuse results from an earlier execution.

After committing and pushing the reviewed source branch, publish the clean build with:

./publish.sh

The script stops if the clean execution fails and only then updates the gh-pages branch.

Numerical conventions

• Time dependence is exp(-i k t), so exp(i k r) is outgoing.
• Frequency-domain notebooks solve for the physical scattered field with boundary data -U_inc.
• The main weak form uses U = Omega^((d-1)/2) exp(i k h) u.
• The characteristic benchmark additionally uses the bounded standard far-field scaling u_char = rho^((d-1)/2) u.