Numerische Mathematik

, Volume 133, Issue 2, pp 203–231 | Cite as

Cut finite element methods for coupled bulk–surface problems

  • Erik Burman
  • Peter Hansbo
  • Mats G. Larson
  • Sara Zahedi


We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and \(L^2\) norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

Mathematics Subject Classification

65N30 65N12 65N15 


  1. 1.
    Booty, M., Siegel, M.: A hybrid numerical method for interfacial fluid flow with soluble surfactant. J. Comput. Phys. 229(10), 3864–3883 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Elliott, C.M., Ranner, T.: Finite element analysis for a coupled bulk-surface partial differential equation. IMA J. Numer. Anal. 33(2), 377–402 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Olshanskii, M.A., Reusken, A., Grande, J.: A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47, 3339–3358 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.: CutFEM: Discretizing geometry and partial differential equations. Internat. J. Numer. Methods Engrg. doi: 10.1002/nme.4823 (in press)
  6. 6.
    Olshanskii, M.A., Reusken, A.: A finite element method for surface PDEs: matrix properties. Numer. Math. 114(3), 491–520 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Burman, E., Hansbo, P., Larson, M.G.: A stable cut finite element method for partial differential equations on surfaces: the Laplace-Beltrami operator. Comput. Methods Appl. Mech. Engrg. 285, 188–207 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62(4), 328–341 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90–114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Massing, A., Larson, M.G., Logg, A., Rognes, M.E.: A stabilized Nitsche fictitious domain method for the Stokes problem. J. Sci. Comput. 61, 604–628 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Olshanskii, M.A., Reusken, A., Xu, X.: A stabilized finite element method for advection-diffusion equations on surfaces. IMA J. Numer. Anal. 34(2), 732–758 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Johansson, A., Larson, M.G.: A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numer. Math. 123(4), 607–628 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Georgievskii, Y., Medvedev, E.S., Stuchebrukhov, A.A.: Proton transport via the membrane surface. Biophys J. 82(6), 2833–2846 (2002)Google Scholar
  14. 14.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 editionGoogle Scholar
  15. 15.
    Dieudonné, J.: Foundations of Modern Analysis. Enlarged and corrected printing, Pure and Applied Mathematics, vol. 10-I. Academic Press, New York-London (1969)Google Scholar
  16. 16.
    Folland, G.B.: Introduction to Partial Differential Equations. Princeton University Press (1995)Google Scholar
  17. 17.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer-Verlag (2008)Google Scholar
  18. 18.
    Ern, A., Guermond, J.L.: Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM: Math. Model. Numer. Anal. 40, 29–48 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Erik Burman
    • 1
  • Peter Hansbo
    • 2
  • Mats G. Larson
    • 3
  • Sara Zahedi
    • 4
  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Department of Mechanical EngineeringJönköping UniversityJönköpingSweden
  3. 3.Department of Mathematics and Mathematical StatisticsUmeå UniversityUmeåSweden
  4. 4.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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