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

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- by David M. Ambrose, Yang Liu and Michael Siegel PDF
- Math. Comp.
**86**(2017), 2745-2775 Request permission

## 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.## References

- Thomas Alazard and Pietro Baldi,
*Gravity capillary standing water waves*, Arch. Ration. Mech. Anal.**217**(2015), no. 3, 741–830. MR**3356988**, DOI 10.1007/s00205-015-0842-5 - David M. Ambrose,
*Well-posedness of two-phase Darcy flow in 3D*, Quart. Appl. Math.**65**(2007), no. 1, 189–203. MR**2313156**, DOI 10.1090/S0033-569X-07-01055-3 - David M. Ambrose,
*The zero surface tension limit of two-dimensional interfacial Darcy flow*, J. Math. Fluid Mech.**16**(2014), no. 1, 105–143. MR**3171344**, DOI 10.1007/s00021-013-0146-1 - David M. Ambrose and Nader Masmoudi,
*Well-posedness of 3D vortex sheets with surface tension*, Commun. Math. Sci.**5**(2007), no. 2, 391–430. MR**2334849** - David M. Ambrose and Nader Masmoudi,
*The zero surface tension limit of three-dimensional water waves*, Indiana Univ. Math. J.**58**(2009), no. 2, 479–521. MR**2514378**, DOI 10.1512/iumj.2009.58.3450 - David M. Ambrose and Michael Siegel,
*A non-stiff boundary integral method for 3D porous media flow with surface tension*, Math. Comput. Simulation**82**(2012), no. 6, 968–983. MR**2903339**, DOI 10.1016/j.matcom.2010.05.018 - David M. Ambrose, Michael Siegel, and Svetlana Tlupova,
*A small-scale decomposition for 3D boundary integral computations with surface tension*, J. Comput. Phys.**247**(2013), 168–191. MR**3066169**, DOI 10.1016/j.jcp.2013.03.045 - Uri M. Ascher, Steven J. Ruuth, and Brian T. R. Wetton,
*Implicit-explicit methods for time-dependent partial differential equations*, SIAM J. Numer. Anal.**32**(1995), no. 3, 797–823. MR**1335656**, DOI 10.1137/0732037 - G. Baker,
*BIT for free surface flows*, Boundary element methods in engineering and sciences, Comput. Exp. Methods Struct., vol. 4, Imp. Coll. Press, London, 2011, pp. 283–322. MR**2792052**, DOI 10.1142/9781848165809_{0}008 - Gregory R. Baker, Daniel I. Meiron, and Steven A. Orszag,
*Boundary integral methods for axisymmetric and three-dimensional Rayleigh-Taylor instability problems*, Phys. D**12**(1984), no. 1-3, 19–31. MR**762803**, DOI 10.1016/0167-2789(84)90511-6 - J. Thomas Beale,
*A convergent boundary integral method for three-dimensional water waves*, Math. Comp.**70**(2001), no. 235, 977–1029. MR**1709144**, DOI 10.1090/S0025-5718-00-01218-7 - J. Thomas Beale, Thomas Y. Hou, and John Lowengrub,
*Convergence of a boundary integral method for water waves*, SIAM J. Numer. Anal.**33**(1996), no. 5, 1797–1843. MR**1411850**, DOI 10.1137/S0036142993245750 - J. Thomas Beale, Thomas Y. Hou, and John S. Lowengrub,
*Growth rates for the linearized motion of fluid interfaces away from equilibrium*, Comm. Pure Appl. Math.**46**(1993), no. 9, 1269–1301. MR**1231428**, DOI 10.1002/cpa.3160460903 - J. T. Beale, T. Y. Hou, J. S. Lowengrub, and M. J. Shelley,
*Spatial and temporal stability issues for interfacial flows with surface tension*, Math. Comput. Modelling**20**(1994), no. 10-11, 1–27. Theory and numerical methods for initial-boundary value problems. MR**1306284**, DOI 10.1016/0895-7177(94)90167-8 - Héctor D. Ceniceros and Thomas Y. Hou,
*Convergence of a non-stiff boundary integral method for interfacial flows with surface tension*, Math. Comp.**67**(1998), no. 221, 137–182. MR**1443116**, DOI 10.1090/S0025-5718-98-00911-9 - Antonio Córdoba, Diego Córdoba, and Francisco Gancedo,
*Porous media: the Muskat problem in three dimensions*, Anal. PDE**6**(2013), no. 2, 447–497. MR**3071395**, DOI 10.2140/apde.2013.6.447 - Ȧke Björck and Germund Dahlquist,
*Numerical methods*, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Translated from the Swedish by Ned Anderson. MR**0368379** - Thomas Y. Hou, John S. Lowengrub, and Michael J. Shelley,
*Removing the stiffness from interfacial flows with surface tension*, J. Comput. Phys.**114**(1994), no. 2, 312–338. MR**1294935**, DOI 10.1006/jcph.1994.1170 - T. Y. Hou, J. S. Lowengrub, and M. J. Shelley,
*The long-time motion of vortex sheets with surface tension*, Phys. Fluids**9**(1997), no. 7, 1933–1954. MR**1455083**, DOI 10.1063/1.869313 - Thomas Y. Hou, Zhen-huan Teng, and Pingwen Zhang,
*Well-posedness of linearized motion for $3$-D water waves far from equilibrium*, Comm. Partial Differential Equations**21**(1996), no. 9-10, 1551–1585. MR**1410841**, DOI 10.1080/03605309608821238 - Thomas Y. Hou and Pingwen Zhang,
*Convergence of a boundary integral method for 3-D water waves*, Discrete Contin. Dyn. Syst. Ser. B**2**(2002), no. 1, 1–34. MR**1877037**, DOI 10.3934/dcdsb.2002.2.1 - Qing Nie,
*The nonlinear evolution of vortex sheets with surface tension in axisymmetric flows*, J. Comput. Phys.**174**(2001), no. 1, 438–459. MR**1869680**, DOI 10.1006/jcph.2001.6926 - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - Gilbert Strang,
*Accurate partial difference methods. II. Non-linear problems*, Numer. Math.**6**(1964), 37–46. MR**166942**, DOI 10.1007/BF01386051 - Sijue Wu,
*Well-posedness in Sobolev spaces of the full water wave problem in 3-D*, J. Amer. Math. Soc.**12**(1999), no. 2, 445–495. MR**1641609**, DOI 10.1090/S0894-0347-99-00290-8 - Xuming Xie,
*Local smoothing effect and existence for the one-phase Hele-Shaw problem with zero surface tension*, Complex Var. Elliptic Equ.**57**(2012), no. 2-4, 351–368. MR**2886746**, DOI 10.1080/17476933.2011.575463

## Additional Information

**David M. Ambrose**- Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania
- MR Author ID: 720777
**Yang Liu**- Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania
- MR Author ID: 1009087
**Michael Siegel**- Affiliation: Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey
- Received by editor(s): September 28, 2015
- Received by editor(s) in revised form: May 30, 2016
- Published electronically: March 3, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 2745-2775 - MSC (2010): Primary 65M12; Secondary 76M25, 76B45, 35Q35
- DOI: https://doi.org/10.1090/mcom/3196
- MathSciNet review: 3667023