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Finite element methods for surface PDEs*

Published online by Cambridge University Press:  02 April 2013

Gerhard Dziuk  and
Charles M. Elliott
Gerhard Dziuk
Affiliation:
Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg im Breisgau, Hermann-Herder-Straβe 10, D–79104 Freiburg im Breisgau, Germany E-mail: [email protected]
Charles M. Elliott
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK E-mail: [email protected]
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Abstract

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

Type
Research Article
Information
Acta Numerica , Volume 22 , May 2013 , pp. 289 - 396
Copyright
Copyright © Cambridge University Press 2013

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Footnotes

*

Colour online for monochrome figures available at journals.cambridge.org/anu.

References

Adalsteinsson, D. and Sethian, J. A. (2003), ‘Transport and diffusion of material quantities on propagating interfaces via level set methods’, J. Comput. Phys. 185, 271–288.CrossRefGoogle Scholar
Aragón, J., Barrio, R. A. and Varea, C. (1999), ‘Turing patterns on a sphere’, Phys. Rev. E 60, 4588–4592.Google Scholar
Barreira, R. (2009), Numerical solution of nonlinear partial differential equations on triangulated surfaces. DPhil thesis, University of Sussex.Google Scholar
Barreira, R., Elliott, C. M. and Madzvamuse, A. (2011), ‘The surface finite element method for pattern formation on evolving biological surfaces’, J. Math. Biology 63, 1095–1119.CrossRefGoogle ScholarPubMed
Barrett, J. W. and Elliott, C. M. (1982), A finite element method on a fixed mesh for the Stefan problem with convection in a saturated porous medium. In Numerical Methods for Fluid Dynamics, Conference proceedings: University of Reading, 29-31 March, 1982 (Morton, K. W. and Baines, M. J., eds), Conference Series (Institute of Mathematics and its Applications), Academic Press, pp. 389–409.Google Scholar
Barrett, J. W. and Elliott, C. M. (1984), ‘A finite-element method for solving elliptic equations with Neumann data on a curved boundary using unfitted meshes’, IMA J. Numer. Anal. 4, 309–325.CrossRefGoogle Scholar
Barrett, J. W. and Elliott, C. M. (1985), ‘Fixed mesh finite element approximations to a free boundary problem for an elliptic equation with an oblique derivative boundary condition’, Comput. Math. Appl. 11, 335–345.CrossRefGoogle Scholar
Barrett, J. W. and Elliott, C. M. (1987 a), ‘Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces’, IMA J. Numer. Anal. 7, 283–300.CrossRefGoogle Scholar
Barrett, J. W. and Elliott, C. M. (1987 b), ‘A practical finite element approximation of a semi-definite Neumann problem on a curved domain’, Numer. Math. 51, 23–36.CrossRefGoogle Scholar
Barrett, J. W. and Elliott, C. M. (1988), ‘Finite-element approximation of elliptic equations with a Neumann or Robin condition on a curved boundary’, IMA J. Numer. Anal. 8, 321–342.CrossRefGoogle Scholar
Barrett, J. W., Garcke, H. and Nürnberg, R. (2007), ‘A parametric finite element method for fourth order geometric evolution equations’, J. Comput. Phys. 222, 441–467.CrossRefGoogle Scholar
Barrett, J. W., Garcke, H. and Nürnberg, R. (2008 a), ‘On the parametric finite element approximation of evolving hypersurfaces in R 3’, J. Comput. Phys. 227, 4281–4307.CrossRefGoogle Scholar
Barrett, J. W., Garcke, H. and Nürnberg, R. (2008 b), ‘Parametric approximation of Willmore flow and related geometric evolution equations’, SIAM J. Sci. Comput. 31, 225–253.CrossRefGoogle Scholar
Barrio, R. A., Maini, P. K., Padilla, P., Plaza, R. G. and Sánchez-Garduno, (2004), ‘The effect of growth and curvature on pattern formation’, J. Dynam. Diff. Equations 16, 1093–1121.Google Scholar
Bastian, P. and Engwer, C. (2009), ‘An unfitted finite element method using discontinuous Galerkin’, Internat. J. Numer. Methods Engng 79, 1557–1576.CrossRefGoogle Scholar
Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klofkorn, R., Oldberger, M. and Sander, O. (2008 a), ‘A generic grid interface for parallel and adaptive scientific computing I: Abstract framework’, Computing 82, 103–119.CrossRefGoogle Scholar
Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klofkorn, R., Oldberger, M. and Sander, O. (2008 b), ‘A generic grid interface for parallel and adaptive scientific computing II: Implementation and tests in DUNE’, Computing 82, 121–138.CrossRefGoogle Scholar
Bergdorf, M., Sbalzarini, I. and Koumoutsakos, P. (2010), ‘A Lagrangian particle method for reaction–diffusion systems on deforming surfaces’, J. Math. Biology 61, 649–663.CrossRefGoogle ScholarPubMed
Berger, M. J., Calhoun, D. A., Helzel, C. and Leveque, R. J. (2009), ‘Logically rectangular finite volume methods with adaptive refinement on the sphere’, Phil. Trans. Royal Soc. London A 367, 4483–4496.Google Scholar
Bertalmío, M., Cheng, L.-T., Osher, S. J. and Sapiro, G. (2001), ‘Variational problems and partial differential equations on implicit surfaces’, J. Comput. Phys. 174, 759–780.CrossRefGoogle Scholar
Blowey, J. F. and Elliott, C. M. (1991), ‘The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy I: Mathematical analysis’, European J. Appl. Math. 2, 233–280.CrossRefGoogle Scholar
Blowey, J. F. and Elliott, C. M. (1992), ‘The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy II: Numerical analysis’, European J. Appl. Math. 3, 147–179.CrossRefGoogle Scholar
Blowey, J. F. and Elliott, C. M. (1993), Curvature dependent phase boundary motion and parabolic obstacle problems. In Degenerate Diffusion (Ni, W.-M., Peletier, L. A. and Vasquez, J. L., eds), Vol. 47 of IMA Volumes in Mathematics and its Applications, Springer, pp. 19–60.CrossRefGoogle Scholar
Burger, M. (2009), ‘Finite element approximation of elliptic partial differential equations on implicit surfaces’, Comput. Vis. Sci. 12, 87–100.CrossRefGoogle Scholar
Caginalp, G. (1989), ‘Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations’, Phys. Rev. A 39, 5887–5896.CrossRefGoogle ScholarPubMed
Calhoun, D. A. and Helzel, C. (2009), ‘A finite volume method for solving parabolic equations on logically Cartesian curved surface meshes’, SIAM J. Sci. Comput. 31, 4066–4099.CrossRefGoogle Scholar
Cermelli, P., Fried, E. and Gurtin, M. E. (2005), ‘Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interfaces’, J. Fluid Mech. 544, 339–351.CrossRefGoogle Scholar
Chaplain, M. A. J., Ganesh, M. and Graham, I. G. (2001), ‘Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth’, J. Math. Biology 42, 387–423.CrossRefGoogle ScholarPubMed
Ciarlet, P. G. (1978), The Finite Element Method for Elliptic Problems, North-Holland.Google Scholar
Copetti, M. I. M. and Elliott, C. M. (1992), ‘Numerical analysis of the Cahn–Hilliard equation with a logarithmic free energy’, Numer. Math. 63, 39–65.CrossRefGoogle Scholar
Crampin, E. J., Gaffney, E. A. and Maini, P. K. (1999), ‘Reaction and diffusion on growing domains: Scenarios for robust pattern formation’, Bull. Math. Biology 61, 1093–1120.CrossRefGoogle ScholarPubMed
Deckelnick, K., Dziuk, G. and Elliott, C. M. (2005), Computation of geometric partial differential equations and mean curvature flow. In Acta Numerica, Vol. 14, Cambridge University Press, pp. 139–232.Google Scholar
Deckelnick, K., Dziuk, G., Elliott, C. M. and Heine, C.-J. (2010), ‘An h-narrow band finite element method for implicit surfaces’, IMA J. Numer. Anal. 30, 351–376.CrossRefGoogle Scholar
Deckelnick, K., Elliott, C. M. and Ranner, T. (2013), Unfitted finite element methods using bulk meshes for surface partial differential equations. In preparation.CrossRefGoogle Scholar
Deckelnick, K., Elliott, C. M. and Styles, V. (2001), ‘Numerical diffusion-induced grain boundary motion’, Interfaces Free Bound. 6, 329–349.Google Scholar
Dedner, A., Madhavan, P. and Stinner, B. (2013), ‘Analysis of the discontinuous Galerkin method for elliptic problems on surfaces’, IMA J. Numer. Anal., to appear.Google Scholar
Demlow, A. (2009), ‘Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces’, SIAM J. Numer. Anal. 47, 805–827.CrossRefGoogle Scholar
Demlow, A. and Dziuk, G. (2007), ‘An adaptive finite element method for the Laplace–Beltrami operator on surfaces’, SIAM J. Numer. Anal. 45, 421–442.CrossRefGoogle Scholar
Demlow, A. and Olshanskii, M. (2012), ‘An adaptive surface finite element method based on volume meshes’, SIAM J. Numer. Anal. 50, 1624–1647.CrossRefGoogle Scholar
Du, Q., Ju, L. and Tian, L. (2011), ‘Finite element approximation of the Cahn–Hilliard equation on surfaces’, Comput. Methods Appl. Mech. Engng 200, 2458–2470.CrossRefGoogle Scholar
Dziuk, G. (1988), Finite elements for the Beltrami operator on arbitrary surfaces. In Partial Differential Equations and Calculus of Variations (Hildebrandt, S. and Leis, R., eds), Vol. 1357 of Lecture Notes in Mathematics, Springer, pp. 142–155.CrossRefGoogle Scholar
Dziuk, G. (1991), ‘An algorithm for evolutionary surfaces’, Numer. Math. 58, 603–611.CrossRefGoogle Scholar
Dziuk, G. and Elliott, C. M. (2007 a), ‘Finite elements on evolving surfaces’, IMA J. Numer. Anal. 25, 385–407.Google Scholar
Dziuk, G. and Elliott, C. M. (2007 b), ‘Surface finite elements for parabolic equations’, J. Comput. Math. 25, 385–407.Google Scholar
Dziuk, G. and Elliott, C. M. (2008), ‘Eulerian finite element method for parabolic PDEs on implicit surfaces’, Interfaces Free Bound. 10, 119–138.CrossRefGoogle Scholar
Dziuk, G. and Elliott, C. M. (2010), ‘An Eulerian approach to transport and diffusion on evolving implicit surfaces’, Comput. Vis. Sci. 13, 17–28.CrossRefGoogle Scholar
Dziuk, G. and Elliott, C. M. (2012), ‘Fully discrete evolving surface finite element method’, SIAM J. Numer. Anal. 50, 2677–2694.CrossRefGoogle Scholar
Dziuk, G. and Elliott, C. M. (2013), ‘L 2 estimates for the evolving surface finite element method’, Math. Comp. 82, 1–24.CrossRefGoogle Scholar
Dziuk, G., Kröner, D. and Müller, T. (2012), Scalar conservation laws on moving hypersurfaces. Technical report, Freiburg.CrossRefGoogle Scholar
Dziuk, G., Lubich, C. and Mansour, D. (2012), ‘Runge–Kutta time discretization of parabolic differential equations on evolving surfaces’, IMA J. Numer. Anal. 32, 394–416.CrossRefGoogle Scholar
Eilks, C. and Elliott, C. M. (2008), ‘Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method’, J. Comput. Phys. 227, 9727–9741.CrossRefGoogle Scholar
Elliott, C. M. and Ranner, T. (2013), ‘Finite element analysis for a coupled bulk–surface partial differential equation’, IMA J. Numer. Anal., to appear.Google Scholar
Elliott, C. M. and Stinner, B. (2009), ‘Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface’, Math. Mod. Methods Appl. Sci. 5, 787–802.CrossRefGoogle Scholar
Elliott, C. M. and Stinner, B. (2010), ‘Modeling and computation of two phase geometric biomembranes using surface finite elements’, J. Comput. Phys. 229, 6585–6612.Google Scholar
Elliott, C. M. and Stinner, B. (2012), ‘Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements’, Commun. Comput. Phys. 13, 325–360.CrossRefGoogle Scholar
Elliott, C. M. and Styles, V. (2003), ‘Computations of bidirectional grain boundary dynamics in thin metallic films’, J. Comput. Phys. 187, 524–543.CrossRefGoogle Scholar
Elliott, C. M. and Styles, V. M. (2012), ‘An ALE ESFEM method for solving PDEs on evolving surfaces’, Milan J. Math. 80, 469–501.CrossRefGoogle Scholar
Elliott, C. M., French, D. A. and Milner, F. A. (1989), ‘A second order splitting method for the Cahn–Hilliard equation’, Numer. Math. 54, 575–590.CrossRefGoogle Scholar
Elliott, C. M., Stinner, B. and Venkataraman, C. (2012), ‘Modelling cell motility and chemotaxis with evolving surface finite elements’, J. Royal Society Interface 9, 3027–3044.CrossRefGoogle ScholarPubMed
Elliott, C. M., Stinner, B., Styles, V. and Welford, R. (2011), ‘Numerical computation of advection and diffusion on evolving diffuse interfaces’, IMA J. Numer. Anal. 31, 786–812.CrossRefGoogle Scholar
Engwer, C. and Heimann, F. (2012), DUNE-UDG: A cut-cell framework for unfitted discontinuous Galerkin methods. In Advances in DUNE, Springer, pp. 89–100.CrossRefGoogle Scholar
Erlebacher, J. and McCue, I. (2012), ‘Geometric characterization of nanoporous metals’, Acta Materialia 60, 6164–6174.CrossRefGoogle Scholar
Erlebacher, J., Aziz, M., Karma, A., Dimitrov, N. and Sieradzki, K. (2001), ‘Evolution of nanoporosity in dealloying’, Nature 410, 450–453.CrossRefGoogle ScholarPubMed
Evans, L. C. (1998), Partial Differential Equations, first edition, Graduate Studies in Mathematics, AMS.Google Scholar
Fife, P. C., Cahn, J. W. and Elliott, C. M. (2001), ‘A free-boundary model for diffusion-induced grain boundary motion’, Interfaces Free Bound. 3, 291–336.CrossRefGoogle Scholar
Ganesan, S. and Tobiska, L. (2009), ‘A coupled arbitrary Lagrangian–Eulerian and Lagrangian method for computation of free-surface flows with insoluble surfactants’, J. Comput. Phys. 228, 2859–2873.CrossRefGoogle Scholar
Ganesan, S., Hahn, A., Held, K. and Tobiska, L. (2012), An accurate numerical method for computation of two-phase flows with surfactants. In European Congress on Computational Methods in Applied Sciences and Engineering (Eberhardsteiner, J.et al., eds), Vol. CD-ROM, TU Vienna. Google Scholar
Gilbarg, D. and Trudinger, N. S. (1998), Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Springer.Google Scholar
Greer, J. B. (2006), ‘An improvement of a recent Eulerian method for solving PDEs on general geometries’, J. Sci. Comput. 29, 321–352.Google Scholar
Greer, J. B., Bertozzi, A. L. and Sapiro, G. (2006), ‘Fourth order partial differential equations on general geometries’, J. Comput. Phys. 216, 216–246.CrossRefGoogle Scholar
Hansbo, A. and Hansbo, P. (2002), ‘An unfitted finite element method, based on Nitsche's method, for elliptic interface problems’, Comput. Methods Appl. Mech. Engng 191, 5537–5552.CrossRefGoogle Scholar
Henderson, A. (2007), Para View Guide: A Parallel Visualization Application, Kitware.Google Scholar
James, A. J. and Lowengrub, J. (2004), ‘A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant’, J. Comput. Phys. 201, 685–722.CrossRefGoogle Scholar
Ju, L. and Du, Q. (2009), ‘A finite volume method on general surfaces and its error estimates’, J. Math. Anal. Appl. 352, 645–668.CrossRefGoogle Scholar
Ju, L., Tian, L. and Wang, D. (2009), ‘A posteriori error estimates for finite volume approximations of elliptic equations on general surfaces’, Comput. Methods Appl. Mech. Engng 198, 716–726.CrossRefGoogle Scholar
Lai, M.-C., Tseng, Y.-H. and Huang, H. (2008), ‘An immersed boundary method for interfacial flows with insoluble surfactant’, J. Comput. Phys. 227, 7279–7293.CrossRefGoogle Scholar
Lowengrub, J., Xu, J.-J. and Voigt, A. (2007), ‘Surface phase separation and flow in a simple model of multicomponent drops and vesicles’, Fluid Dynamics and Material Processing 3, 1–20.Google Scholar
Lubich, C. and Mansour, D. (2012), Variational discretisation of linear wave equation on evolving surfaces. Technical report, Tübingen.Google Scholar
Macdonald, C. B. and Ruuth, S. J. (2008), ‘Level set equations on surfaces via the closest point method’, J. Sci. Comput. 35, 219–240.CrossRefGoogle Scholar
Macdonald, C. B. and Ruuth, S. J. (2009), ‘The implicit closest point method for the numerical solution of partial differential equations on surfaces.’, SIAM J. Sci. Comput. 31, 4330–4350.CrossRefGoogle Scholar
Macdonald, C. B., Brandman, J. and Ruuth, S. J. (2011), ‘Solving eigenvalue problems on curved surfaces using the closest point method.’, J. Comput. Phys. 230, 7944–7956.CrossRefGoogle Scholar
Marenduzzo, D. and Orlandini, E. (2013), ‘Phase separation dynamics on curved surfaces’, Soft Matter 9, 1178–1187.CrossRefGoogle Scholar
Marz, T. and Macdonald, C. B. (2013), ‘Calculus on surfaces with general closest point functions’, SIAM J. Numer. Anal. 50, 3303–3328.CrossRefGoogle Scholar
Mercker, M., Ptashnyk, M., Kühnle, J., Hartmann, D., Weiss, M. and Jäger, W. (2012), ‘A multiscale approach to curvature modulated sorting in biological membranes’, J. Theoret. Biology 301, 67–82.CrossRefGoogle ScholarPubMed
Nedelec, J. (1976), ‘Curved finite element methods for the solution of singular integral equations on surfaces in R 3’, Comput. Methods Appl. Mech. Engng 8, 61–80.CrossRefGoogle Scholar
Novak, I. L., Gao, F., Choi, Y.-S., Resasco, D., Schaff, J. C. and Slepchenko, B. M. (2007), ‘Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology’, J. Comput. Phys. 226, 1271–1290.CrossRefGoogle ScholarPubMed
Olshanskii, M. A. and Reusken, A. (2010), ‘A finite element method for surface PDEs: Matrix properties’, Numer. Math. 114, 491–520.Google Scholar
Olshanskii, M. A., Reusken, A. and Grande, J. (2009), ‘A finite element method for elliptic equations on surfaces’, SIAM J. Numer. Anal. 47, 3339–3358.CrossRefGoogle Scholar
Osher, S. and Fedkiw, R. (2003), Level Set Methods and Dynamic Implicit Surfaces, Vol. 153 of Applied Mathematical Sciences, Springer.CrossRefGoogle Scholar
Rätz, A. and Röger, M. (2012), ‘Turing instabilities in a mathematical model for signaling networks’, J. Math. Biology 65, 1215–1244.CrossRefGoogle Scholar
Rätz, A. and Voigt, A. (2006), ‘PDEs on surfaces: A diffuse interface approach’, Commun. Math. Sci. 4, 575–590.CrossRefGoogle Scholar
Rumpf, M., Schmidt, A.et al. (1990), GRAPE: Graphics programming environment. SFB 256 Report 8, Universität Bonn.Google Scholar
Ruuth, S. J. and Merriman, B. (2008), ‘A simple embedding method for solving partial differential equations on surfaces’, J. Comput. Phys. 227, 1943–1961.CrossRefGoogle Scholar
Schmidt, A. and Siebert, K. G. (2005), Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Vol. 42 of Lecture Notes in Computational Science and Engineering, Springer.Google Scholar
Schnakenberg, J. (1979), ‘Simple chemical reaction systems with limit cycle behaviour’, J. Theoret. Biology 81, 389–400.CrossRefGoogle ScholarPubMed
Schonborn, O. and Desai, R. C. (1997), ‘Phase-ordering kinetics on curved surfaces’, Physica A 239, 412–419.CrossRefGoogle Scholar
Schwartz, P., Adalsteinsson, D., Colella, P., Arkin, A. P. and Onsum, M. (2005), ‘Numerical computation of diffusion on a surface’, Proc. Nat. Acad. Sci. 102, 11151–11156.Google ScholarPubMed
Sethian, J. A. (1999), Level Set Methods and Fast Marching Methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.Google Scholar
Stone, H. A. (1990), ‘A simple derivation of the time dependent convective diffusion equation for surfactant transport along a deforming interface’, Phys. Fluids A 2, 111–112.CrossRefGoogle Scholar
Turing, A. M. (1952), ‘The chemical basis of morphogenesis’, Phil. Trans. Royal Soc. London B 237, 37–72.Google Scholar
Xu, J.-J. and Zhao, H.-K. (2003), ‘An Eulerian formulation for solving partial differential equations along a moving interface’, J. Sci. Comput. 19, 573–594.CrossRefGoogle Scholar
Xu, J.-J., Li, Z., Lowengrub, J. and Zhao, H. (2006), ‘A level-set method for interfacial flows with surfactant’, J. Comput. Phys. 212, 590–616.CrossRefGoogle Scholar
Xu, J.-J., Yang, Y. and Lowengrub, J. (2012), ‘A level-set continuum method for two-phase flows with insoluble surfactant’, J. Comput. Phys. 231, 5897–5909.CrossRefGoogle Scholar