Hyperboloidal Compactification in NGSolve

by M. Wess, Institute of Analysis and Scientific Computing, TU Wien

and A. Zenginoğlu, Institute for Physical Science and Technology, University of Maryland

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This book provides an introduction to the implementation of hyperboloidal compactification in the high-order finite-element library NGSolve, together with executable examples.

Introduction

Hyperboloidal compactification provides a framework for the numerical treatment of wave propagation problems on unbounded domains. By introducing hyperboloidal time slices together with a compactification of the spatial coordinates, future null infinity is mapped to a finite computational boundary while preserving the causal structure of outgoing waves. As a result, radiation can leave the computational domain without artificial reflections, eliminating the need for absorbing boundary conditions or perfectly matched layers.

The method admits both time-domain and frequency-domain formulations. In the time domain, hyperboloidal compactification transforms hyperbolic evolution equations into systems posed on a finite spatial domain and supports long-time simulations that directly capture outgoing radiation. In the frequency domain, the same geometric transformation yields a compactified formulation of the equations, in which the radiation condition is selected by the transformed finite-energy space. This provides a unified approach to the numerical solution of scattering and radiation problems, enabling high-order discretizations on bounded computational domains while maintaining the correct asymptotic behavior at infinity.

Note

This book is an executable introduction and implementation for readers who already know the finite-element treatment of Helmholtz or wave equations. For the complete derivation, analysis, comparisons with PML, and broader numerical study of the hyperboloidal Helmholtz equation, see the Wess–Zenginoğlu preprint.

The book also includes time-domain examples, which are part of ongoing research and will be updated in the future.

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Table of contents

• 1. Exterior Scattering Problem
• 2. Compactifying the Far Field
• 3. Spacetime Picture
• 4. Weak Form Used in NGSolve
• 5. Limitations
• 6. Examples
  • Frequency Domain: Characteristic Compactification of a Sound-Soft Circle
  • Frequency Domain: Poincaré Compactification of a Sound-Soft Circle
  • Frequency Domain: Sound-Soft Circle
  • Frequency Domain: Trapping Square
  • Time Domain: Driven Trapping Square
  • Time Domain: Trapping Square with Discontinuous Galerkin
  • Explicit DG for the Trapping Square

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Reference

M. Wess and A. Zenginoğlu, Finite Elements for Helmholtz Scattering with Infinity as a Computational Boundary, arXiv:2606.25130 (2026).