A Penalty Finite Element Method for a Fluid System Posed on Embedded Surface

  • Maxim A. OlshanskiiEmail author
  • Vladimir Yushutin


The paper introduces a finite element method for the incompressible Navier–Stokes equations posed on a closed surface \(\Gamma \subset \mathbb {R}^3\). The method needs a shape regular tetrahedra mesh in \(\mathbb {R}^3\) to discretize equations on the surface, which can cut through this mesh in a fairly arbitrary way. Stability and error analysis of the fully discrete (in space and in time) scheme is given. The tangentiality condition for the velocity field on \(\Gamma \) is enforced weakly by a penalty term. The paper studies both theoretically and numerically the dependence of the error on the penalty parameter. Several numerical examples demonstrate convergence and conservation properties of the finite element method.


Surface Navier–Stokes problem Fluidic membranes Trace finite element method 


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Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Barrett, J.W., Garcke, H., Nürnberg, R.: A stable numerical method for the dynamics of fluidic membranes. Numerische Mathematik 134, 783–822 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baumgart, T., Hess, S.T., Webb, W.W.: Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature 425, 821 (2003)ADSCrossRefGoogle Scholar
  3. 3.
    Brezzi, F., Pitkäranta, J.: On the Stabilization of Finite Element Approximations of the Stokes Equations, pp. 11–19. Vieweg+Teubner, Wiesbaden (1984)Google Scholar
  4. 4.
    Burman, E., Hansbo, P., Larson, M.G., Massing, A.: Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. ESAIM: Math. Model. Numer. Anal. 53, 2247–2282 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dickinson, E.: Adsorbed protein layers at fluid interfaces: interactions, structure and surface rheology. Colloids Surf. B: Biointerfaces 15, 161–176 (1999)CrossRefGoogle Scholar
  6. 6.
    DROPS package. Accessed 2 Feb 2019
  7. 7.
    Fan, J., Han, T., Haataja, M.: Hydrodynamic effects on spinodal decomposition kinetics in planar lipid bilayer membranes. J. Chem. Phys. 133, 12B604 (2010)CrossRefGoogle Scholar
  8. 8.
    Fries, T.-P.: Higher-order surface fem for incompressible Navier–Stokes flows on manifolds. Int. J. Numer. Methods Fluids 88, 55–78 (2018)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ganesan, S., Matthies, G., Tobiska, L.: Local projection stabilization of equal order interpolation applied to the Stokes problem. Math. Comput. 77, 2039–2060 (2008)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Grande, J., Lehrenfeld, C., Reusken, A.: Analysis of a high-order trace finite element method for PDEs on level set surfaces. SIAM J. Numer. Anal. 56, 228–255 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Groß, S., Jankuhn, T., Olshanskii, M.A., Reusken, A.: A trace finite element method for vector-Laplacians on surfaces. SIAM J. Numer. Anal. 56, 2406–2429 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hansbo, P., Larson, M.G.: A stabilized finite element method for the Darcy problem on surfaces. IMA J. Numer. Anal. p. drw041 37(3), 1274–1299 (2017)Google Scholar
  14. 14.
    Hansbo, P., Larson, M.G., Larsson, K.: Analysis of finite element methods for vector Laplacians on surfaces. arXiv preprint arXiv:1610.06747v2 (2018)
  15. 15.
    Holst, M., Stern, A.: Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Found. Comput. Math. 12, 263–293 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jankuhn, T., Olshanskii, M.A., Reusken, A.: Incompressible fluid problems on embedded surfaces: modeling and variational formulations. Interfaces Free Bound. 20, 353–378 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Koba, H., Liu, C., Giga, Y.: Energetic variational approaches for incompressible fluid systems on an evolving surface. Q. Appl. Math. 75, 359–389 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ledoux, M.: On improved Sobolev embedding theorems. Math. Res. Lett. 10, 659–670 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lehrenfeld, C., Olshanskii, M.A., Xu, X.: A stabilized trace finite element method for partial differential equations on evolving surfaces. SIAM J. Numer. Anal. 56, 1643–1672 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nitschke, I., Reuther, S., Voigt, A.: Discrete exterior calculus (DEC) for the surface Navier–Stokes equation. In: Transport Processes at Fluidic Interfaces, pp. 177–197. Springer (2017)Google Scholar
  21. 21.
    Nitschke, I., Voigt, A., Wensch, J.: A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mech. 708, 418–438 (2012)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Olshanskii, M.A., Quaini, A., Reusken, A., Yushutin, V.: A finite element method for the surface Stokes problem. SIAM J. Sci. Comput. 40, A2492–A2518 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Olshanskii, M.A., Reusken, A.: Trace finite element methods for PDEs on surfaces. In: Geometrically Unfitted Finite Element Methods and Applications, pp. 211–258. Springer (2017)Google Scholar
  24. 24.
    Olshanskii, M.A., Reusken, A., Grande, J.: A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47, 3339–3358 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Reusken, A.: Analysis of trace finite element methods for surface partial differential equations. IMA J. Numer. Anal. 35, 1568–1590 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Reusken, A.: Stream function formulation of surface Stokes equations. IMA J. Numer. Anal. p. dry062 (2018).
  27. 27.
    Reuther, S., Voigt, A.: The interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul. 13, 632–643 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Reuther, S., Voigt, A.: Solving the incompressible surface Navier–Stokes equation by surface finite elements. Phys. Fluids 30, 012107 (2018)ADSCrossRefGoogle Scholar
  29. 29.
    Rodrigues, D.S., Ausas, R.F., Mut, F., Buscaglia, G.C.: A semi-implicit finite element method for viscous lipid membranes. J. Comput. Phys. 298, 565–584 (2015)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Sakai, T.: Riemannian Geometry, vol. 149. American Mathematical Soc., Providence (1996)zbMATHGoogle Scholar
  31. 31.
    Scriven, L.: Dynamics of a fluid interface equation of motion for Newtonian surface fluids. Chem. Eng. Sci. 12, 98–108 (1960)CrossRefGoogle Scholar
  32. 32.
    Slattery, J.C., Sagis, L., Oh, E.-S.: Interfacial Transport Phenomena. Springer, New York (2007)zbMATHGoogle Scholar
  33. 33.
    Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam (1984)zbMATHGoogle Scholar
  34. 34.
    Yavari, A., Ozakin, A., Sadik, S.: Nonlinear elasticity in a deforming ambient space. J. Nonlinear Sci. 26, 1651–1692 (2016)ADSMathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Sechenov UniversityMoscowRussia

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