A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 121)

Abstract

We develop a cut Discontinuous Galerkin method (cutDGM) for a diffusion-reaction equation in a bulk domain which is coupled to a corresponding equation on the boundary of the bulk domain. The bulk domain is embedded into a structured, unfitted background mesh. By adding certain stabilization terms to the discrete variational formulation of the coupled bulk-surface problem, the resulting cutDGM is provably stable and exhibits optimal convergence properties as demonstrated by numerical experiments. We also show both theoretically and numerically that the system matrix is well-conditioned, irrespective of the relative position of the bulk domain in the background mesh.

Notes

Acknowledgements

This work was supported in part by the Kempe foundation (JCK-1612). The author expresses his gratitude to Ceren Gürkan for her help with the set-up of the convergence experiment, to Erik Burman for his great editorial assistance during the preparation of this contribution, and finally, to the two anonymous referees for their valuable comments and suggestions.

References

  1. 1.
    Bastian, P., Engwer, C.: An unfitted finite element method using discontinuous Galerkin. Int. J. Numer. Methods Eng. 79 (12), 1557–1576 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Comput. Methods Appl. Mech. Eng. 199, 2680–2686 (2010)CrossRefMATHGoogle Scholar
  3. 3.
    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)MATHGoogle Scholar
  4. 4.
    Burman, E., Hansbo, P.: Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. ESAIM Math. Model. Num. Anal. 48 (3) (2013)Google Scholar
  5. 5.
    Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.: CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104 (7), 472–501 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Burman, E., Claus, S., Massing, A.: A stabilized cut finite element method for the three field Stokes problem. SIAM J. Sci. Comput. 37 (4), A1705–A1726 (2015). https://doi.org/10.1137/140983574. http://dx.doi.org/10.1137/140983574
  7. 7.
    Burman, E., Hansbo, P., Larson, M.G.: A stabilized cut finite element method for partial differential equations on surfaces: the Laplace–Beltrami operator. Comput. Methods Appl. Mech. Eng. 285, 188–207 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Burman, E., Hansbo, P., Larson, M.G., Massing, A.: Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions (2016). ArXiv e-printsGoogle Scholar
  9. 9.
    Burman, E., Hansbo, P., Larson, M.G., Massing, A.: A cut discontinuous Galerkin method for the Laplace–Beltrami operator. IMA J. Numer. Anal. 37 (1), 138–169 (2016).  https://doi.org/10.1093/imanum/drv068 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Burman, E., Hansbo, P., Larson, M.G., Massing, A., Zahedi, S.: Full gradient stabilized cut finite element methods for surface partial differential equations. Comput. Methods Appl. Mech. Eng. 310, 278–296 (2016). ISSN 0045-7825. https://doi.org/10.1016/j.cma.2016.06.033. http://www.sciencedirect.com/science/article/pii/S0045782516306703.
  11. 11.
    Burman, E., Hansbo, P., Larson, M.G., Zahedi, S.: Cut finite element methods for coupled bulk-surface problems. Numer. Math. 133, 203–231 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ern, A., Guermond, J.-L.: Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM Math. Model. Numer. Anal. 40 (1), 29–48 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for Darcy’s problem in networks of fractures. ESAIM Math. Model. Numer. Anal. 48 (4), 1089–1116 (2013). https://doi.org/10.1051/m2an/2013132.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ganesan,S., Tobiska, L.: A coupled arbitrary lagrangian–eulerian and lagrangian method for computation of free surface flows with insoluble surfactants. J. Comput. Phys. 228 (8), 2859–2873 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)MATHGoogle Scholar
  17. 17.
    Grande, J., Reusken, A.: A higher order finite element method for partial differential equations on surfaces. SIAM J. Numer. Anal. 54 (1), 388–414 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Groß, S., Reusken, A.: Numerical Methods for Two-Phase Incompressible Flows, vol. 40. Springer, Berlin (2011)MATHGoogle Scholar
  19. 19.
    Groß, S., Reusken, A.: Numerical simulation of continuum models for fluid-fluid interface dynamics. Eur. Phys. J. Special Top. 222 (1), 211–239 (2013)CrossRefGoogle Scholar
  20. 20.
    Groß, S., Olshanskii, M.A., Reusken, A.: A trace finite element method for a class of coupled bulk-interface transport problems. ESAIM Math. Model. Numer. Anal. 49 (5), 1303–1330 (2015). https://doi.org/10.1051/m2an/2015013.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Guzmán, J., Olshanskii, M.A.: Inf-sup stability of geometrically unfitted Stokes finite elements. Math. Comput. (2017, to appear).  https://doi.org/10.1090/mcom/3288
  22. 22.
    Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90–114 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hansbo, P., Larson, M.G., Zahedi, S.: Characteristic cut finite element methods for convection–diffusion problems on time dependent surfaces. Comput. Methods Appl. Mech. Eng. 293, 431–461 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for coupled bulk-surface problems on time-dependent domains. Comput. Methods Appl. Mech. Eng. 307, 96–116 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Heimann, F., Engwer, C., Ippisch, O., Bastian, P.: An unfitted interior penalty discontinuous Galerkin method for incompressible Navier–Stokes two–phase flow. Int. J. Numer. Methods Fluids 71 (3), 269–293 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (5), 1667–1691 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Massing, A.: Analysis and implementation of finite element methods on overlapping and fictitious domains. Ph.D. Thesis, Department of Informatics, University of Oslo (2012)Google Scholar
  29. 29.
    Massing, A., Larson, M.G., Logg, A., Rognes, M.E.: A stabilized Nitsche fictitious domain method for the Stokes problem. J. Sci. Comput. 61 (3), 604–628 (2014). https://doi.org/10.1007/s10915-014-9838-9.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Massing, A., Schott, B., Wall, W.A.: A stabilized Nitsche cut finite element method for the Oseen problem. Comput. Methods Appl. Mech. Eng. 328, 262–300 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Massjung, R.: An unfitted discontinuous Galerkin method applied to elliptic interface problems. SIAM J. Numer. Anal. 50 (6), 3134–3162 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Müller, B., Krämer-Eis, S., Kummer, F., Oberlack, M.: A high-order Discontinuous Galerkin method for compressible flows with immersed boundaries. Int. J. Numer. Methods Eng. (2016). ISSN 1097-0207.  https://doi.org/10.1002/nme.5343.  http://dx.doi.org/10.1002/nme.5343
  33. 33.
    Muradoglu, M., Tryggvason, G.: A front-tracking method for computation of interfacial flows with soluble surfactants. J. Comput. Phys. 227 (4), 2238–2262 (2008)CrossRefMATHGoogle Scholar
  34. 34.
    Novak, I.L., Gao, F., Choi, Y.-S., Resasco, D., Schaff, J.C., Slepchenko, B.M.: Diffusion on a curved surface coupled to diffusion in the volume: application to cell biology. J. Comput. Phys. 226 (2), 1271–1290 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Olshanskii, M.A., Reusken, A.: A finite element method for surface PDEs: matrix properties. Numer. Math. 114 (3), 491–520 (2010)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Olshanskii, M.A., Reusken, A., Grande, J.: A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47 (5), 3339–3358 (2009)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Olshanskii, M.A., Reusken, A., Xu, X.: An Eulerian space-time finite element method for diffusion problems on evolving surfaces. SIAM J. Numer. Anal. 52 (3), 1354–1377 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Rätz, A.: Turing-type instabilities in bulk–surface reaction–diffusion systems. J. Comput. Appl. Math. 289, 142–152 (2015)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Saye, R.I.: High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles. SIAM J. Sci. Comput. 37 (2), A993–A1019 (2015)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Schott, B.: Stabilized cut finite element methods for complex interface coupled flow problems. Ph.D. Thesis, Technical University of Munich (2017)Google Scholar
  41. 41.
    Sollie, W.E.H., Bokhove, O., van der Vegt, J.J.W.: Space–time discontinuous Galerkin finite element method for two-fluid flows. J. Comput. Phys. 230 (3), 789–817 (2011)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Winter, M., Schott, B., Massing, A., Wall, W.A.: A Nitsche cut finite element method for the Oseen problem with general Navier boundary conditions. Comput. Methods Appl. Mech. Eng. 330, 220–252 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Mathematical StatisticsUmeå UniversityUmeåSweden

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