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Convergence of a boundary integral method for 3D interfacial Darcy flow with surface tension


Authors: David M. Ambrose, Yang Liu and Michael Siegel
Journal: Math. Comp. 86 (2017), 2745-2775
MSC (2010): Primary 65M12; Secondary 76M25, 76B45, 35Q35
DOI: https://doi.org/10.1090/mcom/3196
Published electronically: March 3, 2017
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Abstract: We study convergence of a boundary integral method for 3D interfacial flow with surface tension when the fluid velocity is given by Darcy's Law. The method is closely related to a previous method developed and implemented by Ambrose, Siegel, and Tlupova, in which one of the main ideas is the use of an isothermal parameterization of the free surface. We prove convergence by proving consistency and stability, and the main challenge is to demonstrate energy estimates for the growth of errors. These estimates follow the general lines of estimates for continuous problems made by Ambrose and Masmoudi, in which there are good estimates available for the curvature of the free surface. To use this framework, we consider the curvature and the position of the free surface each to be evolving, rather than attempting to determine one of these from the other. We introduce a novel substitution which allows the needed estimates to close.


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Additional Information

David M. Ambrose
Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania

Yang Liu
Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania

Michael Siegel
Affiliation: Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey

DOI: https://doi.org/10.1090/mcom/3196
Received by editor(s): September 28, 2015
Received by editor(s) in revised form: May 30, 2016
Published electronically: March 3, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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