The Dual Cell Method in NGSolve

by M. Wess, J. Schöberl

TU Wien, Institute of Analysis and Scientific Computing,

based on joint work with B. Kapidani and L. Codecasa

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This book is designed to provide an introduction and examples to the implementation of the Dual Cell Method in the high-order finite element library NGSolve.

The Dual Cell Method (DCM) is a Galerkin Method for the simulation of time-domain waves (e.g., electromagnetic or acoustiv waves) in mixed formulation. It is a Disconitinuous Galerkin variant where the two wave-fields are approximated by conforming functions on meshes dual to each other. Thus the respective ansatz functions feature discontinuities on different element boundaries.

For a full mathematical introduction to the method we refer to [KCSchoberl21, WKCS24] and to [CKSW24] for a shorter read.

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Table of Contents

• 1. Installation
• 2. The dual cell method for the time-domain Maxwell system
• 3. Examples
  • 3.1. Basics on dual cell spaces
  • 3.2. Mass Lumping for dual cell spaces
  • 3.3. Computing the gradient of a Gaussian peak in dual cell spaces
  • 3.4. A plane wave on a square
  • 3.5. Ring resonator
  • 3.6. Electromagnetic 3d waveguide
  • 3.7. Bent electromagnetic 3d waveguide with PML

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References

[CKSW24]

Lorenzo Codecasa, Bernard Kapidani, Joachim Schöberl, and Markus Wess. Mass-lumped high-order cell methods for the time-dependent maxwell's equations. In Laurent Gizon, editor, Book of Abstracts, The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2024), pp. 353 – 354. Berlin, Germany, 2024. Edmond MPDL. doi:10.17617/3.MBE4AA.

[KCSchoberl21]

Bernard Kapidani, Lorenzo Codecasa, and Joachim Schöberl. An arbitrary-order cell method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations. J. Comput. Phys., 433:Paper No. 110184, 20, 2021. doi:10.1016/j.jcp.2021.110184.

[WKCS24]

Markus Wess, Bernard Kapidani, Lorenzo Codecasa, and Joachim Schöberl. Mass lumping the dual cell method to arbitrary polynomial degree for acoustic and electromagnetic waves. Journal of Computational Physics, pages 113196, 2024. doi:https://doi.org/10.1016/j.jcp.2024.113196.