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$ L^2$-estimates for the evolving surface finite element method

Authors: Gerhard Dziuk and Charles M. Elliott
Journal: Math. Comp. 82 (2013), 1-24
MSC (2010): Primary 65M60, 65M15; Secondary 35K99, 35R01, 35R37, 76R99
Published electronically: April 13, 2012
MathSciNet review: 2983013
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the evolving surface finite element meth-
od for the advection and diffusion of a conserved scalar quantity on a moving surface. In an earlier paper using a suitable variational formulation in time dependent Sobolev space we proposed and analysed a finite element method using surface finite elements on evolving triangulated surfaces. An optimal order $ H^1$-error bound was proved for linear finite elements. In this work we prove the optimal error bound in $ L^2(\Gamma (t))$ uniformly in time.

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  • 1. D. Adalsteinsson and J. A. Sethian, Transport and diffusion of material quantities on propagating interfaces via level set methods, Journal of Computational Physics 185 (2003), no. 1, 271-288. MR 2010161 (2004i:65072)
  • 2. M. Bertalmío, Li-Tien Cheng, S. J. Osher, and G. Sapiro, Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys. 174 (2001), no. 2, 759-780. MR 1868103 (2002j:65105)
  • 3. 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), no. 6, 4066-4099. MR 2566584 (2011a:65276)
  • 4. K. Deckelnick, G. Dziuk, and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica 14 (2005), 139-232. MR 2168343 (2006h:65159)
  • 5. 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), 351-376. MR 2608464 (2011c:65256)
  • 6. A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal. 47 (2009), 805-827. MR 2485433 (2010a:65233)
  • 7. A. Demlow and G. Dziuk, An adaptive finite element method for the Laplace-Beltrami operator on surfaces, SIAM Journal on Numerical Analysis 45 (2007), 421-442. MR 2285862 (2008c:65320)
  • 8. G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial differential equations and calculus of variations (S. Hildebrandt and R. Leis, eds.), Lecture Notes in Mathematics, vol. 1357, Springer Berlin Heidelberg New York London Paris Tokyo, 1988, pp. 142-155. MR 976234 (90i:65194)
  • 9. G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA Journal Numerical Analysis 27 (2007), 262-292. MR 2317005 (2008c:65253)
  • 10. -, Surface finite elements for parabolic equations, J. Comp. Mathematics 25 (2007), 385-407. MR 2337402 (2008j:74065)
  • 11. -, Eulerian finite element method for parabolic pdes on implicit surfaces, Interfaces and Free Boundaries 10 (2008), 119-138. MR 2383539 (2009a:65250)
  • 12. -, An Eulerian approach to transport and diffusion on evolving implicit surfaces, Computing and Visualization in Science 13 (2010), 17-28. MR 2565107 (2010k:76109)
  • 13. 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), no. 23, 9727-9741. MR 2469030 (2009i:80013)
  • 14. C. M. Elliott and B. Stinner, Modeling and computation of two phase geometric biomembranes using surface finite elements, J. Comput. Phys. 229 (2010), 6585-6612. MR 2660322 (2011j:74092)
  • 15. -, Analysis of a diffuse interface approach to an advection diffusion equation on a moving surfacei, Math. Mod. Meth. Appl. Sci. 5 (2009), 787-802. MR 2531040 (2011a:35251)
  • 16. Charles M. Elliott, Björn Stinner, Vanessa Styles, and Richard Welford, Numerical computation of advection and diffusion on evolving diffuse interfaces, IMA J. Numer. Anal. 31 (2011), 786-812. MR 2832780
  • 17. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1998.
  • 18. John B. Greer, An improvement of a recent Eulerian method for solving PDEs on general geometries, J. Sci. Comput. 29 (2006), no. 3, 321-352. MR 2272322
  • 19. John B. Greer, Andrea L. Bertozzi, and Guillermo Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys. 216 (2006), no. 1, 216-246. MR 2223442 (2007e:65090)
  • 20. M. E. Gurtin, Configurational forces as basic concepts in continuum physics, Applied Mathematical Sciences, vol. 137, Springer, Berlin-Heidelberg, 2000. MR 1735282 (2002f:74001)
  • 21. S. Hildebrandt, Analysis 2, Springer-Verlag, Berlin, Heidelburg, New York, 1982.
  • 22. Lili Ju and Qiang Du, A finite volume method on general surfaces and its error estimates, J. Math. Anal. Appl. 352 (2009), no. 2, 645-668. MR 2501909 (2010a:65212)
  • 23. Lili Ju, Li Tian, and Deshang Wang, A posteriori error estimates for finite volume approximations of elliptic equations on general surfaces, Comput. Methods Appl. Mech. Engrg. 198 (2009), 716-726. MR 2498524 (2010b:65234)
  • 24. 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., Finite volumes for complex applications (Hoboken, NJ,) (R.Eymard and J.-M. Hérard, eds.), John Wiley & Sons, Inc., 2008, pp. Hoboken, NJ,. MR 2451453
  • 25. C. B. Macdonald and S. J. Ruuth, Level set equations on surfaces via the Closest Point Method, Journal of Scientific Computing 35 (2008), 219-240. MR 2429939 (2009i:65141)
  • 26. -, The implicit closest point method for the numerical solution of partial differential equations on surfaces., SIAM J. Sci. Comput. 31 (2009), 4330-4350. MR 2594984 (2011c:65166)
  • 27. M. A. Olshanskii, A. Reusken, and J. Grande, A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal. 47 (2009), 3339-3358. MR 2551197 (2010k:65265)
  • 28. Maxim A. Olshanskii and Arnold Reusken, A finite element method for surface pdes: matrix properties, Numer. Math. 114 (2010), no. 3, 491-520 (English). MR 2570076 (2010m:65289)
  • 29. S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, Appl. Math. Sci, vol. 153, Springer-Verlag, 2003. MR 1939127 (2003j:65002)
  • 30. A. Ratz and A. Voigt, Pdes on surfaces - a diffuse interface approach, Communications in Mathematical Sciences 4 (2006), no. 3, 575-590. MR 2247931 (2007g:35120)
  • 31. Steven J. Ruuth and Barry Merriman, A simple embedding method for solving partial differential equations on surfaces, Journal of Computational Physics 227 (2008), no. 3, 1943-1961. MR 2450979 (2009h:35239)
  • 32. J. A. Sethian, Level set methods and fast marching methods,, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 1999. MR 1700751 (2000c:65015)
  • 33. Jian-Ju Xu and H.-K. Zhao, An Eulerian formulation for solving partial differential equations along a moving interface, Journal of Scientific Computing 19 (2003), 573-594. MR 2028859 (2004j:65129)

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Additional Information

Gerhard Dziuk
Affiliation: Abteilung für Angewandte Mathematik, University of Freiburg, Hermann-Herder-Straße 10, D–79104 Freiburg i. Br., Germany

Charles M. Elliott
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Keywords: Surface finite elements, advection diffusion, moving surface, error analysis
Received by editor(s): July 15, 2010
Received by editor(s) in revised form: July 5, 2011, and July 29, 2011
Published electronically: April 13, 2012
Additional Notes: The work was supported by Deutsche Forschungsgemeinschaft via SFB/TR 71
This research was also supported by the UK Engineering and Physical Sciences Research Council (EPSRC), Grant EP/G010404.
Article copyright: © Copyright 2012 American Mathematical Society

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