Skip to main content
SpringerLink
Log in

An ALE ESFEM for Solving PDEs on Evolving Surfaces

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

Numerical methods for approximating the solution of partial differential equations on evolving hypersurfaces using surface finite elements on evolving triangulated surfaces are presented. In the ALE ESFEM the vertices of the triangles evolve with a velocity which is normal to the hypersurface whilst having a tangential velocity which is arbitrary. This is in contrast to the original evolving surface finite element method in which the nodes move with a material velocity. Numerical experiments are presented which illustrate the value of choosing the arbitrary tangential velocity to improve mesh quality. Simulations of two applications arising in material science and biology are presented which couple the evolution of the surface to the solution of the surface partial differential equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adalsteinsson D ., Sethian J.A: Transport and diffusion of material quantities on propagating interfaces via level set methods. Journal of Computational Physics. 185, 271–288 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barreira R, Elliott C, Madzvamuse A: The surface finite element method for pattern formation on evolving biological surfaces. Journal of Mathematical Biology. 63, 1095–1119 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. J. W. Barrett and C. M. Elliott, A finite-element method for solving elliptic equations with Neumann data on a curved boundary using unfitted meshes, IMA Journal of Numerical Analysis, 4 (1984), pp. 309–325.

    Google Scholar 

  4. Barrett J. W., Garcke H., Nürnberg R.: On the variational approximation of combined second and fourth order geometric evolution equations. SIAM J. Sci. Comput., 29, 1006–1041 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. W. Barrett, H. Garcke, and R. Nürnberg , A parametric finite element mthod for fourth order geometric evolution equations, J. Comp. Phys., 222 (2007), pp. 441–467.

    Google Scholar 

  6. J. W. Barrett, H. Garcke, and R. Nürnberg, Numerical approximation of anisotropic geometric evolution equations in the plane, IMA J. Numer. Anal., 28 (2008), pp. 292–330.

    Google Scholar 

  7. J. W. Barrett, H. Garcke, and R. Nürnberg, On the parametric finite element approximation of evolving hypersurfaces in R3, J. Comp. Phys., 227 (2008), pp. 4281–4307.

    Google Scholar 

  8. M. J. Berger, D. A. Calhoun, C. Helzel, and R. J. Leveque, adaptive refinement on the sphere logically rectangular finite volume methods with adaptive refinement on the sphere, Philosophical Transactions of the Royal Society of London A, 367 (2009), pp. 4483–4496.

  9. M. Bertalmío, L.-T. Cheng, S. J. Osher, and G. Sapiro, Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys., 174 (2001), pp. 759–780.

    Google Scholar 

  10. Cahn J.W., Fife P, Penrose O.: A phase-field model for diffusion-induced grain-boundary motion. Acta Materialia. 45, 4397–4413 (1997)

    Article  Google Scholar 

  11. D. A. Calhoun and C. Helzel, A finite volume method for solving parabolic equations on logically cartesian curved surface meshes, SIAM J. Sci. Comput., 31 (2009), pp. 4066–4099.

    Google Scholar 

  12. M. A. J. Chaplain, M. Ganesh, and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth, Journal of Mathematical Biology, 42 (2001), pp. 387–423.

    Google Scholar 

  13. E. J. Crampin, E. A. Gaffney, and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bull. Math. Biology, 61 (1999), pp. 1093–1120.

    Google Scholar 

  14. K. Deckelnick, G. Dziuk, and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), pp. 139–232.

    Google Scholar 

  15. K. Deckelnick, G. Dziuk, C. M. Elliott, and C.-J. Heine, An h-narrow band finite element method for implicit surfaces, IMA J. Numer. Anal., 30 (2010), pp. 351–376.

    Google Scholar 

  16. K. Deckelnick and C. Elliott, An existence and uniqueness result for a phase-field model of diffusion-induced grain-boundary motion, Proc. R. Soc. Edin., 131A (2001), pp. 1323–1344.

  17. K. Deckelnick and C. M. Elliott, Finite element error bounds for curve shrinking with prescribed normal contact to a fixed boundary, IMA J. Numer. Anal., 18 (1998), pp. 635–654.

    Google Scholar 

  18. K. Deckelnick, C. M. Elliott, and V. Styles, Numerical diffusion-induced grain boundary motion, Interfaces and Free Boundaries, 6 (2001), pp. 329–349.

  19. A. Dedner, P. Madhaven, and B. Stinner, Analysis of the discontinuous Galerkin method for elliptic problems on surfaces, IMA J. Numer. Anal., to appear (2012).

  20. A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal., 47 (2009), pp. 805–827.

    Google Scholar 

  21. Demlow , Dziuk G.: An adaptive finite element method for the Laplace- Beltrami operator on surfaces. SIAM Journal on Numerical Analysis. 45, 421–442 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, in Partial differential equations and calculus of variations, S. Hildebrandt and R. Leis, eds., vol. 1357 of Lecture Notes in Mathematics, Springer Berlin Heidelberg New York London Paris Tokyo, 1988, pp. 142–155.

  23. G. Dziuk, An algorithm for evolutionary surfaces, Numerische Mathematik, 58 (1991), pp. 603–611.

  24. G. Dziuk, Computational parametric Willmore flow, Numerische Mathematik, 111 (2008), pp. 55–80.

  25. G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA Journal Numerical Analysis, 25 (2007), pp. 385–407.

    Google Scholar 

  26. G. Dziuk and C. M. Elliott, Surface finite elements for parabolic equations, J. Comp. Mathematics, 25 (2007), pp. 385–407.

    Google Scholar 

  27. G. Dziuk and C. M. Elliott, Eulerian finite element method for parabolic pdes on implicit surfaces, Interfaces and Free Boundaries, 10 (2008), pp. 119–138.

    Google Scholar 

  28. G. Dziuk and C. M. Elliott, An Eulerian approach to transport and diffusion on evolving implicit surfaces, Computing and Visualization in Science, 13 (2010), pp. 17–28.

  29. G. Dziuk and C. M. Elliott, Fully discrete evolving surface finite element method, SIAM J. Numer. Anal., to appear (2012).

  30. G. Dziuk and C. M. Elliott, L 2 estimates for the evolving surface finite element method, Math. Comp., (2012).

  31. G. Dziuk, C. Lubich, and D. Mansour, Runge-Kutta time discretization of parabolic differential equations on evolving surfaces, IMA Journal of Numerical Analysis, 32 (2012), pp. 394–416.

    Google Scholar 

  32. C. Eilks and C. M. Elliott, Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method, Journal of Computational Physics, 227 (2008), pp. 9727–9741.

    Google Scholar 

  33. C. M. Elliott and T. Ranner, Finite element analysis for a coupled bulk-surface partial differential equation, IMA J. Numer. Anal., to appear (2012).

  34. C. M. Elliott and B. Stinner, Analysis of a diffuse interface approach to an advection diffusion equation on a moving surfacei, Math. Mod. Meth. Appl. Sci., 5 (2009), pp. 787–802.

    Google Scholar 

  35. C. M. Elliott and B. Stinner, Modeling and computation of two phase geometric biomembranes using surface finite elements, J Comput Phys, 229 (2010), pp. 6585–6612.

    Google Scholar 

  36. C. M. Elliott, B. Stinner, V. Styles, and R. Welford, Numerical computation of advection and diffusion on evolving diffuse interfaces, IMA J. Numer. Anal., 31 (2011), pp. 786–812.

    Google Scholar 

  37. C. M. Elliott, B. Stinner, and C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements, Journal of the Royal Society Interface, 9 (2012), pp. 3027–3044.

    Google Scholar 

  38. C. M. Elliott and V. Styles, Computations of bidirectional grain boundary dynamics in thin metallic films, Journal of Computational Physics, 187 (2003), pp. 524–543.

    Google Scholar 

  39. P. C. Fife, J. W. Cahn, and C. M. Elliott, A free-boundary model for diffusioninduced grain boundary motion, Interfaces Free Bound., 3 (2001), pp. 291–336.

  40. P. C. Fife and X. Wang, Chemically induced grain boundary dynamics, forced motion by curvature, and the appearance of double seams, European J. Applied Mathematics, 13 (2002), pp. 25–52.

    Google Scholar 

  41. H. Garcke and V. M. Styles, Bi-directional diffusion induced grain boundary motion with triple junctions, Interfaces and Free Boundaries, 6 (2004), pp. 271–294.

    Google Scholar 

  42. Greer J. B.: An improvement of a recent Eulerian method for solving PDEs on general geometries. J. Sci. Comput. 29, 321–352 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. J. B. Greer, A. L. Bertozzi, and G. Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys., 216 (2006), pp. 216–246.

    Google Scholar 

  44. C. Handwerker, Diffusion-induced grain boundary migration in thin films, in Diffusion Phenomena in Thin Films and Microelectronic Materials, D. Gupta and P. Ho, eds., Noyes Pubs. Park Ridge, N.J., 1988, pp. 245–322.

  45. A. Henderson, ParaView Guide, A Parallel Visualization Application, Kitware Inc., 2007.

  46. A. J. James and J. Lowengrub, A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, Journal of Computational Physics, 201 (2004), pp. 685–722.

    Google Scholar 

  47. L. Ju and Q. Du, A finite volume method on general surfaces and its error estimates, J. Math. Anal. Appl., 352 (2009), pp. 645–668.

    Google Scholar 

  48. L. Ju, L. Tian, and D. Wang, A posteriori error estimates for finite volume approximations of elliptic equations on general surfaces, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 716–726.

    Google Scholar 

  49. M. Lenz, S. F. Nemadjieu, and M. Rumpf, Finite volume method on moving surfaces v. Proceedings of the 5th international symposium held in Aussois, June 2008., in Finite volumes for complex applications, R.Eymard and J.-M. H´erard, eds., Hoboken, NJ,, 2008, John Wiley & Sons, Inc., pp. Hoboken, NJ,.

  50. M. Lenz, S. F. Nemadjieu, and M. Rumpf, A convergent finite volume scheme for diffusion on evolving surfaces, SIAM J. Numer. Anal., 49 (2011), pp. 15–37.

  51. C. B. Macdonald and S. J. Ruuth, Level set equations on surfaces via the Closest Point Method, Journal of Scientific Computing, 35 (2008), pp. 219–240.

    Google Scholar 

  52. C. B. Macdonald and S. J. Ruuth, The implicit closest point method for the numerical solution of partial differential equations on surfaces., SIAM J. Sci. Comput., 31 (2009), pp. 4330–4350.

    Google Scholar 

  53. U. F. Mayer and G. Simonett, Classical solutions for diffusion-induced grainboundary motion, J. Math. Anal. Appl., 234 (1999), pp. 660–674.

    Google Scholar 

  54. M. Neilson, J. Mackenzie, S. Webb, and R. Insall, Modelling cell movement and chemotaxis using pseudopod-based feedback, SIAM Journal on Scientific Computing, 33 (2011), pp. 1035–1057.

    Google Scholar 

  55. M. A. Olshanskii and A. Reusken, A finite element method for surface pdes: matrix properties, Numer. Math., 114 (2010), pp. 491–520.

  56. M. A. Olshanskii, A. Reusken, and J. Grande, A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal., 47 (2009), pp. 3339–3358.

    Google Scholar 

  57. S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, vol. 153 of Appl. Math. Sci, Springer Verlag, 2003.

  58. A. Ratz and A. Voigt, Pdes on surfaces - a diffuse interface approach, Communications in Mathematical Sciences, 4 (2006), pp. 575–590.

  59. Ruuth S. J., Merriman B: A simple embedding method for solving partial differential equations on surfaces. Journal of Computational Physics. 227, 1943–1961 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  60. A. Schmidt and K. G. Siebert, Design of adaptive finite element software: The finite element toolbox ALBERTA, vol. 42 of Lecture notes in computational science and engineering, Springer, 2005.

  61. J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, Journal of Theoretical Biology, 81 (1979), pp. 389–400.

    Google Scholar 

  62. J. A. Sethian, Level set methods and fast marching methods,, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 1999.

  63. J.-J. Xu and H.-K. Zhao, An Eulerian formulation for solving partial differential equations along a moving interface, Journal of Scientific Computing, 19 (2003), pp. 573–594.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Charles M. Elliott

  2. Department of Mathematics, University of Sussex, Brighton, BN1 9QH, UK

    Vanessa Styles

Authors
  1. Charles M. Elliott
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vanessa Styles
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charles M. Elliott.

Additional information

The work of V. Styles was supported by the UK Engineering and Physical Sciences Research Council EPSRC Grant EP/D078334/1. The work of C. M. Elliott was supported by the UK Engineering and Physical Sciences Research Council EPSRC Grant EP/G010404.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Elliott, C.M., Styles, V. An ALE ESFEM for Solving PDEs on Evolving Surfaces. Milan J. Math. 80, 469–501 (2012). https://doi.org/10.1007/s00032-012-0195-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-012-0195-6

Mathematics Subject Classification (2010)

Keywords

Search

Navigation