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Phase-field approximations of the Willmore functional and flow

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Abstract

We discuss in this paper phase-field approximations of the Willmore functional and the associated \({\mathrm L}^{2}\)-flow. After recollecting known results on the approximation of the Willmore energy and its \({\mathrm L}^{1}\) relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence is proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.

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Acknowledgments

The authors thank Luca Mugnai, Selim Esedoḡlu, Petru Mironescu, and Giovanni Bellettini for fruitful discussions.

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Authors and Affiliations

  1. CNRS UMR 5208, INSA de Lyon, Institut Camille Jordan, Université de Lyon, 20, avenue Albert Einstein, 69621, Villeurbanne Cedex, France

    Elie Bretin

  2. CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, Université de Lyon, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Simon Masnou

  3. Laboratoire Jean Kuntzmann, Université Joseph Fourier, Tour IRMA, BP 53, 51, rue des Mathématiques, 38041, Grenoble Cedex 9, France

    Édouard Oudet

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Bretin, E., Masnou, S. & Oudet, É. Phase-field approximations of the Willmore functional and flow. Numer. Math. 131, 115–171 (2015). https://doi.org/10.1007/s00211-014-0683-4

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