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Discretization of the Koch Snowflake Domain with Boundary and Interior Energies

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Fractals in Engineering: Theoretical Aspects and Numerical Approximations

Part of the book series: SEMA SIMAI Springer Series ((ICIAM2019SSSS,volume 8))

Abstract

We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. We compute eigenvalues and eigenfunctions, and demonstrate the localization of high energy eigenfunctions on the boundary via a modification of an argument of Filoche and Mayboroda. Hölder continuity and uniform approximation of eigenfunctions are also discussed.

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Acknowledgements

The authors are grateful to Michael Hinz, Maria Rosaria Lancia, Svitlana Mayboroda, Anna Rozanova-Pierrat, and Tatiana Toro for helpful discussions.

The authors are grateful to Kevin Marinelli for the support in implementing the numerical codes.

Research supported in part by NSF DMS Grants 1659643 and 1613025.

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Authors and Affiliations

  1. Colorado College, Department of Mathematics and Computer Science, Colorado Springs, CO, USA

    Malcolm Gabbard

  2. California State University Long Beach, Department of Physics and Astronomy, Long Beach, CA, USA

    Carlos Lima

  3. University of Connecticut, Department of Mathematics, Storrs, CT, USA

    Gamal Mograby, Luke Rogers & Alexander Teplyaev

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  1. Malcolm Gabbard
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  2. Carlos Lima
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  3. Gamal Mograby
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  4. Luke Rogers
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  5. Alexander Teplyaev
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Correspondence to Alexander Teplyaev .

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Editors and Affiliations

  1. Dipartimento di Scienze di base ed Applicate per l’ingegneria, Sapienza Università di Roma, Roma, Italy

    Maria Rosaria Lancia

  2. Department of Mathematics, CentraleSupélec, Université Paris Saclay, Gif-sur-Yvette, France

    Anna Rozanova-Pierrat

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Gabbard, M., Lima, C., Mograby, G., Rogers, L., Teplyaev, A. (2021). Discretization of the Koch Snowflake Domain with Boundary and Interior Energies. In: Lancia, M.R., Rozanova-Pierrat, A. (eds) Fractals in Engineering: Theoretical Aspects and Numerical Approximations. SEMA SIMAI Springer Series(), vol 8. Springer, Cham. https://doi.org/10.1007/978-3-030-61803-2_4

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