An Eulerian approach to transport and diffusion on evolving implicit surfaces

Regular article


In this article we define a level set method for a scalar conservation law with a diffusive flux on an evolving hypersurface Γ(t) contained in a domain \({\Omega \subset \mathbb R^{n+1}}\) . The partial differential equation is solved on all level set surfaces of a prescribed time dependent function Φ whose zero level set is Γ(t). The key idea lies in formulating an appropriate weak form of the conservation law with respect to time and space. A major advantage of this approach is that it avoids the numerical evaluation of curvature. The resulting equation is then solved in one dimension higher but can be solved on a fixed grid. In particular we formulate an Eulerian transport and diffusion equation on evolving implicit surfaces. Using Eulerian surface gradients to define weak forms of elliptic operators naturally generates weak formulations of elliptic and parabolic equations. The finite element method is applied to the weak form of the conservation equation yielding an Eulerian Evolving Surface Finite Element Method. The computation of the mass and element stiffness matrices, depending only on the gradient of the level set function, are simple and straightforward. Numerical experiments are described which indicate the power of the method. We describe how this framework may be employed in applications.


  1. 1.
    Dziuk G., Elliott C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, 262–292 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Dziuk G., Elliott C.M.: Surface finite elements for parabolic equations. J. Comp. Math. 25, 385–407 (2007)MathSciNetGoogle Scholar
  3. 3.
    Dziuk, G., Elliott, C.M.: Eulerian finite element method for parabolic PDEs on implicit surfaces. Interfaces Free Boundaries 10 (2008)Google Scholar
  4. 4.
    Deckelnick, K., Dziuk, G., Elliott, C.M., Heine, C.-J.: An h-narrow band finite element method for elliptic equations on implicit surfaces (in preparation)Google Scholar
  5. 5.
    Bertalmio, M., Memoli, F., Cheng, L.T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces:bye bye triangulated surfaces? Geometric level set methods in imaging, vision and graphics Springer New York, pp 381–397 (2003)Google Scholar
  6. 6.
    Bertalmio M., Cheng L.T., Osher S., Sapiro G.: Variational problems and partial differential equations on implicit surfaces. J. Comp. Phys. 174, 759–780 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Greer J.B.: An improvement of a recent Eulerian method for solving PDEs on general geometries. J. Sci. Comput. 29, 321–352 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Greer J., Bertozzi A., Sapiro G.: Fourth order partial differential equations on general geometries. J. Comput. Phys. 216(1), 216–246 (2006)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Burger, M.: Finite element approximation of elliptic partial differential equations on implicit surfaces. Comput. Visual. Sci. (2007). doi:10.1007/s00791-007-0081-x
  10. 10.
    Adalsteinsson D., Sethian J.A.: Transport and diffusion of material quantities on propagating interfaces via level set methods. J. Comp. Phys. 185, 271–288 (2003)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Xu J.-J., Zhao H.-K.: An Eulerian formulation for solving partial differential equations along a moving interface. J. Sci. Comput. 19, 573–594 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    James A.J., Lowengrub J.: A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comp. Phys. 201, 685–722 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Stone H.A.: A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2, 111–112 (1990)CrossRefGoogle Scholar
  14. 14.
    Cahn J.W., Fife P., Penrose O.: A phase field model for diffusion induced grain boundary motion. Acta Mater. 45, 4397–4413 (1997)CrossRefGoogle Scholar
  15. 15.
    Deckelnick K., Elliott C.M., Styles V.: Numerical diffusion induced grain boundary motion. Interfaces Free Boundaries 3, 393–414 (2001)MATHMathSciNetGoogle Scholar
  16. 16.
    Fife P., Cahn J.W., Elliott C.M.: A free boundary model for diffusion induced grain boundary motion. Interfaces Free Boundaries 3, 291–336 (2001)MATHMathSciNetGoogle Scholar
  17. 17.
    Mayer U.F., Simonnett G.: Classical solutions for diffusion induced grain boundary motion. J. Math. Anal. 234, 660–674 (1999)MATHCrossRefGoogle Scholar
  18. 18.
    Leung C.H., Berzins M.: A computational model for organism growth based on surface mesh generation. J. Comp. Phys. 188, 75–99 (2003)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Jin, H., Yezzi, A.J., Soatto, S.: Region based segmentation on evolving surfaces with application to 3D reconstruction of shape and piecewise constant radiance. UCLA Preprint (2004)Google Scholar
  20. 20.
    Schmidt, A., Siebert, K.G.: Design of adaptive finite element software. The finite element toolbox ALBERTA. Springer Lecture Notes in Computational Science and Engineering, vol. 42. Springer, Berlin (2004)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Abteilung für Angewandte MathematikUniversity of FreiburgFreiburg i. Br.Germany
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations