Volume 27, 2021
|Number of page(s)||28|
|Published online||22 March 2021|
A new diffuse-interface approximation of the Willmore flow
Department of Mathematics, TU Dortmund,
* Corresponding author: email@example.com
Accepted: 2 January 2021
Standard diffuse approximations of the Willmore flow often lead to intersecting phase boundaries that in many cases do not correspond to the intended sharp interface evolution. Here we introduce a new two-variable diffuse approximation that includes a rather simple but efficient penalization of the deviation from a quasi-one dimensional structure of the phase fields. We justify the approximation property by a Gamma convergence result for the energies and a matched asymptotic expansion for the flow. Ground states of the energy are shown to be one-dimensional, in contrast to the presence of saddle solutions for the usual diffuse approximation. Finally we present numerical simulations that illustrate the approximation property and apply our new approach to problems where the usual approach leads to an undesired behavior.
Mathematics Subject Classification: 35R35 / 35K65 / 65N30
Key words: Free boundary problem / Willmore flow / phase-field model / diffuse interface / finite elements
© EDP Sciences, SMAI 2021
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